National College of Business Administration & Economics

Lahore

pL −BOUNDEDNESS OF INTEGRAL OPERATORS

INVOLVING GENERALIZED HYPERGEOMETRIC

FUNCTIONS

BY

SHAHID MUBEEN

DOCTOR OF PHILOSOPHY IN

APPLIED MATHEMATICS

October, 2011

ii

NATIONAL COLLEGE OF BUSINESS ADMINISTRATION & ECONOMICS

pL −BOUDEDNESS OF INTEGRAL OPERATORS INVOLVING GENERALIZED HYPERGEOMETRIC

FUNCTIONS BY

SHAHID MUBEEN

A Dissertation Submitted to School of Computer Sciences

In Partial Fulfillment of the

Requirements for the Degree of

DOCTOR OF PHILOSOPHY IN

APPLIED MATHEMATICS

October, 2011

iii

IN THE NAME OF ALLAH, THE MOST BENEFICIENT,

THE MOST MERCIFUL

iv

NATIONAL COLLEGE OF BUSINESS ADMINISTRATION & ECONOMICS,

LAHORE

pL −BOUDEDNESS OF INTEGRAL OPERATORS INVOLVING GENERALIZED HYPERGEOMETRIC

FUNCTIONS BY

SHAHID MUBEEN

A dissertation submitted to the School of Computer Sciences, in partial

fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

IN APPLIED MATHEMATICS

Dissertation Committee: Chairman

Member

Member

_

Rector National College of Business Administration & Economics

v

DECLARATION

This is to certify that this research work is my own work and it has not

been submitted for obtaining a similar degree from any other university

and/or institution.

SHAHID MUBEEN October, 2011

vi

DEDICATED TODEDICATED TODEDICATED TODEDICATED TO

MY PARENTMY PARENTMY PARENTMY PARENTSSSS

ANDANDANDAND

MYMYMYMY WIFE WIFE WIFE WIFE Dr. Dr. Dr. Dr. MARIAMARIAMARIAMARIA

vii

ACKNOWLEDGEMENT

Thanks to Almighty ALLAH whose infinite blessings have enabled me to

complete this humble piece of dissertation. I am also indebted to my teachers,

seniors and fellows at NCBA & E.

First of all, I am grateful to my supervisor, Professor

Dr. G. M. Habibullah, National College of Business Administration and

Economics, Lahore, for guiding me in the right direction; without his help it

would never been possible to accomplish this dissertation. I am indebted to

Professor Dr. Munir Ahmad, Rector NCBA & E, for his support to

successfully pursue my research work. I also thank my colleague Mr. Adnan

Khan for encouragement and appreciation in completing this task.

The financial support of the Higher Education Commission, Pakistan in

facilitating me to take up this research work, is most gratefully

acknowledged.

In the end, I pay rich tributes to my sisters, my brothers, my wife and my

daughter Meerab for providing me all kinds of support, in particular the

continuous prayers and encouragement of my father Mr. Abdul Ghafoor

Jhanda and my mother for my success in achieving the set goals.

viii

RESEARCH COMPLETION CERTIFICATE Certified that the research work contained in this thesis entitled

“ pL −Boundedness of Integral Operators Involving Generalized

Hypergeometric Functions” has been carried out and completed by Mr.

Shahid Mubeen under my supervision during his Ph. D programme.

(Dr. G. M. Habibullah)

National College of Business

Administration and Economics,

Lahore

ix

SUMMARY

Studies on integral operators involving the hypergeometric functions 2 1F and the confluent hypergeometric functions 1 1F as kernel have been a popular subject area. Researchers have discussed the boundedness of integral operators in 1L and , 1pL p > , and determined their inversions. Often, they have investigated the necessary and sufficient conditions for such an inversion by using mapping properties. We have used, in this thesis, such relations to study pL − boundedness of integral operators that involve generalized k −hypergeometric functions.

We have first extended integral representation of the confluent

k −hypergeometric functions from 1 1F to 2 2F and to , 2m mF m > . We have also extended integral representation of k −hypergeometric functions from 2 1F to 3 2F and to 1 , 2m mF m+ > . We have then formulated integral operators involving generalized k −hypergeometric functions both m mF and 1 , 1m mF m+ ≥ as kernel.

We have introduced extensions of Erdélyi-Kober fractional

integrals to be called k − fractional integrals, which are based upon definition of k −gamma function. We have also proved a few properties of k − fractional integrals to gain its formalized knowledge. We have shown that integral operators, earlier considered, are compositions of known operators, k − fractional integrals and variants of Laplace transform. Using Hardy’s inequality, we have proved the pL − boundedness , 1p > , of all these integral operators including k − fractional integrals. We have also proved that each of these operators is bounded in , 1pL p > , with a weight function.

x

TABLE OF CONTENTS Page DECLARATION

v

ACKNOWLEDGEMENT

vii

RESEARCH COMPLETION CERTIFICATE

viii

SUMMARY

ix

1. INTRODUCTION

1

1.1 Background 1.2 Literature Review

1 2

2. k −ANALOGUE OF HYPERGEOMETRIC FUNCTIONS 6

2.1 The Hypergeometric Functions

2.2 The Generalized Hypergeometric Functions 2.3 Integral Representations of Hypergeometric and Confluent Hypergeometric Functions 2.4 Other Integral Representations of Hypergeometric Functions 2.5 Results on Confluent Hypergeometric Functions 2.6 k − Hypergeometric Functions 2.7 Integral Representations of k −Hypergeometric Functions 2.8 Integral Representations of Confluent k −Hypergeometric Functions

6 6 7 8 17 19 20 24

3. k −FRACTIONAL INTEGRALS 27

3.1 Riemann- Liouville k −Fractional Integral

3.2 Properties of k −Fractional Integrals 27 28

4. pL −BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING 1 , , 1m m kF m+ ≥ AS KERNEL

34

4.1 Basic Results 4.1.1 pL − Spaces

34 34

xi

4.2 Erdélyi - Kober Operators 4.3 Boundedness of Integrals ( ),

, m kM fα β with Variants of Homogeneous Kernel 4.4 Boundedness of Integral Operators Involving

Hypergeometric Functions 1 , , 1m m k mF+ ≥

38 39 43

5. pL −BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING , , 1m m kF m ≥ AS KERNEL

46

5.1 Boundedness of Integrals ( ),m kH fβ Involving Exponential Kernel

5.2 Boundedness of Integral Operators Involving Hypergeometric Functions , , 1m m k mF ≥

46 50

6. pL −BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING GENERALIZED HYPERGEOMETRIC

FUNCTIONS 1 , , 1m m k mFσ+ ≥ AS KERNEL

52

6.1 Boundedness of Integrals ( ), ,,m kM fσ α β with Variants of

Homogeneous Kernel 6.2 Boundedness of Integral Operators Involving

Hypergeometric Functions 1 , , 1m m k mFσ+ ≥

52 57

7. pL −BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING GENERALIZED HYPERGEOMETRIC

FUNCTIONS , , 1m m k mFσ ≥ AS KERNEL

60

7.1 Boundedness of Integrals ,, m kHσ β Involving Exponential

Functions as Kernel 7.2 Boundedness of Integral Operators Involving Hypergeometric

Functions , , 1m m k mFσ ≥

60 64

xii

8. pL −BOUNDEDNESS OF INTEGRAL OPERATORS WITH WEIGHT FUNCTION INVOLVING GENERALIZED

HYPERGEOMETRIC FUNCTION 1 , m m kFσ+ AND

, , 1m m k mFσ ≥ AS KERNEL

67

8.1 Boundedness of Integrals ( ), ,, m kM fσ α β with Weight

Function 8.2 Boundedness of Integral Operators Involving Hypergeometric Functions 1 , , 1m m k mFσ

+ ≥ with Weight Function xν

8.3 Boundedness of Integrals ,, m kHσ β with Weight Function

8.4 Boundedness of Integral Operators Involving Hypergeometric Functions , , 1m m k mFσ ≥ with Weight Function xν

67 70 71 72

REFRENCES

74

APPENDIX 79

1

CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

Traditionally, Laplace, Mellin and Fourier transforms are used to solve

differential and integral equations. However, fractional integrals have been played an important role in solving a large number of differential and integral equations.

Many researchers have shown deep interest in integral operators involving

various hypergeometric functions. Newly established properties of hypergeometric functions and polynomials facilitated solutions of many differential and integral equations involving these functions.

Erdélyi and Love have initiated research on issues which involved

hypergeometric functions 2 1F , so did Saxena, Srivastava and Kilbas, and Habibullah has followed them to consider integral equations involving confluent hypergeometric functions 1 1F . Okikiolu proved the boundedness of similar integral operators in 1L and , 1pL p > , and he did so by proving that these operators could be represented as composition of linear operators and used their mapping properties to derive inversion processes and hence to show their boundedness. In addition to a variety of integral operators involving Fredholm type kernels, integral operators of convolution type involving hypergeometric functions have also been studied by various authors. Karapetiants and Samko proved that two types of operators are related by a simple relation. Numerous forms of fractional integral operators have been the critical source of facilitation for solutions and have enriched the study of integral operators.

Recently, Driver and Johnston gave a simple integral representation of some hypergeometric functions 1m mF+ . This representation would help to study 1m mF+ and integral operators involving these hypergeometric functions. Similarly one could study m mF . The direction of using these representations has been source of inspiration of the current work.

2

1.2 LITERATURE REVIEW

Hypergeometric functions and polynomials generated by hypergeometric functions have been extensively studied for years. We could mention Euler, Gauss, Kummer, Riemann, Schwarz, Abramowitz, Andrews and Slater who contributed to advancement of special functions. Integral representations of hypergeometric functions have been excellent source for their application and for studying their behavior. Recently, Driver and Johnston (2006) have found new integral representations of some special generalized hypergeometric functions that have facilitated investigations in the present work.

Earlier, Erdélyi (1953), Rainville (1963), Wang et al. (1989), Mathai

(1993), Andrews et al. (2004), Mathai and Haubold (2008) and Cuyt at al. (2008) and other collected a large number of properties of hypergeometric functions. Saad and Hall (2003) showed that many integrals containing products of confluent hypergeometric functions follow directly from one single integral which has a very simple formula in terms of Appell’s double series. Virchenko et al. (2001) defined a new generalized hypergeometric function, using a special case of Wright's function. Chiavetta (2007a and 2007b) determined the summation of a series of hypergeometric functions using Schäfke’s method. Ahmad and Saboor (2009) gave another notation to generalized power series hypergeometric functions and applied it to statistical analysis. Virchenko and Ovcharenko (2011) established the generalization of Laplace, Stieltjes, and potential integral transforms with the generalized (according to Wright) hypergeometric functions. Medhat and Arjun (2011) provided the generalizations of the classical summation theorems such as of Gauss, Kummer and Bailey for the series 2 1 F and 3 2 F in general cases.

Fractional derivatives and integrals had been popular research directions for many years. There were so many researchers who studied fractional calculus; we could mention Fourier, Abel, Liouville, Riemann, Letnikov, Hardy, Weyl, Littlewood, Kober, Erdélyi, Love, Samko, Kilbas, Srivastava, Trujillo, Amsterdam and Miller. Finding properties and uses of fractional integration began with Kober (1940). It had, henceforth, been studied by numerous researchers for example Erdélyi (1950), Saxena (1967), Lowndes (1970). Habibullah and Choudhary (1993) proved some simple and applicable properties of fractional integration. Saigo and Kilbas (1994) generalized the fractional integrals and derivatives in Hölder spaces. Kilbas et al. (2002) studied the generalized fractional integral transforms involving the Gauss hypergeometric functions as kernel by generalizing Love’s work. Many other researchers studied generalizations of Erdélyi-Kober operators for example

3

Nishimoto (1991), Samko et al. (2002) and Kilbas et al. (2006). Castell (2004) investigated a relationship of fractional derivatives and the inverse Fourier transform of radial functions. Kilbas and Sebastian (2008) studied the generalized fractional integration of Bessel functions of the first kind. Srivastava and Tomovski (2009) considered fractional integral operator containing a generalized Mittag-Leffler function operator given by Prabhakar (1971), which are more suitable for applied problems, see Mathai et al. (2010). Virchenko et al. (2010) gave a generalization of Gauss hypergeometric function, and investigated its basic properties. Further, they defined fractional integral operators and their inverses in terms of the Mellin transform. Agarwal and Jain (2011) obtained additional fractional calculus formulae, using series expansion method, for polynomials which were introduced by Srivastava (1972). Higgins (1963) found an inversion formula for integral equation with Gegenbauer transformation as kernel. Srivastava (1963 and 1966) discussed a class of integral equations involving ultraspherical and Laguerre polynomials.

Erdélyi (1963, 1964) used fractional integrals to study integral transform

given by

( ) ( ) ( )1

2 2 2x x

g x x t P f t dtt

µ µν

α

− = − ∫ .

Wimp (1965) discussed a pair of integral transforms involving hypergeometric functions and thus started study of pairs of integral equations. With the help of Erdélyi–Kober fractional integrals, Love (1967) determined the necessary and sufficient condition for the solution of the equation

( ) ( )( ) ( )

1

2 10

,; ; ;1 , 0cx x t x

g xc t

F a b c f t dt x−−

= Γ − >∫

where ( )2 1 ; ; ;F a b c x is the usual hypergeometric function. Kalla and Saxena (1969) discussed the integral operators involving hypergeometric functions. Habibullah (1970) used integer-value fractional integral to solve an integral equation involving Shively’s polynomials. Okikiolu (1971) used fractional integral to find inversion of integral operator involving Bessel functions. Habibullah (1971) used fractional integration to investigate a solution of the integral equation of the type

( ) ( )( ) ( ) ( )

1

0

,; ; , 0, 0cx x t

g xc

a c t x f t dt a c−−

=Γ

Φ − > >∫

where ( ); ;a c t xΦ − is the confluent hypergeometric function and determined the necessary and sufficient condition for this solution of the integral equation.

4

Srivastava (1976) solved a number of integral equations involving confluent hypergeometric functions as kernel. Okikiolu (1966) proved the boundedness of generalized Fourier transforms in pL -space. Habibullah (1977) also investigated in , 1pL p ≥ , the boundedness of integral transform of the type

( ) ( ) ( ) ( )12 1

0

., ; ;x

bg x xt F a b c x f t dt−= ∫

Saigo (1977/78) studied integral operators involving the Gauss hypergeometric functions. Andersen and Heinig (1983) obtained new inequalities for a class of convolution operators, various fractional integrals and the Laplace transform. Saxena and Ram (1990) discussed the multidimensional generalized Erdélyi-Kober operators associated with the Gauss’s hypergeometric functions. Karapetiants and Samko (1999) gave a self – contained representation of the Fredholm theory of one–and multi-dimensional integral operators of the kind

( ) ( ) ( ), , , , 0 ,n

y a

x k x y y dy x R x a aλϕ ϕ<

= ∈ < < < ∞∫

with the homogeneous kernel of degree n− , that is, ( ) ( ), , , , , 0.n nk tx ty t k x y x y R t−= ∈ >

Goyal et al. (1991) derived a number of different expressions for the

certain compositions of fractional integral operators involving a product of generalized hypergeometric functions and a general class of polynomials with essentially arbitrary coefficients. Saxena and Saigo (1998) obtained a fractional integral formula for the H-function II. Kilbas et al. (2000) studied the Meijer integral transform with the modified Bessel function of the third kind ( )k zη of

complex order η as the kernel. Srivastava and Saxena (2001) discussed the applications of fractional integral operators. Kilbas et al. (2004) gave the relation between generalized Mittag-Leffler function and generalized fractional integral operators.

Vi Nhan and Duc (2008) gave some new type of convolution inequalities

in weighted pL − spaces and their important applications to partial differential and integral transforms. Belarbi and Dahmani (2009) established integral inequalities for the Chebyshev functional in the case of two synchronous functions by using the Riemann-Liouville fractional integrals. Aliev (2010) obtained a relation between the Fourier, Bessel and Riemann-Liouville integral transforms and gave an application of this relation to weighted norm inequalities. Saha and Arora (2010) investigated the relations that exist between the Riemann-

5

Liouville fractional calculus and multi-index Dzrbashjan-Gelfond-Leontiev differentiation and integration with multi-index Mittag-Leffler function. Arcadii (2011) used a weighted inequality for two functions to estimate a natural connection between solutions of Volterra equations and related convolution integral equations. Salahuddin (2011) developed summation formulae based on half argument by the help of Gauss second summation theorem. Zareen (2011) obtained new generalizations of the Hardy type integral inequality by using a fairly elementary analysis. Haubold et al. (2011) derived relations of Mittag-Leffler functions with Riemann-Liouville fractional operators.

There have been a lot of investigations in generalizing classical gamma function. For example, Saxena et al. (2007) defined and studied generalization of the generalized gamma-type functions. Recently, in a series of research publications, Diaz et al. (2005, 2007 and 2010) have introduced k −gamma and k −beta functions and proved a number of their properties where we are interested in. They have also studied k −zeta function and k −hypergeometric functions based on Pochhammer k −symbols for factorial functions. It has been followed by works of Mansour (2009), Kokologiannaki (2010), Kransniqi (2010) and Merovci (2010) elaborating and strengthening study of k −gamma and k −beta functions. These studies on k − formulations have generated interest in developing the work presented here after.

6

CHAPTER 2

k −ANALOGUE OF HYPERGEOMETRIC FUNCTIONS

In this chapter, following the Pochhammer k −symbols and the definitions

of k −hypergeometric functions introduced by Diaz et al. (2005, 2007 and 2010), we prove properties of generalized k −hypergeometric functions. We prove integral representations of k −hypergeometric functions. In the same vein, we also determine results on generalized confluent k −hypergeometric functions that can be used later. We begin with the classical hypergeometric functions by noting the assumption that all numbers we encounter henceforth are real unless and until otherwise mentioned. 2.1 THE HYPERGEOMETRIC FUNCTIONS

Let ij be a function from N into N such that 1 , i ij j i N+< ∀ ∈ , where

N is a set of natural numbers and also m N∈ . The confluent hypergeometric functions with two parameters , β γ , one

parameter β in numerator and one parameterγ in denominator, is given by ( )( )1 1

0

; ; , , 0, 1, 2,...,!

jj

j j

xF x

j

βββ γ γ

γ γ

∞

=

= ∀ ≠ − −

∑ for all finite x .

The hypergeometric functions with three parameters , , ,α β γ two

parameters , α β in numerator and one parameter γ in denominator, is defined by

( ) ( )( )2 1

0

,; ; , , , 0, 1, 2,..., 1,

!

jj j

j j

xF x x

j

α βα βα β γ γ

γ γ

∞

=

= ∀ ≠ − − <

∑

where ( ) ( )( ) ( ) ( )0 1 2 ... 1 ; 1; 1m

m mα α α α α α= + + + − ≥ = ,

(See Rainville (1963)). 2.2 THE GENERALIZED HYPERGEOMETRIC FUNCTIONS

The generalized hypergeometric functions with r s+ parameters, r

parameters in numerator and s parameters in denominator, is defined by

7

( )

( )1 2 1

1 2 01

, ,..., ;

, ,..., ; !

j

j

rju

r ur s s

s j vv

xF x

j

αα α αβ β β β

∞=

==

∏ =

∏∑ ,

, , 0, 1, 2,..., 1, 1,2,..., and 1,2,..., .u v v x u r v sα β β∀ ≠ − − < = = It is known that

i) if r s≤ , the series converges for all finite x ;

ii) if 1r s= + , the series converges for 1x < and diverges for 1x > ;

iii) if 1r s> + , the series diverges for 0x ≠ unless the series terminates.

2.3 INTEGRAL REPRESENTATIONS OF HYPERGEOMETRIC AND

CONFLUENT HYPERGEOMETRIC FUNCTIONS

A number of integral representations of the hypergeometric and confluent

hypergeometric functions have been known for years and a few of these are stated in the following lemmas. Lemma 2.3.1: If 0γ β> > , then for all finite x

( )( ) ( ) ( )

111

1 10

; 1 xtF x t t e dtγ ββγβγ β γ β

− −−Γ = − Γ Γ −

∫ ,

(See Rainville (1963)). Lemma 2.3.2: If 0, 1xγ β> > < , then

( )( ) ( )

( ) ( )1 11

2 10

, ; 1 1F x t t xt dt

γ β αβα β γγ β γ β

− − −−Γ = − − Γ Γ −

∫ ,

(See Rainville (1963)).

Driver and Johnston (2006) have, subsequently, proved integral representations of special case of generalized hypergeometric functions as stated below.

8

Lemma 2.3.3: If 0, 1, 1m xγ β> > ≥ < , then

1

1 1, , ,...,

; 1 1

, ,...,m m

mm m mF x

mm m m

β β βα

γ γ γ+

+ + −

+ + −

=( )

( ) ( ) ( ) ( )1

11

0

1 1 mt t xt dtαγ ββγ

β γ β

−− −−Γ− −

Γ Γ − ∫ ,

(See Driver and Johnston (2006)).

Using similar arguments, we determine a convenient integral representation of generalized hypergeometric functions , 1m mF m ≥ annunciated in the next lemma (See Habibullah and Mubeen (2011)). Lemma 2.3.4: If 0, 1mγ β> > ≥ , then for all finite x

1 1, ,...,

; 1 1

, ,...,m m

mm m mF x

mm m m

β β β

γ γ γ

+ + −

+ + −

=( )

( ) ( ) ( )1

11

0

1mxtt t e dtγ ββγ

β γ β− −−Γ

−Γ Γ − ∫ .

2.4 OTHER INTEGRAL REPRESENTATIONS OF

HYPERGEOMETRIC FUNCTIONS In this section, we determine an integral representation of generalized hypergeometric functions by using Legendre’s multiplication formula. Also, we use this integral representation to obtain some results on confluent hypergeometric functions m mF . We take Legendre’s multiplication formula (See Rainville (1963)).

( ) ( ) ( )1 12 21

1

12

m m ms

sm m

mαα π α− −

=

− + = Γ

Γ∏ (2.4.1)

to obtain the integral representation of generalized hypergeometric functions as given below.

9

Theorem 2.4.1: If 0, 1c b x> > < , then

3 21 1

, , ; , ;2 2

F a b b c c x + +

( )( ) ( ) ( ) ( )

12 2 12 1 2

0

21 1 .

2 2 2

ac bbct t xt dt

b c b

−− −−Γ= − −Γ Γ − ∫ (2.4.2)

Proof: Suppose that 1x < . Then

3 21 1

, , ; , ;2 2

F a b b c c x + +

( ) ( ) ( )( ) ( )

12

10 2

!

jj j j

j j j

a b b xjc c

∞

=

+=

+∑

( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

1122

1 102 2

!

jj

j

a b j b jc c xjb b c j c j

∞

=

Γ + Γ + +Γ Γ +=Γ Γ + Γ + Γ + +∑

since ( ) ( )( )

j

jαα

αΓ +

=Γ

.

Now, we use (2.4.1) and obtain the following expressions.

3 21 1

, , ; , ;2 2

F a b b c c x + +

( )

( ) ( )( ) ( ) ( )

( )0

2 2 2 222 2 2 2 2 !

jj

j

a b j c bc xb c b c j j

∞

=

Γ + Γ −Γ=Γ Γ − Γ +∑

( )

( ) ( ) ( ) ( )1

2 2 12 2 1

0 0

21

2 2 2 !

jc bb j

jj

c xa t t dt

b c b j

∞− −+ −

=

Γ= −

Γ Γ − ∑ ∫

( )

( ) ( ) ( ) ( )( )21

2 2 12 1

00

21

2 2 2 !

j

c bbj

j

xtct t a dt

b c b j

∞− −−

=

Γ = − Γ Γ −

∑∫

( )( ) ( ) ( ) ( )

12 2 12 1 2

0

21 1 .

2 2 2

ac bbct t xt dt

b c b

−− −−Γ= − −Γ Γ − ∫

Theorem 2.4.2: If 1

2 0, 0, c b a c a> > > > , then

3 21 1

, , ; , ;12 2

F a b b c c + +

( ) ( )( ) ( ) ( )2 12 2 2

,2 ;2 ; 12 2 2c c b a

F a b c ac a c b

Γ Γ − −= − −Γ − Γ −

.

10

Proof: Put 1x = in (2.4.2). We, then, get the following expressions.

3 21 1

, , ; , ;12 2

F a b b c c + +

( )( ) ( ) ( ) ( )

12 2 12 1 2

0

21 1

2 2 2

ac bbct t t dt

b c b

−− −−Γ= − −Γ Γ − ∫

( )( ) ( ) ( ) ( )

12 2 12 1

0

21 1

2 2 2c b a abc

t t t dtb c b

− − − −−Γ= − +Γ Γ − ∫

( )( ) ( ) ( ) 1

1

12 2 12 1

10

21

2 2 2c b a jb

j

act t t dt

jb c b

∞− − −− −Γ

= − Γ Γ − ∑∫

( )( ) ( ) ( )1

1

12 2 12 1

10 0

21

2 2 2c b ab j

j

act t dt

jb c b

∞− − −+ −

=

−Γ = − Γ Γ −

∑ ∫

( )( ) ( )

( ) ( )( )

1

1

1 10

2 2 2 22 2 2 2j

ac b j c b ajb c b c a j

∞

=

−Γ Γ + Γ − − = Γ Γ − Γ − +

∑

( ) ( )( ) ( )

( ) ( )( )

( ) 11 1

1 110

22 2 2 12 2 2 2 !

jj j

j j

a bc c b ac a c b c a j

∞

=

Γ Γ − − −=Γ − Γ − −∑

( ) ( )( ) ( ) ( )2 12 2 2

,2 ;2 ; 12 2 2c c b a

F a b c ac a c b

Γ Γ − −= − −Γ − Γ −

.

Corollary 2.4.3: If 1

2 0, c b c n> > > , then

3 21 1

, , ; , ;12 2

F n b b c c − + +

( )( ) ( )2 12 2

,2 ;2 ; 12

n

n

c bF n b c n

c

−= − + − .

Theorem 2.4.4: If 0, 1c b x> > < , then

4 31 2 1 2

, , , ; , , ;3 3 3 3

F a b b b c c c x + + + +

( )( ) ( ) ( ) ( )

13 3 13 1 3

0

31 1 .

3 3 3

ac bbct t xt dt

b c b

−− −−Γ= − −Γ Γ − ∫ (2.4.3)

Proof: Suppose that 1x < . Then

4 31 2 1 2

, , , ; , , ;3 3 3 3

F a b b b c c c x + + + +

11

( ) ( ) ( ) ( )( ) ( ) ( )

1 23 3

1 20 3 3

!

jj j j j

j j j j

a b b b xjc c c

∞

=

+ +=

+ +∑

( ) ( ) ( )( ) ( ) ( )

1 23 31 23 3

c c c

b b b

Γ Γ + Γ +=Γ Γ + Γ +

( ) ( ) ( ) ( )

( ) ( ) ( )1 23 3

1 20 3 3

!

jj

j

a b j b j b j xjc j c j c j

∞

=

Γ + Γ + + Γ + +×

Γ + Γ + + Γ + +∑

since ( ) ( )( )

j

jαα

αΓ +

=Γ

.

Now, we use (2.4.1) and obtain the following expressions.

4 31 2 1 2

, , , ; , , ;3 3 3 3

F a b b b c c c x + + + +

( )

( ) ( )( ) ( ) ( )

( )0

3 3 3 333 3 3 3 3 !

jj

j

a b j c bc xb c b c j j

∞

=

Γ + Γ −Γ=Γ Γ − Γ +∑

( )

( ) ( ) ( ) ( )1

3 3 13 3 1

0 0

31

3 3 3 !

jc bb j

jj

c xa t t dt

b c b j

∞− −+ −

=

Γ= −

Γ Γ − ∑ ∫

( )

( ) ( ) ( ) ( )( )31

3 3 13 1

00

31

3 3 3 !

j

c bbj

j

xtct t a dt

b c b j

∞− −−

=

Γ = − Γ Γ −

∑∫

( )

( ) ( ) ( ) ( )1

3 3 13 1 3

0

31 1 .

3 3 3

ac bbct t xt dt

b c b

−− −−Γ= − −Γ Γ − ∫

Theorem 2.4.5: If 1

3 0, 0, c b a c a> > > > , then

4 31 2 1 2

, , , ; , , ;13 3 3 3

F a b b b c c c + + + +

( ) ( )( ) ( )

( )( )

1

1 110

33 3 33 3 3 3

j

j j

bac c b ajc a c b c a

∞

=

−Γ Γ − − = Γ − Γ − −

∑

( )2 1 1 1 1,3 ;3 ; 1F j b j c a j× − + − + − .

12

Proof: Put 1x = in (2.4.3) to get the following expressions.

4 31 2 1 2

, , , ; , , ;13 3 3 3

F a b b b c c c + + + +

( )( ) ( ) ( ) ( )

13 3 13 1 3

0

31 1

3 3 3

ac bbct t t dt

b c b

−− −−Γ= − −Γ Γ − ∫

( )( ) ( ) ( ) ( )

13 3 13 1 2

0

31 1

3 3 3

ac b abct t t t dt

b c b

−− − −−Γ= − + +Γ Γ − ∫

( )( ) ( ) ( ) ( ) 11

1

13 3 13 1

10 0

31 1

3 3 3c b a jb j

j

act t t dt

jb c b

∞− − −+ −

=

−Γ = − + Γ Γ −

∑ ∫

( )

( ) ( )3

3 3 3c

b c bΓ

=Γ Γ −

( )1

1 2

1 2

13 3 11 3 1

1 20 0

1j

c b ab j j

j j

a jt t dt

j j

∞− − −+ + −

=

− × −

∑ ∑ ∫

( )( ) ( )

1 10

33 3 3 j

acjb c b

∞

=

−Γ = Γ Γ −

∑

( ) ( )( )

1

2

1 1 2

2 1 20

3 3 33

j

j

j b j j c b aj c a j j=

Γ + + Γ − − × Γ − + + ∑

( ) ( )( ) ( )

1 10

3 3 33 3 3 j

ac c b ajc a c b

∞

=

−Γ Γ − − = Γ − Γ −

∑

( )( )

11 2

2 1 2

1

20

3

3

jj j

j j j

bj

j c a+

= +

× −

∑

( ) ( )( ) ( )

( )( )

1

1 110

33 3 33 3 3 3

j

j j

bac c b ajc a c b c a

∞

=

−Γ Γ − − = Γ − Γ − −

∑

( ) ( )( )

( ) 212 2

2 2

1 1

1 20

3 13 !

jjj j

j j

j b j

c a j j=

− + −×

− +∑

( ) ( )( ) ( )

( )( )

1

1 110

33 3 33 3 3 3

j

j j

bac c b ajc a c b c a

∞

=

−Γ Γ − − = Γ − Γ − −

∑

( )2 1 1 1 1,3 ;3 ; 1F j b j c a j× − + − + − .

13

Corollary 2.4.6: If 13 0, c b c n> > < , then

4 31 2 1 2

, , , ; , , ;13 3 3 3

F n b b b c c c − + + + +

( )( )3 3

3n

n

c b

c

−=

( )( ) ( )1

1 1

2 1 1 1 110

3,3 ;3 ; 1

3

nj

j j

bnF j b j c n j

j c n=

× − + + + − + ∑ .

Theorem 2.4.7: If 0, 1, 1c b x m> > < ≥ , then

11 1 1 1

, , ,..., ; , ,..., ;m mm m

F a b b b c c c xm m m m+

− − + + + +

( )

( ) ( ) ( ) ( )1

11

0

1 1 .amc mbmb mmc

t t xt dtmb mc mb

−− −−Γ= − −Γ Γ − ∫ (2.4.4)

Proof: Suppose that 1x < . Then

11 1 1 1

, , ,..., ; , ,..., ;m mm m

F a b b b c c c xm m m m+

− − + + + +

( ) ( ) ( ) ( )( ) ( ) ( )

11

110

...

!...

m jm mj j j jm

j m mj j j

a b b b xjc c c

−∞

−=

+ +=

+ +∑

( ) ( ) ( )( ) ( ) ( )

11

11

...

...

mm m

mm m

c c c

b b b

−

−

Γ Γ + Γ +=Γ Γ + Γ +

( ) ( ) ( ) ( )

( ) ( ) ( )11

110

...

!...

m jm mj

mj m m

a b j b j b j xjc j c j c j

−∞

−=

Γ + Γ + + Γ + +×

Γ + Γ + + Γ + +∑

since ( ) ( )( )

j

jαα

αΓ +

=Γ

.

Now, we use (2.4.1) and obtain the following expressions.

11 1 1 1

, , ,..., ; , ,..., ;m mm m

F a b b b c c c xm m m m+

− − + + + +

( )

( ) ( )( ) ( ) ( )

( )0 !

jj

j

a mb mj mc mbmc xmb mc mb mc mj j

∞

=

Γ + Γ −Γ=Γ Γ − Γ +∑

( )

( ) ( ) ( ) ( )1

11

0 0

1!

jmc mbmb mj

jj

mc xa t t dt

mb mc mb j

∞− −+ −

=

Γ= −

Γ Γ − ∑ ∫

14

( )( ) ( ) ( ) ( )

( )111

00

1!

jmmc mbmb

jj

xtmct t a dt

mb mc mb j

∞− −−

=

Γ = − Γ Γ −

∑∫

( )

( ) ( ) ( ) ( )1

11

0

1 1 .amc mbmb mmc

t t xt dtmb mc mb

−− −−Γ= − −Γ Γ − ∫

Theorem 2.4.8: If 1 0, 0, , 2mc b a c a m> > > > ≥ , then

11 1 1 1

, , ,..., ; , ,..., ;1m mm m

a b b b c c cm m m m

F+− −

+ + + +

( ) ( )( ) ( )mc mc mb amc a mc mb

Γ Γ − −=Γ − Γ −

31 2

1 2 3 2

2 31

3 21 20 0 0 0

... ...m

m

jj jm

mj j j j

j ja j

j jj j

−

−

∞−

−= = = =

− ×

∑ ∑ ∑ ∑

( ) ( ) ( )

( ) ( ) ( )1 2 2

1 2 2

1 1 2 3

1 1 2 3

... ...

... ...m

m

mj j j

mj j j

mb mb j mb j j j

mc a mc a j mc a j j j−

−

−

−

+ + + + +×

− − + − + + +

( )2 1 2 1 2 2 1 2 2, ... ; ... ; 1m m mF j mb j j j mc a j j j− − −× − + + + + − + + + + − . Proof: Using 1x = in (2.4.4), we get the following expressions.

11 1 1 1

, , ,..., ; , ,..., ;1m mm m

a b b b c c cm m m m

F+− −

+ + + +

( )( ) ( ) ( ) ( )

111

0

1 1 .amc mbmb mmc

t t t dtmb mc mb

−− −−Γ= − −Γ Γ − ∫

( )

( ) ( )mc

mb mc mbΓ

=Γ Γ −

( ) ( )1

11 2 1

0

1 1 ...amc mb amb mt t t t t dt

−− − −− −× − + + + +∫

( )

( ) ( )mc

mb mc mbΓ

=Γ Γ −

( ) ( ) 11

1

111 2 2

10 0

1 1 ...jmc mb amb j m

j

at t t t t dt

j

∞− − −+ − −

=

− × − + + + +

∑ ∫

( )( ) ( )

mcmb mc mb

Γ=Γ Γ −

1

1 2

1

1 20

j

j j

a j

j j

∞

=

− ×

∑∑

15

( ) ( ) 21 2

111 2 3

0

1 1 ...jmc mb amb j j mt t t t t dt− − −+ + − −× − + + + +∫

( )

( ) ( )mc

mb mc mbΓ

=Γ Γ −

1 2

1 2 3

21

31 20 0

j j

j j j

ja j

jj j

∞

= =

− ×

∑∑ ∑

( ) ( ) 31 2 3

111 2 4

0

1 1 ...jmc mb amb j j j mt t t t t dt− − −+ + + − −× − + + + +∫ .

Continue this process, we arrive at the following result

11 1 1 1

, , ,..., ; , ,..., ;1m mm m

a b b b c c cm m m m

F+− −

+ + + +

( )

( ) ( )mc

mb mc mbΓ

=Γ Γ −

31 2

1 2 3 2

2 31

3 21 20 0 0 0

... ...m

m

jj jm

mj j j j

j ja j

j jj j

−

−

∞−

−= = = =

− ×

∑ ∑ ∑ ∑

( ) ( ) 21 2 3 2

11... 1

0

1 1 mm mc mb a jmb j j j jt t t dt−− − − −+ + + + + −× − +∫

( )

( ) ( )mc

mb mc mbΓ

=Γ Γ −

31 2

1 2 3 2

2 31

3 21 20 0 0 0

... ...m

m

jj jm

mj j j j

j ja j

j jj j

−

−

∞−

−= = = =

− ×

∑ ∑ ∑ ∑

( )2

1 2 3 1

1

112 ... 1

10 0

1m

m

m

jmc mb am mb j j j j

mj

jt t dt

j

−−

−

− − −− + + + + + −

−=

× −

∑ ∫

( )

( ) ( )mc

mb mc mbΓ

=Γ Γ −

31 2

1 2 3 2

2 31

3 21 20 0 0 0

... ...m

m

jj jm

mj j j j

j ja j

j jj j

−

−

∞−

−= = = =

− ×

∑ ∑ ∑ ∑

( ) ( )

( )2

1

2 1 2 3 1

1 1 2 3 10

...

...

m

m

jm m

m mj

j mb j j j j mc mb aj mc a j j j j

−

−

− −

− −=

Γ + + + + + Γ − − × Γ − + + + + + ∑

( ) ( )( ) ( )mc mc mb amc a mc mb

Γ Γ − −=Γ − Γ −

16

31 2

1 2 3 2

2 31

3 21 20 0 0 0

... ...m

m

jj jm

mj j j j

j ja j

j jj j

−

−

∞−

−= = = =

− ×

∑ ∑ ∑ ∑

( ) ( )

( )( ) 12

1 2 3 11

1 1 2 3 1

2 ...

10 ...

1!

mmmm

m m

jjm j j j jj

mj j j j j

j mb

mc a j

−−−−

− −

− + + + +

−= + + + +

− −×

Γ −∑

( ) ( )( ) ( )mc mc mb amc a mc mb

Γ Γ − −=Γ − Γ −

31 2

1 2 3 2

2 31

3 21 20 0 0 0

... ...m

m

jj jm

mj j j j

j ja j

j jj j

−

−

∞−

−= = = =

− ×

∑ ∑ ∑ ∑

( ) ( ) ( )

( ) ( ) ( )1 2 2

1 2 2

1 1 2 3

1 1 2 3

... ...

... ...m

m

mj j j

mj j j

mb mb j mb j j j

mc a mc a j mc a j j j−

−

−

−

+ + + + +×

− − + − + + +

( ) ( )

( )( ) 12

1 1

1 1

2 1 2 3 2

1 2 3 2 10

... 1... !

mmm m

m m

jjm mj j

m mj j

j mb j j j j

mc a j j j j j

−−− −

− −

− −

− −=

− + + + + + −×

− + + + + +∑

( ) ( )( ) ( )mc mc mb amc a mc mb

Γ Γ − −=Γ − Γ −

31 2

1 2 3 2

2 31

3 21 20 0 0 0

... ...m

m

jj jm

mj j j j

j ja j

j jj j

−

−

∞−

−= = = =

− ×

∑ ∑ ∑ ∑

( ) ( ) ( )

( ) ( ) ( )1 2 2

1 2 2

1 1 2 3

1 1 2 3

... ...

... ...m

m

mj j j

mj j j

mb mb j mb j j j

mc a mc a j mc a j j j−

−

−

−

+ + + + +×

− − + − + + +

( )2 1 2 1 2 2 1 2 2, ... ; ... ; 1m m mF j mb j j j mc a j j j− − −× − + + + + − + + + + − . Corollary 2.4.9: If 1 0, , 2mc b c n m> > > ≥ , then

11 1 1 1

, , ,..., ; , ,..., ;1m mm m

n b b b c c cm m m m

F+− −

− + + + +

( )( )

n

n

mc mb

mc

−=

31 2

1 2 3 2

2 31

3 21 20 0 0 0

... ...m

m

jj jnm

mj j j j

j jn j

j jj j

−

−

−

−= = = =

×

∑ ∑ ∑ ∑

( ) ( ) ( )

( ) ( ) ( )1 2 2

1 2 2

1 1 2 3

1 1 2 3

... ...

... ...m

m

mj j j

mj j j

mb mb j mb j j j

mc n mc n j mc n j j j−

−

−

−

+ + + + +×

+ + + + + + +

( )2 1 2 1 2 2 1 2 2, ... ; ... ; 1m m mF j mb j j j mc n j j j− − −× − + + + + + + + + + − .

17

2.5 RESULTS ON CONFLUENT HYPERGEOMETRIC FUNCTIONS In this section, we evaluate a number of results on confluent

hypergeometric functions by using the results in section 2.4. Result 2.5.1: If 0c b> > , then

2 2 , ; , ;2 21 1xe F b b c c x−

+ +

( ) ( ) ( )( ) ( )

12

10 0 2

! !

n jj j

n j j j

b bx xn jc c

∞ ∞

= =

+ − = + ∑ ∑

( )( )

( ) ( )( ) ( )

12

10 0 2

! !

n jn jj j

n j j j

b bx xn j jc c

−∞

= =

+−=

− +∑∑

( ) ( ) ( )( ) ( )

( )12

10 0 2

1! !

nnj j j

n j j j

n b b xj nc c

∞

= =

− + − = +

∑∑

( )3 2

0

1 1, , ; , ;1

2 2 !

n

n

xF n b b c c

n

∞

=

− = − + +

∑

( )( ) ( ) ( )

2 10

2 2,2 ;2 ; 1

2 !

nn

n n

c b xF n b c n

c n

∞

=

− −= − + −∑ .

Hence, we obtain the following result

2 2 , ; , ,;2 21 1

F b b c c x

+ +

( )( ) ( ) ( )

2 10

2 2,2 ;2 ; 1

2 !

nx n

n n

c b xe F n b c n

c n

∞

=

− −= − + −∑ .

Result 2.5.2: If 0c b> > , then

3 31 2 1 2

, , ; , , ;3 3 3 3

xe F b b b c c c x− + + + +

18

( ) ( ) ( ) ( )( ) ( ) ( )

1 23 3

1 20 0 3 3

! !

n jj j j

n j j j j

b b bx xn jc c c

∞ ∞

= =

+ + − = + +

∑ ∑

( )( )

( ) ( ) ( )( ) ( ) ( )

1 23 3

1 20 0 3 3

! !

n jn jj j j

n j j j j

b b bx xn j jc c c

−∞

= =

+ +−=

− + +∑∑

( ) ( ) ( ) ( )( ) ( ) ( )

( )1 23 3

1 20 0 3 3

1! !

nn j j j j

n j j j j

n b b b xj nc c c

∞

= =

− + + − = + +

∑∑

( )4 3

0

1 2 1 2, , , ; , , ;1

3 3 3 3 !

n

n

xF n b b b c c c

n

∞

=

− = − + + + +

∑

( )( )

( )( )

1

1 110 0

33 3

3 3

njn

n jn j

bc b n

jc c n

∞

= =

− = + ∑ ∑

( ) ( )2 1 1 1 1,3 ;3 ; 1

!

nxF j b j c n j

n−

× − + + + − .

Hence,

3 31 2 1 2

, , ; , , ;3 3 3 3

F b b b c c c x

+ + + +

( )( )

( )( )

1

1 110 0

33 3

3 3

njz n

n jn j

bc b ne

jc c n

∞

= =

− = + ∑ ∑

( ) ( )2 1 1 1 1,3 ;3 ; 1 .

!

nxF j b j c n j

n−

× − + + + −

Result 2.5.3: If 0, 2c b m> > ≥ , then

1 1 1 1, ,..., ; , ,..., ;x

m mm m

e F b b b c c c xm m m m

− − −+ + + +

( ) ( ) ( ) ( )

( ) ( ) ( )11

110 0

...

! !...

mn jm mj j jm

n j m mj j j

b b bx xn jc c c

−∞ ∞

−= =

+ + − = + +

∑ ∑

( )( )

( ) ( ) ( )( ) ( ) ( )

11

110 0

...

! !...

mn jn jm mj j jm

n j m mj j j

b b bx xn j jc c c

−−∞

−= =

+ +−=

− + +∑∑

19

( ) ( ) ( ) ( )( ) ( ) ( )

( )11

110 0

... 1! !...

m nn m mj j j jm

n j m mj j j

n b b b xj nc c c

−∞

−= =

− + + − = + +

∑∑ .

This implies that 1 1 1 1

, ,..., ; , ,..., ;xm m

m me F b b b c c c x

m m m m− − −

+ + + +

( )1

0

1 1 1 1, , ,..., ; , ,..., ;1

!

n

m mn

xm mF n b b b c c c

m m m m n

∞

+=

−− − = − + + + +

∑ .

Consequently, we conclude 1 1 1 1

, ,..., ; , ,..., ;m mm m

F b b b c c c xm m m m

− −+ + + +

( )( )0

x n

n n

mc mbe

mc

∞

=

−= ∑

31 2

1 2 3 2

2 31

3 21 20 0 0 0

... ...m

m

jj jnm

mj j j j

j jn jj jj j

−

−

−

−= = = =

×

∑ ∑ ∑ ∑

( ) ( ) ( )

( ) ( ) ( )1 2 2

1 2 2

1 1 2 3

1 1 2 3

... ...

... ...m

m

mj j j

mj j j

mb mb j mb j j j

mc n mc n j mc n j j j−

−

−

−

+ + + + +×

+ + + + + + +

( )2 1 2 1 2 2 1 2 2, ... ; ... ; 1m m mF j mb j j j mc n j j j− − −× − + + + + + + + + + −

( )

.!

nxn−

×

2.6 k − HYPERGEOMETRIC FUNCTIONS

At this juncture it is relevant to introduce the Pochhammer k −symbols defined by Diaz and Teruel (2005). Let

( ) ( )1

,0

, 0m

m kj

x x jk k−

=

= + >∏ , ( ) ( )1

0

, when 1,m

mj

x x j k−

=

= + =∏

so that we can commence the study of generalized k −hypergeometric functions given by

20

( ) ( ) ( )( ) ( ) ( )

( )

( )

,

,

1 1 2 2 1

1 1 2 2 01

, , , ,..., ,;

, , , ,..., , !

j ku

j kv

rju

r r ur s s

s s j vv

k k k xF x

k k k j

αα α αβ β β β

∞=

==

∏ =

∏∑

, , 0, 1, 2,..., 1, 0, 1,2,..., and 1,2,..., .u v v x k u r v sα β β∀ ≠ − − < > = = 2.7 INTEGRAL REPRESENTATIONS OF k −HYPERGEOMETRIC

FUNCTIONS

Diaz et al. (2007 and 2010) and others (See Mansour (2009),

Kokologiannaki (2010), Kransniqi (2010) and Merovci (2010)) proved a number of identities of k −gamma function, k −beta function and Pochhammer k −symbols. They also gave integral representations of k −gamma function and k −beta function. In this section, we determine integral representations of various k −hypergeometric functions to be used later in this work.

The k −gamma function is defined by

( ) ( )( )

1

,

!lim ,

xkm

km

m k

m k mkx

x

−

→∞Γ =

( ) ( )1

,0

, 0m

m kj

x x jk k−

=

= + >∏ is the Pochhammer k −symbols for factorial

functions. It has been shown that the Mellin transform of the exponential

function ktke− is the k −gamma function, explicitly given by

( ) ( )1 1

0

, 0, 0

kx txxk k

k kx k t e dt x k∞− −−Γ = Γ = > >∫ .

Clearly, ( ) ( )1

lim kk

x x→

Γ = Γ , ( ) ( )1xk x

k kx k −Γ = Γ and ( ) ( )k kx k x xΓ + = Γ .

This gives rise to k −beta function defined by

( )1

1 1

0

1, (1 ) , 0, 0

yxk k

kB x y t t dt x yk

− −= − > >∫

so that ( ) 1, ,k

x yB x y B

k k k =

and ( ) ( ) ( )( )

, k kk

k

x yB x y

x yΓ Γ

=Γ +

.

21

We take some following results which could be used in this study:

( ) ( ) ( )22 2 2 ,, ,

2 j x x kj kj k j k

x+ = ;

( ) ( ) ( ) ( )( )1, , , ,

...x m kmj x x k

m m mmj k j k j k j kx m

+ −+= ;

( ) ( )( ),

k

k

x jkj k x

xΓ +Γ

= ;

( ) ( )( )2

2 ,k

k

x jkj k x

xΓ +Γ

= ;

( ) ( )( ),

k

k

x mjkmj k x

xΓ +Γ

= ;

( ) ( ),0

1!

k

j

j kj

xkx

j

α

α∞

−

=

= −∑ ;

0 !

jx

j

xe

j

∞

=

=∑ .

We now prove the k −analogue of Theorem 2.3.2. Theorem 2.7.1: If 0, 0kγ β> > > and 1x < , then

( ) ( )( )2 1,, , ,

;,k

k kF x

k

α βγ

( )( ) ( ) ( )

11 1

0

(1 ) 1 kk kk

k k

t t kxt dtk

β γ β αγ

β γ β

− −− −Γ= − −

Γ Γ − ∫ . (2.7.1)

Proof: First note that for any positive integer j , we have

( )( )

( )( )

( )( )

,

,

k kj k

k kj k

jk

jk

β β γ

γ β γ

Γ + Γ= ×

Γ Γ +

( )( ) ( )

( );kk

k k

B jkγ

β γ ββ γ β

Γ= + −Γ Γ −

( )

( ) ( ) ( )1

11

0

1 kkk j

k k

t t dtk

γ ββγ

β γ β

− −+ −Γ= −

Γ Γ − ∫ . (2.7.2)

Then, for 1x < , the left hand side of (2.7.1) becomes

( ) ( )( )2 1,, , ,

;,k

k kF x

k

α βγ

=( ) ( )

( ), ,

0 , !

jj k j k

j j k

xj

α β

γ

∞

=∑ .

22

Hence by using (2.7.2), we obtain the following expression.

( ) ( )( )2 1,, , ,

;,k

k kF x

k

α βγ

( )

( ) ( )( )

( ) ( )111 ,

00

1!

kk

jk j k

jk k

xtt t dt

jk

γ ββ αγ

β γ β

− ∞−−

=

Γ = − Γ Γ −

∑∫

( )( ) ( ) ( )

11 1

0

(1 ) 1 kk kk

k k

t t kxt dtk

β γ β αγ

β γ β

− −− −Γ= − −

Γ Γ − ∫ .

Theorem 2.7.2: If 0, 0kγ β> > > and 1x < , then

( ) ( ) ( )( ) ( )

2 23 2,

2 2

, , , , ,;

, , ,

k

k k

k k kF x

k k

β β

γ γ

α +

+

( )( ) ( ) ( )

11 1 2

0

(1 ) 1 kk kk

k k

t t kxt dtk

αβ γ βγ

β γ β

− −− −Γ= − −

Γ Γ − ∫ . (2.7.3)

Proof: First note that for any positive integer j , we have

( )( )

( )( )

( )( )

2 ,

2 ,

2

2k kj k

k kj k

jk

jk

β β γ

γ β γ

Γ + Γ= ×

Γ Γ +

( )( ) ( )

( )2 ;kk

k k

B jkγ

β γ ββ γ β

Γ= + −Γ Γ −

( )

( ) ( ) ( )1

12 1

0

1 kkk j

k k

t t dtk

γ ββγ

β γ β

− −+ −Γ= −

Γ Γ − ∫ . (2.7.4)

Then for 1x < , the left hand side of (2.7.3) becomes

( ) ( ) ( )( ) ( )

2 23 2,

2 2

, , , , ,;

, , ,

k

k k

k k kF x

k k

β β

γ γ

α +

+

=( ) ( ) ( )

( ) ( )2 2, , ,

0 2 2, ,!

kjj k j k j k

kj

j k j k

xj

β β

γ γ

α +∞

+=∑

23

( ) ( )( ), 2 ,

0 2 , !

jj k j k

j j k

xj

α β

γ

∞

=

=∑ .

Hence, by using (2.7.4), we obtain the following

( ) ( ) ( )( ) ( )

2 23 2,

2 2

, , , , ,;

, , ,

k

k k

k k kF x

k k

β β

γ γ

α +

+

( )( ) ( ) ( )

11 1 2

0

(1 ) 1 kk kk

k k

t t kxt dtk

αβ γ βγ

β γ β

− −− −Γ= − −

Γ Γ − ∫ .

Theorem 2.7.3: If 0, 0, 1k mγ β> > > ≥ and 1x < , then

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

1

1 , 1

, , , , , ,..., ,;

, , , ,..., ,

m kkm m m

m m k m kkm m m

k k k kF x

k k k

γβ β

γγ γ

α + −+

+ + −+

( )( ) ( ) ( )

11 1

0

(1 ) 1 kk kk m

k k

t t kxt dtk

αβ γ βγ

β γ β

− −− −Γ= − −

Γ Γ − ∫ . (2.7.5)

Proof: First note that for any positive integer j , we have

( )( )

( ) ( )( ) ( )

,

,

mj k k k

k kmj k

mjkmjk

β γ ββ γγ

Γ Γ +=Γ Γ +

( )( ) ( )

( ),kk

k k

B mjkγ

β γ ββ γ β

Γ= + −Γ Γ −

( )

( ) ( ) ( )1

11

0

1 kkk mj

k k

t t dtk

γ ββγ

β γ β

− −+ −Γ= −

Γ Γ − ∫ . (2.7.6)

Using (2.7.5), we get

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

1

1 , 1

, , , , , ,..., ,;

, , , ,..., ,

m kkm m m

m m k m kkm m m

k k k kF x

k k k

γβ β

γγ γ

α + −+

+ + −+

=( ) ( ) ( ) ( )

( ) ( ) ( ), , , ,

0, , ,

...

!...

k kjm m mj k j k j k j k

k kj m m mj k j k j k

xj

β β β

γ γ γ

α + +∞

+ +=∑

24

( ) ( )( ), ,

0 , !

jj k mj k

j mj k

xj

α β

γ

∞

=

=∑

( )( ) ( ) ( )

11 1

0

(1 ) 1 kk kk m

k k

t t kxt dtk

αβ γ βγ

β γ β

− −− −Γ= − −

Γ Γ − ∫ .

Corollary 2.7.4: If 0, 0, 1k mγ β> > > ≥ and 1x < , then

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

1

1 , 1

,1 , , , , ,..., ,;

, , , ,..., ,

m kkm m m m

m m k m kkm m m

k k kF x

k k k

γβ βα

γγ γ

+ −+

+ + −+

( )( ) ( ) ( )

11 1

0

(1 ) 1 mk kk m

k k

t t xt dtk

αβ γ βγ

β γ β

− −− −Γ= − −

Γ Γ − ∫ .

2.8 INTEGRAL REPRESENTATIONS OF CONFLUENT

k −HYPERGEOMETRIC FUNCTIONS

In this section, we introduce a simple integral representation of certain confluent k −hypergeometric functions. Theorem 2.8.1: If 0, 0kγ β> > > , then for all finite x

( )( )1 1,,;

,kk

F xk

βγ

=( )

( ) ( )1

1 1

0

(1 )k kk xt

k k

t t e dtk

β γ βγ

β γ β

−− −Γ−

Γ Γ − ∫ . (2.8.1)

Proof: First note that for any positive integer j , we have the following expressions

( )( )

( )( )

( )( )

,

,

k kj k

k kj k

jk

jk

β β γ

γ β γ

Γ + Γ= ×

Γ Γ +

( )

( ) ( )( );k

kk k

B jkγ

β γ ββ γ β

Γ= + −Γ Γ −

( )

( ) ( ) ( )1

11

0

1 kkk j

k k

t t dtk

γ ββγ

β γ β

− −+ −Γ= −

Γ Γ − ∫ . (2.8.2)

Then for all finite x , the left hand side of (2.8.1) becomes

25

( )( )1 1,,;

,kk

F xk

βγ

=( )( )

,

0 , !

jj k

j j k

xj

β

γ

∞

=∑ .

Hence by using (2.8.2), we obtain the following expression.

( )( )1 1,,;

,kk

F xk

βγ

( )( ) ( )

( ) ( )111

00

1!

kk

jk

jk k

xtt t dt

jk

γ ββγ

β γ β

− ∞−−

=

Γ = − Γ Γ −

∑∫

( )

( ) ( )1

1 1

0

(1 )k kk xt

k k

t t e dtk

β γ βγ

β γ β

−− −Γ= −

Γ Γ − ∫ .

Theorem 2.8.2: If 0, 0kγ β> > > , then for all finite x

( ) ( )( ) ( )2 2

2 2,

2 2

, , ,;

, , ,

k

k k

k kF x

k k

β β

γ γ

+

+

( )

( ) ( )2

11 1

0

(1 )k kk xt

k k

t t e dtk

β γ βγ

β γ β

−− −Γ= −

Γ Γ − ∫ . (2.8.3)

Proof: For any positive integer j , we have

( )( )

( )( )

( )( )

2 ,

2 ,

2

2k kj k

k kj k

jk

jk

β β γ

γ β γ

Γ + Γ= ×

Γ Γ +

( )( ) ( )

( )2 ;kk

k k

B jkγ

β γ ββ γ β

Γ= + −Γ Γ −

( )

( ) ( ) ( )1

12 1

0

1 kkk j

k k

t t dtk

γ ββγ

β γ β

− −+ −Γ= −

Γ Γ − ∫ (2.8.4)

Then for all finite x , the left hand side of (2.8.3) becomes

( ) ( )( ) ( )2 2

2 2,

2 2

, , ,;

, , ,

k

k k

k kF x

k k

β β

γ γ

+

+

=( ) ( )( ) ( )2 2, ,

0 2 2, ,!

kj

j k j kk

jj k j k

xj

β β

γ γ

+∞

+=∑

( )( )

2 ,

0 2 , !

jj k

j j k

xj

β

γ

∞

=

=∑ .

26

From (2.8.4), we obtain the following expression.

( ) ( )( ) ( )2 2

2 2,

2 2

, , ,;

, , ,

k

k k

k kF x

k k

β β

γ γ

+

+

( )( ) ( )

21

1 1

0

(1 )k kk xt

k k

t t e dtk

β γ βγ

β γ β

−− −Γ= −

Γ Γ − ∫ .

Theorem 2.8.3: If 0, 0, 1k mγ β> > > ≥ , then for all finite x

( ) ( ) ( )( )( ) ( ) ( )( )

1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m

m m k m kkm m m

k k kF x

k k k

γβ β

γγ γ

+ −+

+ −+

( )

( ) ( )1

1 1

0

(1 )m

k kk xt

k k

t t e dtk

β γ βγ

β γ β

−− −Γ= −

Γ Γ − ∫ . (2.8.5)

Proof: First note that for any positive integer j , we get

( )( )

( ) ( )( ) ( )

,

,

mj k k k

k kmj k

mjkmjk

β γ ββ γγ

Γ Γ +=Γ Γ +

( )

( ) ( )( ),k

kk k

B mjkγ

β γ ββ γ β

Γ= + −Γ Γ −

( )

( ) ( ) ( )1

11

0

1 kkk mj

k k

t t dtk

γ ββγ

β γ β

− −+ −Γ= −

Γ Γ − ∫ (2.8.6)

Now, using (2.8.5), we get

( ) ( ) ( )( )( ) ( ) ( )( )

1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m

m m k m kkm m m

k k kF x

k k k

γβ β

γγ γ

+ −+

+ −+

=( ) ( ) ( )( ) ( ) ( )

, , ,

0, , ,

...

!...

k kjm m mj k j k j k

k kj m m mj k j k j k

xj

β β β

γ γ γ

+ +∞

+ +=∑

( )( )

,

0 , !

jmj k

j mj k

xj

β

γ

∞

=

=∑

( )( ) ( )

11 1

0

(1 )m

k kk xt

k k

t t e dtk

β γ βγ

β γ β

−− −Γ= −

Γ Γ − ∫ .

27

CHAPTER 3

k −FRACTIONAL INTEGRALS

With the introduction of k −gamma function, it is natural to redefine the usual fractional integrals to align with it and to attempt to prove a few of its properties analogous to known fractional integrals. 3.1 RIEMANN- LIOUVILLE k −FRACTIONAL INTEGRAL

Most of the functions involving or based on gamma function can be

refined by using k −gamma function. For example, k −zeta and k −Mittag-Leffler functions could be defined respectively by the formulae

( )( )0

1, , , 0, 1k s

j

x s k x sx jk

ζ∞

=

= > >+

∑

and

( ) ( ),

0

1, , 0

j

kkj

xE x

k jα β α β

α β

∞

=

= >Γ +∑ .

The k −gamma also leads to another interesting direction, a variant of

Riemann-Liouville fractional integral defined by

( )( ) ( ) ( ) ( )1

0

1, 0 .k

x

kk

I f x x t f t dt t xk

αα

α−= − < < < ∞

Γ ∫

It will henceforth be called k − fractional integral. Note that when 1k→ , it reduces to the classical Riemann-Liouville fractional integral

( )( ) ( ) ( ) ( )1

0

1, 0 .

x

I f x x t f t dt t xαα

α−= − < < < ∞

Γ ∫

Thinking simple, let 0C be the class of all functions which are continuous and

integrable on the interval (0,∞ ). Then, kI fα exists in 0C if 0f C∈ . Define

formally ( )kI fα , 0,α < to be the solution, if exists, of the equation kf I gα−= .

Clearly, k kI f I gα α= implies f g= (See Kober (1940)).

28

3.2 PROPERTIES OF k −FRACTIONAL INTEGRALS

In this section, we prove a number of useful properties of k − fractional integrals. Theorem 3.2.1: Let 00, 0, f Cα β> > ∈ , then

( )( )( ) ( )( ) ( )( )k k k kI I f x I f x I f xα β α β β α+ += = . (3.2.1)

Proof: Note that by changing the order of integration (4.1.5), we have

( )( )( ) ( ) ( ) ( )( )( )1

0

1k

x

k k kk

I I f x x t I f t dtk

αα β β

α−= −

Γ ∫

( ) ( ) ( ) ( ) ( )1 1

0 0

1 1k k

x t

k kx t t u f u du dt

k k

βα

α β− −

= − − Γ Γ

∫ ∫

( ) ( )

( ) ( ) ( )1 12

0 0

1k k

x t

k k

x t t u f u du dtk

βα

α β− −

= − − Γ Γ

∫ ∫

( ) ( )

( ) ( ) ( )1 12

0

1k k

x x

k k u

f u x t t u dt duk

βα

α β− −

= − − Γ Γ

∫ ∫

( ) ( )

( )20

1 x

k k

f uk α β

=Γ Γ ∫

( ) ( ) ( )1 1k k

x

u

x u t u t u dt duβα − −

× − − − − ∫

( ) ( )

( ) ( )12

0

1k

x

k k

x u f uk

α

α β−= −

Γ Γ ∫

( )( ) ( )

111 .

k

k

x

u

t ut u dt du

x u

αβ

−−

− × − − − ∫ (3.2.2)

Consider the integral( )( ) ( )

11 1

k

k

x

u

t ut u dt

x u

αβ

−− −

− − −

∫ .

Put ( )( )t u

sx u−

=−

in the above integral to get the following equation

29

( )( ) ( ) ( ) ( )

1 11 1 1

0

1 1k

k k k k

x

u

t ut u dt x u s s dt

x u

αβ β βα

−− − − −

− − = − − −

∫ ∫ .

Equation (3.2.2) then becomes

( )( )( )k kI I f xα β

( ) ( )( ) ( ) ( )

11 1 1

20 0

11k k k k

x

k k

x u f u s s dt duk

β βα α

α β+ − − −

= − − Γ Γ

∫ ∫

( ) ( )( ) ( ) ( ) ( )

( )1

20

1k k

xk k

kk k

kx u f u du

k

βα α βα βα β

+ − Γ Γ= −

Γ +Γ Γ ∫

( ) ( ) ( )1

0

1k k

x

kx u f u du

k

βα

α β+ −= −

Γ + ∫

( )( ) ( )( )k kI f x I f xα β β α+ += = .

Theorem 3.2.2: Suppose that , , and k k kk k kI I I I I Iα β α β β α exist, then k kk kI I I Iα β β α= for all real numbers and α β . Proof: Case (i): If 0, 0α β≥ ≥ , then it holds by Theorem 3.2.1. Case (ii): If 0, 0α β< < , then

( ) ( ) ( ) ( )k kk kI I f x g x I f x I g xα β β α−= ⇒ =

( ) ( )kkf x I I g xβ α− −⇒ =

( ) ( )k kf x I I g xα β− −⇒ = by case (i)

( ) ( )k kI f x I g xα β−⇒ =

( ) ( )kkI I f x g xβ α⇒ = .

Hence, ( ) ( )k kk kI I f x I I f xα β β α= . Case (iii): If 0, 0α β< > , then

( ) ( ) ( ) ( )k kI f x g x f x I g xα α−= ⇒ =

( ) ( )kk kI f x I I g xβ β α−⇒ = by case (i)

( ) ( )kk kI f x I I g xβ α β−⇒ =

( ) ( ) ( ) ( ), where k k kk kI I f x I I f x g x I f xα β β α α⇒ = = .

30

Case (iv): If 0, 0α β> < , then interchange the role of α andβ in (iii) to obtain ( ) ( ) k kk kI I f x I I f xα β β α= .

Theorem 3.2.3: Let ( )( ) ( ) ( ) ( )1

0

1.k

x

kk

I f x x t f t dtk

αα

α−= −

Γ ∫

If 0α > , then 1k

kI tβα − =

( )( )

1k kk

kx

βαβα β

+ −ΓΓ +

.

Proof: We have( ) ( ) 11 1

0

1 kk k

x

kk

I t x t t dtk

β βαα

α−− − = −

Γ ∫ . (3.2.3)

Substitute t ux= in (3.2.3), we get

1kkI t

βα − ( ) ( )

111 1

0

1k k

k k

k

xu u du

k

βαβα

α

+ −− −= −

Γ ∫

( )

( ) ( )( )

1k kk k

k k

xβα

α βα α β

+ − Γ Γ=Γ Γ +

( )

( )1k kk

kx

βαβα β

+ −Γ=Γ +

.

Theorem 3.2.4: Suppose that ( ) ( ), k kI f x I f xα β exists, then

( ) ( )k k kI I f x I f xα β α β+= for all real numbers and α β . Proof: Case (i): If 0, 0α β≥ ≥ , then it holds by Theorem 3.2.1. Case (ii): If ( )0, 0 and k kI I f xα βα β< < exists, then

( ) ( ) ( ) ( )k kk kI I f x g x I f x I g xα β β α−= ⇒ =

( ) ( )kkf x I I g xβ α− −⇒ =

( ) ( ) ( )kf x I g xβ α− +⇒ = by case (i)

( ) ( ) ( )kf x I g xα β− +⇒ =

( ) ( )kI f x g xα β+⇒ = .

Hence, ( ) ( )k k kI I f x I f xα β α β+= .

31

Case (iii): If ( )0, 0, 0 and kI f xα βα β α β +> < + > exists, then

( ) ( ) ( ) ( )k k k kI f x g x I I f x I g xα β β α β β+ − + −= ⇒ =

( ) ( )k kI f x I g xα β β β+ − −⇒ =

( ) ( )k kI f x I g xα β−⇒ =

( ) ( )kkI I f x g xβ α⇒ = .

Hence, ( ) ( ) ( )kk kI I f x g x I f xβ α α β+= = . Case (iv): If ( )0, 0, 0 and kI f xα βα β α β +> < + < exists, then

( ) ( ) ( ) ( ) ( )k kI f x g x f x I g xα βα β − ++ = ⇒ =

( ) ( )k kI f x I g xα β−⇒ =

( ) ( )kkI I f x g xβ α⇒ = .

Hence, ( ) ( ) ( )kk kI I f x g x I f xβ α α β+= = . Case (v and vi): For ( )0, 0, 0, 0 and kI f xα βα β α β α β +< > + > + < exists, then interchange the role of α andβ in (iii and iv).

Theorem 3.2.5: Let ( )( ) ( ) ( ) ( )1

0

1.k

x

kk

I f x x t f t dtk

αα

α−= −

Γ ∫

If 0α > , then

( ) 1kkI x u

βα − −

( )( ) ( ) 1k kk

kx u

βαβα β

+ −Γ= −Γ +

.

Proof: We have

( ) 1kkI x u

βα − −

( ) ( ) ( )1 11

k k

x

k u

x t t u dtk

βα

α− −= − −

Γ ∫ (3.2.4)

( ) ( ) ( )( ) ( )1 11 k k

x

k u

x u t u t u dtk

βα

α− −= − − − −

Γ ∫

( )

( )( )( ) ( )

1111

kkk

x

k u

x u t ut u dt

k x u

ααβ

α

−−− − −

= − − Γ −

∫ .

Put t ux= in Equation (3.2.4) to get

32

( ) 1kkI x u

βα − −

( )

( ) ( )1 1

1 1

0

11

k kk k

k

x us s dt

k

βαβα

α

+ −− −−

= −Γ ∫

( )( )

( ) ( )( )

1k kk k

k k

x uβα

α βα α β

+ −− Γ Γ=

Γ Γ +

( )( ) ( ) 1k kk

kx u

βαβα β

+ −Γ= −Γ +

.

Theorem 3.2.6: Let ( )( )( )

( ) ( ) ( )1,

0

k

k k

x

kk

xI f x x t t f t dt

k

α ββαα β

α

+−−= −

Γ ∫ .

If 0 0, 0, 0, f Cα β λ> > > ∈ , then

( )( )( ) ( )( ), , ,k k kI I f x I f xλ α β α β α λ β+ += .

Proof:

( )( )( ), ,k kI I f xλ α β α β+

( )

( ) ( ) ( )( )( )1 ,

0

kk k

x

kk

xx t t I f t dt

k

α β λα βλ α β

λ

+ ++−

−= −Γ ∫

( )

( ) ( )( ) ( ) ( )1 1

20 0

kk k k

x t

k k

xx t t u u f u du dt

k

α β λβλ α

α λ

+ +−− −

= − − Γ Γ

∫ ∫

( )

( ) ( )( ) ( ) ( )1 1

20

kk kk

x x

k k u

xu f u x t t u dt du

k

α β λβ λ α

α λ

+ +−− −

= − − Γ Γ

∫ ∫

( )

( ) ( )2

k

k k

xk

α β λ

α λ

+ +−

=Γ Γ

( ) ( ) ( )( ) ( )1 1

0

k kk

x x

u

u f u x u t u t u dt duβ λ α− −

× − − − −

∫ ∫

( )

( ) ( )( )( ) 1

20

kkk

x

k k

xu f u x u

k

α β λβ λ

α λ

+ +−−= −

Γ Γ ∫

( )1

11k

k

x

u

t ut u dt du

x u

λα

−− − × − − −

∫ . (3.2.5)

33

Consider the integral ( )1

11k

k

x

u

t ut u dt

x u

λα

−−− − − − ∫ .

Using ( )( )t u

sx u−

=−

in the above integral, we have the following equation

( ) ( ) ( )11

1 1 1

0

1 1k

k k kk

x

u

t ut u dt s s x u dt

x u

λα λ αα

−− − −− − − = − − − ∫ ∫ .

Equation (3.2.5) becomes

( )( )( ), ,k kI I f xλ α β α β+

( )

( ) ( )( ) ( ) ( )

11 1 1

20 0

1k

k k kk k

x

k k

xx u u f u s s ds du

k

α β λβα λ λ α

α λ

+ +−+ − − −

= − − Γ Γ

∫ ∫

( )

( ) ( )( ) ( ) ( ) ( )

( )1

20

kk k k

xk k

kk k

kxx u u f u du

k

α β λβα λ α λ

α λα λ

+ +−+ − Γ Γ

= − Γ +Γ Γ

∫

( )

( ) ( ) ( )1

0

kk k k

x

k

xx u u f u du

k

α β λβα λ

α λ

+ +−+ −= −

Γ + ∫

( )( ),kI f xα λ β+= .

34

CHAPTER 4

pL −BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING 1 , , 1m m kF m+ ≥ AS KERNEL

In this chapter, we consider integral operators involving homogeneous

functions as kernel and discuss the pL − boundedness of these integral operators by using properties of hypergeometric functions 1 , , 1m m kF m+ ≥ as kernel.

4.1 BASIC RESULTS

We, first, define some terms and state some basic results on boundedness

of integral operators in pL . These are extensions of famous Hardy’s inequality. 4.1.1 pL - SPACES

Let ,R R+ denote the set of real numbers and the set of positive real

numbers respectively.

( )pL R is the class of measurable functions whose p th powers, 1p ≥ , are

integrable on R so that ( )pf L R∈ if and only if p

Rf dx∫ is finite, where p is

any finite positive real number. The positive number pf is defined by

( )1 ppp R

f f dx= ∫ . (4.1.1)

Functions in pL have the following properties: i. (Hölder’s Inequality). If ( ) pf L R∈ , where 1 p≤ < ∞ , and ( )pg L R′∈ ,

( ) ( )1 1 1p p′+ = , then ( )1fg L R∈ , and pp pR

fg dx f g ′≤∫ .

For 1 p≤ < ∞ , the expressions are equal if and only if there is a constant

C such that ( ) ( )p pg x C f x= for all x in R .

ii. If ( )pf L R∈ , ( )qg L R∈ , where 0, 0p q> > , ( ) ( )1 1 1r p q= + , then

( ) rfg L R∈ and r p qfg f g≤ . (4.1.2)

35

iii. (Minkowski’s Inequality). IfY ( ) pf L R∈ , ( )pg L R∈ , where 1 p≤ < ∞ ,

then ( )( ) pf g L R+ ∈ and p p pf g f g+ ≤ + . (4.1.3)

iv. (Minkowski’s Integral Inequality). Let X and Y be subsets of R , and let ( ),f x y be a measurable function such that ( )(., ) pf y L X∈ for each

point y Y∈ . Then, for 1 p≤ < ∞ ,

( ) ( )( )1 1

, ,p pp p

X Y Y Xf x y dy dx f x y dx dy

≤

∫ ∫ ∫ ∫ (4.1.4)

whenever the expressions are finite. v. (Fubini’s Theorem). Let ( ),f x y be non-negative, measurable function

X Y× , ( ) ( , )xf y f x y= is measurable on Y and ( ) ( , )yf x f x y= is measurable on X ,

then ( )( )( ) ( )( ), ,X Y Y X

f x y dx dy f x y dx dy≤∫ ∫ ∫ ∫ . (4.1.5)

Lemma 4.1.2: Let 1p > . Then ( ) pf L R+∈ , there exists a function 2 1, ( )( ) ( )p

xV V f x x f−= defined on R+ such that ( )2 V f f=

and ( ) : p pV f L L→ .

A function ( ),x tψ defined on 2R is said to be homogeneous of degree

µ if ( ) ( ) , ,hx ht h x tµψ ψ= . Theorem 4.1.3 (Hardy et al. (1952)): Let p be real numbers such that

1p > and let ( ),x tψ be a function which is measurable in the variable t and for

each fixed x in R+ such that, for 0h ≠ ,

( ) ( )1 , ,hx ht h x tψ ψ−= .

If ( ) ( ) ( ) ( )1, , , ,. and .,p pf L R g L R x f t gν ψ ψ′+ +> ∈ ∈ are integrable.

(a) Let ( )( ) ( ) ( )0

,f x f t x t dtψ∞

Ψ = ∫ .

Then

(i) ( ) ( ): p pL R L R+ +Ψ →

36

(ii) ( )( ) ( )1 1

0 0

,

p pp p

x f x dx C x f x dtν ν∞ ∞

Ψ ≤ ∫ ∫

where. ( ) ( ) ( ) ( ) ( ) ( )1 1 1

0 0

1, ,1p p p pC u u du u u duν νψ ψ∞ ∞

− − + −= =∫ ∫

(b) Further, if the operator *Ψ is defined by

( )( ) ( ) ( )*

0

,g t g x x t dxψ∞

Ψ = ∫ .

Then

(i) ( ) ( )* : p pL R L R′ ′+ +Ψ →

(ii) ( )( ) ( )1 1

*

0 0

p pp p

t g t dt C t g t dtν ν

′ ′∞ ∞′ ′ Ψ ≤

∫ ∫

Proof: (a) (i) When 0ν = (ii) implies (i).

(ii) Consider ( )( ) ( ) ( )0

,f x f t x t dtψ∞

Ψ = ∫ .

Now

( )( ) ( ) ( )0

,f x f t x t dtψ∞

Ψ = ∫

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )1 1 11 1 1 1 1 1

0

, ,p pp p p pf t t t x t x t dtν ν ψ ψ

∞−− + − + −≤ ∫

( )( ) ( )( )

( ) ( )( ) ( ) ( )1 1 1

1 1 1 1

0 0

, ,

p ppp pt x t dt t f t x t dtν νψ ψ

−∞ ∞− + + −

= ∫ ∫ . (4.1.6)

Consider the following integral

( )( ) ( )1

0

,pt x t dtν ψ∞

− +∫ . (4.1.7)

Substitute t ux= in (4.1.7) to have

( )( ) ( ) ( )( ) ( )( ) ( )1 1 1

0 0

, 1,p p pt x t dt x u u duν ν νψ ψ∞ ∞

− + − + − +=∫ ∫

( )( )1 px Cν− += ,

where ( )( ) ( )1

0

1,pC u u duν ψ∞

− += ∫ .

37

Equation (4.1.6) becomes

( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )1

1 1 1 1 1 11 1

0

,

ppp p ppf x C x t f t x t dtν ν ψ

∞− + − + −−

Ψ ≤ ∫

( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )1 1 1 1 1 11

0

,p pp ppf x C x t f t x t dtν ν ψ

∞− + − + −−Ψ ≤ ∫

( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )1 1 1 1 1 11

0

,p pp ppx f x C x t f t x t dtν ν νν ψ

∞− + − + + −−Ψ ≤ ∫

( ) ( ) ( ) ( ) ( ) ( )1 1 1 11

0

,pp p p ppC x t f t x t dtν ν ν ψ

∞+ − + − −−≤ ∫ ,

so that

( )( )0

px f x dxν

∞

Ψ∫

( ) ( ) ( ) ( ) ( ) ( )1 1 1 11

0 0

, .pp p p ppC t f t x x t dx dtν ν ν ψ

∞ ∞+ − − + −−

≤

∫ ∫ (4.1.8)

Consider the following integral

( ) ( ) ( )1 1

0

,p px x t dxν ψ∞

+ −∫ . (4.1.9)

Putting x ut= in (4.1.9), we have

( ) ( ) ( )1 1

0

,p px x t dxν ψ∞

+ −∫

( ) ( ) ( ) ( ) ( )1 1 1 1

0

,1p p p pt u u duν ν ψ∞

+ − + −= ∫

( ) ( )1 1p pt Cν+ −= ,

where ( ) ( ) ( )1 1

0

,1p pC u u duν ψ∞

+ −= ∫ .

Then, (4.1.8) becomes

( )( )0

px f x dxν

∞

Ψ∫

( ) ( ) ( ) ( ) ( )1 1 1 11

0

pp p p ppC t f t t C dtν ν ν∞

+ − − + −− ≤ ∫

( )0

ppC t f t dtν∞

= ∫ .

38

This implies that

( )( ) ( )1 1

0 0

p pp p

x f x dx C t f t dtν ν∞ ∞

Ψ ≤ ∫ ∫

or

( )( ) ( )1 1

0 0

p pp p

x f x dx C x f x dtν ν∞ ∞

Ψ ≤ ∫ ∫ .

Similarly, we can prove the second part of this theorem. 4.2 ERDÉLYI - KOBER OPERATORS

Following 3.1.1, we now define Erdélyi-Kober k − fractional integral

operator ,kI fλη as given below and prove pL − boundedness of one of its special

cases that is used later. For 0λ > , 0k > , write

( )( ) ( )( )

( ) ( )( ) 1,

0

( ) , 0xk k

k kk

k

xI f x x t t f t dt t x

k

η λλλ η

η α

− +−= − < < < ∞

Γ ∫ .

Note that ,0kI fλ = kI fλ and { },, kkI I fβ αηη λ+ = ,kI fα β

η+ .

Theorem 4.2.1: Let ( ) ( ) 1, 0, 0 and , 2 1p k k pγ β λ γ β η> − > > = − = − .

If ( ) ( )( )( ) ( )( )

( ) ( ) ( )( ) ( )2 1

1 2 1, 2 1

0

( )xp k

k pk p

k

xI f x x t t f t dt

k

γ βγ βγ β

γ β

− + − −− − −−

− = −Γ − ∫ ,

where 0 t x< < < ∞ , then ( ) ( ), 2 1 : p pk pI f L Lγ β−

− → and there exists a constant

( ), , ,C C p kβ γ′ ′= such that ( ) ( ), 2 1k p ppI f C fγ β−

−′≤ .

Proof: Consider

( ) ( )( )( ) ( )( )

( ) ( ) ( )( ) ( )2 1

1 2 1, 2 1

0

( )xp k

k pk p

k

xI f x x t t f t dt

k

γ βγ βγ β

γ β

− + − −− − −−

− = −Γ − ∫ .

( ) ( ) ( )( ) ( ) ( ) ( )( )2 1 12 1Suppose that , .p k kpx t x t x tγ β γ βψ − + − − − −−= −

Then ( ) ( ) ( )( ) ( ) ( ) ( )( )1 2 1 12 1, p k kphx ht h x t x tγ β γ βψ − − + − − − −−= −

( )1 ,h x tψ−= .

( )Since , is a homogeneous function of degree 1x tψ − , by Theorem 4.1.3(a(i)),

39

there exists a constant ( ), , ,C C p kβ γ′ ′= such that

( ) ( ), 2 1k p ppI f C fγ β−

−′≤ ,

where

( )1

1

0

1,pC t t dtψ−′ = ∫

= ( ) ( ) ( )( )1

12 11

0

1 kppt t t dtγ β− −−− −∫

= ( ) ( ) ( )( )1

11 1

0

1 kpt t dtγ β− −− − < ∞∫ , if 1, 0, 0p kγ β> − > > .

Hence, ( ) ( ), 2 1 : p pk pI f L Lγ β−

− → .

4.3 BOUNDEDNESS OF INTEGRALS ( ),

,m kM fα β WITH VARIANTS

OF HOMOGENEOUS KERNEL

In this section, we consider integral operators of the type

( )( ) ( ) ( ) ( ) ( ) ( ) ( )2 1,,

0

1 , 0mk p k m m

m kM f x x t x t f t dt xαβ βα β

∞ −− −= + >∫ .

We prove their boundedness in pL . We consider other relations that are useful in our study of integral operators involving hypergeometric functions. We begin with the following result. Theorem 4.3.1: Let

( )( ) ( ) ( ) ( ) ( ) ( ) ( )2 1,,

0

1 , 0mk p k m m

m kM f x x t x t f t dt xαβ βα β

∞ −− −= + >∫ .

( ) ( ) ( )If 1, 1 , 0 1, 0p k p k kβ α β α> − < < < < > ,

( ),,then : p p

m kM f L Lα β → and there exists a constant ( )1 1 , , ,C C p kα β= such

that ( ),1, m k pp

M f C fα β ≤ .

Proof: Note that

( )( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( )2 1,,

0

1 , 0mk p k m m

m kM V f x x t x t V f t dt xαβ βα β

∞ −− −= + >∫

40

( ) ( ) ( ) ( ) ( ) ( )2 1 2

0

1 1mk p k p m mx t x t f t dt

αβ β∞ −− − −= +∫ .

Substitute 1y t= to get

( )( )( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 1,,

0

mk p p k m mm kM V f x x y x y f y dy

αβ α βα β∞ −− + − −= +∫ .

Suppose that ( ) ( ) ( ) ( ) ( ) ( )2 2 1 , .mk p p k m mx y x y x y

αβ α βψ−− + − −= +

Now, ( ) ( ) ( ) ( ) ( ) ( )1 2 2 1,mk p p k m mhx hy h x y x y

αβ α βψ−− − + − −= +

= ( )1 ,h x yψ− .

Since ( ),x yψ is a homogeneous function of degree 1− , by Theorem

4.1.3(a(i)), there exists a constant ( )1 1 , , ,C C p kα β= such that

( )( ) ( ),1 1,m k ppp

M V f C V f C fα β ≤ = ,

where

( )11

0

1,pC y y dyψ∞

−= ∫

( ) ( ) ( )1 1

0

1mp k my y dy

αα β∞ −+ − −= +∫

( ) ( ) ( )1

1 1

0

1mp k my y dy

αα β −+ − −= +∫

( ) ( ) ( )1 1

1

1mp k my y dy

αα β∞ −+ − −+ + < ∞∫ ,

( ) ( ) ( )if 1, 1 , 0 1, 0p k p k kβ α β α> − < < < < > .

Thus, ( )( ),, : p p

m kM V f L Lα β → .

Also, since ( ) : p pV f L L→ , ( )( ),, : p p

m kM V f L Lα β → and

( ) ( )( ), , 2, ,m k m kM f M V fα β α β= .

Then ( ) ( )( ) ( ), , 21 1, , m k m k ppp p

M f M V f C V f C fα β α β= ≤ = .

Hence, ( ),, : p p

m kM f L Lα β → and there exists a constant ( )1 1 , , ,C C p kα β= such that

( ),1, m k pp

M f C fα β ≤ .

41

Lemma 4.3.2: Let

( )( ) ( ) ( ) ( ) ( ) ( ) ( )1 2* ,,

0

1 , 0mk k p m m

m kM g x x t x t g t dt xαβ βα β

∞ −− −= + >∫ .

If ( ) ( ) ( ) ( ) ( )1, 1 ,0 1, 1 1 1, 0p k p k p p kβ α β α′ ′ ′> − < < < < + = > , then

( )* ,, : p p

m kM g L Lα β ′ ′→ and there exists a constant ( )1 1 , , ,C C p kα β′ ′ ′= such that

( )* ,1,m k pp

M g C gα β′′

′≤ .

Proof: Proof of this lemma is identical as above. Now, we prove the following product formula. Theorem 4.3.3: Let

( )( ) ( ) ( ) ( ) ( ) ( ) ( )2 1,,

0

1 , 0mk p k m m

m kM f x x t x t f t dt xαβ βα β

∞ −− −= + >∫

and

( )( ) ( ) ( ) ( ) ( ) ( ) ( )1 2* ,,

0

1 , 0mk k p m m

m kM g x x t x t g t dt xαβ βα β

∞ −− −= + >∫ .

If ( )pf L R+∈ and ( ) pg L R′ +∈ , then

( ) ( )( ) ( ) ( )( ), * ,, ,

0 0m k m kg t M f t dt f t M g t dtα β α β

∞ ∞

=∫ ∫ .

Proof: An application of Fubini’s theorem yields the result. Theorem 4.3.4: For 0x > , let

( )( ) ( )1 2 1( ) ( ) for 0 ; k px t x t t t xγ βφ − − −= − < < < ∞

0= for t x≥ . Then

( )( )* ,, xm kM tα β φ =

( ) ( )( )

( ) ( )1 1k kk k

k

kx tγ ββ γ β

γ− −Γ Γ −

Γ

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )

1

1 , 1

,1 , , , , ,..., ,;

, , , ,..., ,

m kkm m m m m m

m m k m kkm m m

k k kF x t

k k k

γβ βα

γγ γ

+ −+

+ + −+

× −

.

Proof: Consider the following integral representation given in Lemma 2.7.4

42

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

1

1 , 1

,1 , , , , ,..., ,;

, , , ,..., ,

m kkm m m m

m m k m kkm m m

k k kF x

k k k

γβ βα

γγ γ

+ −+

+ + −+

( )

( ) ( )( ) ( )( ) ( )

111

0

(1 ) 1mk kk m

k k

u u xu duk

αγ ββγ

β γ β

−− −−Γ= − −

Γ Γ − ∫ .

Replacing by m mx x t− , we have

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

1

1 , 1

,1 , , , , ,..., ,;

, , , ,..., ,

m kkm m m m m m

m m k m kkm m m

k k kF x t

k k k

γβ βα

γγ γ

+ −+

+ + −+

−

( )

( ) ( )( ) ( )( ) ( )

111

0

(1 ) 1mkk m m mk

k k

u u x t u duk

αγ ββγβ γ β

−− −−Γ= − +

Γ Γ − ∫ .

Substituting u y x= to get

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )

1

1 , 1

,1 , , , , ,..., ,;

, , , ,..., ,

m kkm m m m m m

m m k m kkm m m

k k kF x t

k k k

γβ βα

γγ γ

+ −+

+ + −+

−

=( )

( ) ( )( ) ( ) ( ) ( )( ) ( )11 1

0

1m

xkk k m mk

k kx y x y t y dy

k

αγ βγ βγβ γ β

−− −− −Γ− +

Γ Γ − ∫ .

This implies that

( ) ( ) ( )( ) ( )11

0

1m

xkk m my x y t y dy

αγ ββ −− −− − +∫

=( ) ( )

( )( ) 1kk k

k

kx γβ γ β

γ−Γ Γ −

Γ

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )

1

1 , 1

,1 , , , , ,..., ,;

, , , ,..., ,

m kkm m m m m m

m m k m kkm m m

k k kF x t

k k k

γβ βα

γγ γ

+ −+

+ + −+

× −

. (4.3.1)

Now, consider * ,,m kM α β as in Theorem 4.3.2 to obtain the following expression.

( )( )* ,,m kM g tα β ( ) ( ) ( ) ( ) ( ) ( )1 2

0

1 , 0mk k p m mt y t y g y dy t

αβ β −∞

− −= + >∫ .

43

Then, by taking ( ) ( )xg t tφ= , we get

( )( )* ,, xm kM tα β φ

( ) ( ) ( ) ( ) ( ) ( )( ) ( )11 2 2 1

0

1m

xkk k p pm mt y t y x y y dy

α γ ββ β − − −− − − = + − ∫

( ) ( ) ( ) ( )( ) ( )11 1

0

1m

xkk k m mt y x y t y dy

αγ ββ β −− −− −= − +∫ .

By using Equation (4.3.1), we finally obtain

( )( )* ,, xm kM tα β φ =

( ) ( )( )

( ) ( )1 1k kk k

k

kx tγ ββ γ β

γ− −Γ Γ −

Γ

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )

1

1 , 1

,1 , , , , ,..., ,;

, , , ,..., ,

m kkm m m m m m

m m k m kkm m m

k k kF x t

k k k

γβ βα

γγ γ

+ −+

+ + −+

× −

.

4.4 BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING

HYPERGEOMETRIC FUNCTIONS 1 , , 1m m k mF+ ≥

We now formulate integral operators involving k −hypergeometric functions of the type 1 , , 1m m kF m+ ≥ . We use the results proved in Section 4.2

and 4.3 to prove the pL − boundedness of these integral operators involving k −hypergeometric functions 1 , , 1m m kF m+ ≥ .

Theorem 4.4.1: Let

( )( ), ,,m kS f xα β γ ( ) ( )( ) 12 1

0

kpx xt β∞

−− += ∫

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

( )1

1 , 1

,1 , , , , ,..., ,; .

, , , ,..., ,

m kkm m m m m m

m m k m kkm m m

k k kF x t f t dt

k k k

γβ βα

γγ γ

+ −+

+ + −+

× −

If ( ) ( ) ( ) 1, 1 , 0 1, 0, 0p k p k kβ α β α γ β> − < < < < − > > , then

a) ( ) ( ) ( ){ }, , ,, ,, 2 1 km k m kk pS I Mf A fα β γ γ β α β−

−= where ( )( )

kk

kA

γβ

Γ=Γ

,

b) ( ), ,, :m k

P PS f L Lα β γ → and explicitly

44

c) there exists a constant kC such that ( ), ,, km k pp

S f C fα β γ ≤ .

Proof: We employ Theorem 4.2.1 to get

( ) ( ){ }( ),,, 2 1 m kk pI M f xγ β α β−

−

=( ) ( )( )

( ) ( ) ( )( ) ( ) ( )( )2 1

1 2 1 ,,

0

xp k

k pm k

k

xx t t M f t dt

k

γ βγ β α β

γ β

− + − −− − −−

Γ − ∫ .

Use of Theorem 4.3.3 and Theorem 4.3.4 yields

( ) ( ){ }( ),,, 2 1 m kk pI M f xγ β α β−

−

=( ) ( )( )

( ) ( ) ( ) ( )( ) ( )2 1

1 2 1* ,,

0

p k

k pm k

k

xf t M x t t dt

k

γ βγ βα β

γ β

∞− + − −− − − −

Γ − ∫

=( ) ( )( )

( ) ( ) ( )( )2 1

* ,,

0

p k

xm kk

xf t M t dt

k

γ βα β φ

γ β

∞− + − −

Γ − ∫

=( ) ( )( )

( )( ) ( )

( )( ) ( )

2 11 1

0

p kk kk k

k k

kxx t

k

γ βγ ββ γ β

γ β γ

∞− + − −− −Γ Γ −

Γ − Γ∫

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

( )1

1 , 1

,1 , , , , ,..., ,;

, , , ,..., ,

m kkm m m m m m

m m k m kkm m m

k k kF x t f t dt

k k k

γβ βα

γγ γ

+ −+

+ + −+

× −

( )( )

( ) ( )( ) 12 1

0

kpk

kx xt ββ

γ

∞−− +Γ

=Γ ∫

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

( )1

1 , 1

,1 , , , , ,..., ,;

, , , ,..., ,

m kkm m m m m m

m m k m kkm m m

k k kF x t f t dt

k k k

γβ βα

γγ γ

+ −+

+ + −+

× −

( )( ) ( )( ), ,

,k

m kk

S f xα β γβγ

Γ=Γ

.

Hence, ( )( ), ,,m kS f xα β γ =

( )( ) ( ) ( ){ }( ),

,, 2 1k

m kk pk

I M f xγ β α βγβ

−−

ΓΓ

( ) ( ){ }( ),,, 2 1k m kk pA I M f xγ β α β−

−=

where( )( )

kk

kA

γβ

Γ=Γ

.

45

Since ( ) ( ), 2 1 : p pk pI f L Lγ β−

− → by Theorem 4.2.1 and it, therefore, follows from

Theorem 4.3.1 that if ( ) ( ) ( )1, 1 ,0 1, 0, 0p k P k kβ α β α γ β> − < < < < − > > ,

then

( ) ( ) ( )( ) ( ), , , ,, , ,, 2 1

1 .

k km k m k m kk pp pp

k kp p

S f A I M f A C M f

A C C f C f

α β γ γ β α β α β−−

′= ≤

′≤ =

Hence, ( ), ,, : P P

m kS f L Lα β γ → and there exists a constant 1k kC A C C′= such that

( ), ,, km k pp

S f C fα β γ ≤ .

46

CHAPTER 5

pL −BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING , , 1m m kF m ≥ AS KERNEL

In this chapter, we consider integral operators involving exponential

functions as kernel and discuss the pL -boundedness of these integral operators by using properties of k −hypergeometric functions , , 1m m kF m ≥ as kernel. 5.1 BOUNDEDNESS OF INTEGRALS ( ),m kH fβ INVOLVING

EXPONENTIAL KERNEL

In this section, we consider integral operators of the type

( )( ) ( ) ( ) ( ) ( ) ( )2 1,

0

, 0m mk p k x t

m kH f x x t e f t dt xβ ββ∞

− − −= >∫ .

We prove their boundedness in pL and other relations that are useful in our study of integral operators involving k −hypergeometric functions. We begin with the following result.

Theorem 5.1.1: Let

( )( ) ( ) ( ) ( ) ( ) ( )2 1,

0

, 0m mk p k x t

m kH f x x t e f t dt xβ ββ∞

− − −= >∫ .

If ( ) ( ) 1, 1 , 0p k p kβ> > > , ( ),then : p pm kH f L Lβ → and there exists a

constant ( )2 2 , ,C C p kβ= such that

( ) 2,m k ppH f C fβ ≤ .

Proof: Note that

( )( )( ) ( ) ( ) ( ) ( )( )( ) ( )2 1,

0

, 0m mk p k x t

m kH V f x x t e V f t dt xβ ββ∞

− − −= >∫

( ) ( ) ( ) ( ) ( )2 1 2

0

1m mk p k p x tx t e f t dtβ β

∞− − − −= ∫ .

Substitute 1y t= . Then

47

( )( )( ) ( ) ( ) ( ) ( ) ( )2 2 1,

0

m mk p p k x ym kH V f x x y e f y dyβ ββ

∞− − − −= ∫

Suppose that ( ) ( ) ( ) ( ) ( )2 2 1 ,m mk p p k x yx y x y eβ βψ − − − −= .

Now, ( ) ( ) ( ) ( ) ( )1 2 2 1,m mk p p k x yhx hy h x y eβ βψ − − − − −=

= ( )1 ,h x yψ− .

Since ( ),x yψ is a homogeneous function of degree 1− , by Theorem

4.1.3(a(i)), there exists a constant ( )2 2 , ,C C p kβ= such that

( )( ) ( )2 2,m k pppH V f C V f C fβ ≤ =

where

( )12

0

1,pC y y dyψ∞

−= ∫

( ) ( )1 1 1

0

mp k yy e dyβ∞

− − −= < ∞∫ ,

if ( ) ( ) 1, 1 , 0p k p kβ> > > . Thus, ( )( ), : p pm kH V f L Lβ → .

Also, since ( ) : p pV f L L→ and ( ) ( )( )2, , m k m kH f H V fβ β= , therefore

( ) ( )( ) ( )22 2, ,m k m k ppp p

H f H V f C V f C fβ β= ≤ = .

Hence, ( ), : p pm kH f L Lβ → and there exists a constant ( )2 2 , ,C C p kβ= such

that

( ) 2, m k ppH f C fβ ≤ .

Lemma 5.1.2: Let

( )( ) ( ) ( ) ( ) ( ) ( )1 2*,

0

, 0m mk k p x t

m kH g x x t e g t dt xβ ββ∞

− − −= >∫ .

If ( ) ( ) ( ) ( )1, 1 , 0, 1 1 1p k p k p pβ′ ′ ′> > > + = , then

( )*, : p p

m kH g L Lβ ′ ′→ and there exists a constant ( )2 2 , ,C C p kβ′ ′ ′= such that

( )*2,m k pp

H g C gβ′′

′≤ .

Proof: Proof of this lemma is identical as above.

48

Now, consider the following product formula. Theorem 5.1.3: Let

( )( ) ( ) ( ) ( ) ( ) ( )2 1,

0

, 0m mk p k x t

m kH f x x t e f t dt xβ ββ∞

− − −= >∫

and

( )( ) ( ) ( ) ( ) ( ) ( )1 2*,

0

, 0m mk k p x t

m kH g x x t e g t dt xβ ββ∞

− − −= >∫ .

If ( )pf L R+∈ and ( ) pg L R′ +∈ , then

( ) ( )( ) ( ) ( )( )*, ,

0 0m k m kg t H f t dt f t H g t dtβ β

∞ ∞

=∫ ∫ .

Proof: The change of order of integration yields the result. Theorem 5.1.4: For 0, 0kγ β> > > , let

( )( ) ( )1 2 1( ) ( ) for 0 ; k px t x t t t xγ βφ − − −= − < < < ∞

0= for t x≥ . Then

( )( )*, xm kH tβ φ =

( ) ( )( )

( ) ( )1 1k kk k

k

kx tγ ββ γ β

γ− −Γ Γ −

Γ

( ) ( ) ( )( )( ) ( ) ( )( )

1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m m m

m m k m kkm m m

k k kF x t

k k k

ββ β

γγ γ

+ −+

+ −+

× −

.

Proof: Consider the following integral representation given in Theorem 2.8.3

( ) ( ) ( )( )( ) ( ) ( )( )

1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m

m m k m kkm m m

k k kF x

k k k

ββ β

γγ γ

+ −+

+ −+

( )

( ) ( )( ) ( )( )

111

0

(1 )mk kk xu

k k

u u e duk

γ ββγ

β γ β− −−Γ

= −Γ Γ − ∫ .

Replacing by m mx x t− , we have

49

( ) ( ) ( )( )( ) ( ) ( )( )

1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m m m

m m k m kkm m m

k k kF x t

k k k

ββ β

γγ γ

+ −+

+ −+

−

( )

( ) ( )( ) ( )( )

111

0

(1 )m m mkk x t u

k k

k u u e duk

γ ββγ

β γ β− −− −Γ

= −Γ Γ − ∫ .

Substitute xu y= to get

( ) ( ) ( )( )( ) ( ) ( )( )

1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m m m

m m k m kkm m m

k k kF x t

k k k

ββ β

γγ γ

+ −+

+ −+

−

=( )

( ) ( )( ) ( ) ( ) ( )( ) 11 1

0

m mx

kk k yk

k k

tx y x y dyk

eγ βγ βγβ γ β

− −− − −Γ−

Γ Γ − ∫

( ) ( ) ( )( ) 11

0

m mx

kk yty x y dyeγ ββ − −− −−∫ =( ) ( )

( )( ) 1kk k

k

kx γβ γ β

γ−Γ Γ −

Γ

( ) ( ) ( )( )( ) ( ) ( )( )

1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m m m

m m k m kkm m m

k k kF x t

k k k

ββ β

γγ γ

+ −+

+ −+

× −

. (5.1.1)

Now, consider *,m kH β as in Theorem 5.1.2. We, then, obtain

( )( )*,m kH g tβ ( ) ( ) ( ) ( ) ( )1 2

0

, 0m mk k p t yt y e g y dy tβ β

∞− − −= >∫ .

Then taking ( ) ( )xg t tφ= , we get

( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )11 2 2 1*,

0

m mx

kk k p pt yxm kH t t y e x y y dyγ ββ ββ φ − −− − −− = −

∫

( ) ( ) ( ) ( )( ) 11 1

0

m mx

kk k t yt y x y e dyγ ββ β − −− − −= −∫ .

By using (5.1.1), we obtain

50

( )( )*, xm kH tβ φ =

( ) ( )( )

( ) ( )1 1k kk k

k

kx tγ ββ γ β

γ− −Γ Γ −

Γ

( ) ( ) ( )( )( ) ( ) ( )( )

1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m m m

m m k m kkm m m

k k kF x t

k k k

β β

γγ γ

+ −+

+ −+

× −

.

5.2 BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING

HYPERGEOMETRIC FUNCTIONS , , 1m m k mF ≥

We now formulate integral operators involving k −hypergeometric functions of the type , , 1m m kF m ≥ . We use the results proved in Section 5.1 to

prove the pL − boundedness of these integral operators involving k −hypergeometric functions , , 1m m kF m ≥ .

Theorem 5.2.1: Let

( )( ),,m kG f xβ γ

( ) ( )( ) 12 1

0

kpx xt β∞

−− += ∫

( ) ( ) ( )( )( ) ( ) ( )( )

1

, 1

, , , ,..., ,; .

, , , ,..., ,

m kkm m m m m

m m k m kkm m m

k k kF x t dt

k k k

ββ β

γγ γ

+ −+

+ −+

× −

If ( ) ( ) 1, 0, 1 , 0p k p kγ β β> − > > > , then

a) ( ) ( ) ( ){ },, ,, 2 1 km k m kk pG I Hf A fβ γ γ β β−

−= where ( )( )

kk

kA

γβ

Γ=Γ

,

b) ( ),, G :m k

p pf L Lβ γ → and explicitly

c) there exists a constant kC′ such that ( ),, km k pp

G f C fβ γ ′≤ .

Proof: We employ Theorem 4.2.1 to get

( ) ( ){ }( ),, 2 1 m kk pI H f xγ β β−−

=( ) ( )( )

( ) ( ) ( )( ) ( ) ( )( )2 1

1 2 1,

0

p k

k pm k

k

xx t t H f t dt

k

γ βγ β β

γ β

∞− + − −− − −−

Γ − ∫ .

51

Now, we use Theorem 5.1.3 and Theorem 5.1.4 to get

( ) ( ){ }( ),, 2 1 m kk pI H f xγ β β−−

=( ) ( )( )

( ) ( ) ( ) ( )( ) ( )2 1

1 2 1*,

0

p k

k pm k

k

xf t H x t t dt

k

γ βγ ββ

γ β

∞− + − −− − − −

Γ − ∫

=( ) ( )( )

( ) ( ) ( )( )2 1

*,

0

p k

xm kk

xf t H t dt

k

γ ββ φ

γ β

∞− + − −

Γ − ∫ .

( ) ( )( )

( )( ) ( )

( )( ) ( )

2 11 1

0

p kk kk k

k k

kxx t

k

γ βγ ββ γ β

γ β γ

∞− + − −− −Γ Γ −

=Γ − Γ∫

( ) ( ) ( )( )( ) ( ) ( )( )

( )1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m m m

m m k m kkm m m

k k kF x t f t dt

k k k

γβ β

γγ γ

+ −+

+ −+

× −

( )( )

( ) ( )( ) 12 1

0

kpk

kx xt ββ

γ

∞−− +Γ

=Γ ∫

( ) ( ) ( )( )( ) ( ) ( )( )

( )1

, 1

, , , ,..., ,;

, , , ,..., ,

m kkm m m m m

m m k m kkm m m

k k kF x t f t dt

k k k

γβ β

γγ γ

+ −+

+ −+

× −

( )( ) ( )( ),

,k

m kk

G f xβ γβγ

Γ=Γ

.

Hence, ( )( ),,m kG f xβ γ =

( )( ) ( ) ( ){ }( ),, 2 1

km kk p

kI H f xγ β ββ

γ−

−

ΓΓ

( ) ( ){ }( ),,, 2 1k m kk pA I H f xγ β α β−

−=

where( )( )

kk

kA

γβ

Γ=Γ

.

Since ( ) ( ), 2 1 : p pk pI f L Lγ β−

− → by Theorem 4.2.1 and it, therefore, follows from

Theorem 5.1.1 that if ( ) ( ) 1, 0, 1 , 0p k p kγ β β> − > > > , then

( ) ( )( ) ( ), ,2, , ,k k k km k k m k m k p pp pp

G f A I H f A C H f A C C f C fβ γ β γ β β′ ′ ′= ≤ ≤ = .

Hence, ( ),, : p p

m kG f L Lβ γ → and there exists a constant 2k kC A C C′ ′= such that

( ),, km k pp

G f C fβ γ ′≤ .

52

CHAPTER 6

pL −BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING GENERALIZED HYPERGEOMETRIC

FUNCTIONS 1 , , 1m m k mFσ+ ≥ AS KERNEL

In this chapter, we consider integral operators involving generalized

k −hypergeometric functions of the type 1 , , 1m m k mFσ+ ≥ as kernel and discuss

the pL − boundedness of these integral operators by using properties of generalized k −hypergeometric functions 1 , , 1m m k mFσ

+ ≥ .

6.1 BOUNDEDNESS OF INTEGRALS ( ), ,

,m kM fσ α β WITH VARIANTS

OF HOMOGENEOUS KERNEL

We now consider the integral operator

( )( ) ( ) ( ) ( ) ( ) ( ) ( )2 1, ,,

0

1 , 0mk p k m m

m kM f x x t x t f t dt xα σσβ σβσ α β σ σ −

∞− −= + >∫ .

We shall, then, prove its boundedness in pL and other relations that are useful in our study of integral operators involving generalized k −hypergeometric functions.

We now define general Erdélyi-Kober k − fractional integral operator as

follows and prove pL − boundedness of one of its special cases to be used later. For 0λ > , 0, 0k σ> > , define

( )( ) ( )( )

( ) ( )( ) ( )1 1, ,

0

( ) , 0xk k k k

kk

xI f x x t t f t dt t x

k

σδ σλ λ σδ σλ σ σσ δ

σλ

− + − + −= − < < < ∞Γ ∫ .

Theorem 6.1.1: Let ( ) ( ) 1, 0, 0, 0 and , 2p k k pγ β σ λ γ β σδ σ> − > > > = − + = .

If ( ) ( )( ), , 2k pI f xγ βσ−

( ) ( )( )

( ) ( ) ( )( ) ( )2 1 2 1

0

( )xp k k p

k

xx t t f t dt

k

σ σ γ β γ βσ σσγ β

− + − − − − −= −Γ − ∫ ,

53

then ( ) ( ), , 2 : p pk pI f L Lγ βσ− → and there exists a constant ( ), , , ,C C p kβ γ σ′′ ′′=

such that

( ) ( ), , 2k p ppI f C fγ β

σ− ′′≤ .

Proof: Consider

( ) ( )( )( ) ( )( )

( ) ( ) ( )( ) ( )2 1 2 1

, , 20

( ) .xp k k p

k pk

xI f x x t t f t dt

k

σ σ γ β γ βγ β σ σσ

σγ β

− + − − − − −− = −Γ − ∫

( ) ( ) ( )( ) ( ) ( )( ) ( )12 2 1Suppose that ,kp k px t x x t t

γ βσ σ γ β σ σψ− −− + − − −= − .

Then ( ) ( )1 , ,hx ht h x tψ ψ−= .

Since ( ),x yψ is a homogeneous function of degree 1− , then by Theorem

4.1.3(a(i)), there exists a constant ( ), , , ,C C k pβ γ σ′′ ′′= such that

( ) ( ), , 2k p ppI f C fγ β

σ− ′′≤

where

( )1

1

0

1,pC t t dtψ−′′ = ∫

= ( ) ( ) ( )( )1 11 1

0

1kpt t dt

γ βσ − −− − < ∞∫ ,

if 1, 0, 0, 0p kγ β σ> − > > > .

Hence, ( ) ( ), , 2 : p pk pI f L Lγ βσ− → .

Now, consider the integral operator ( ), ,,m kM fσ α β as given below.

Theorem 6.1.2: Let

( )( ) ( ) ( ) ( ) ( ) ( )2 1, ,,

0

1mk p k m m

m kM f x x t x t f t dtα σσβ σβσ α β σ σ −

∞− −= +∫ .

If ( ) ( ) ( )1, 1 , 0 1, 0, 0p k p k kσβ α σβ α σ> − < < < < > > ,

( ), ,,then : p p

m kM f L Lσ α β → and there exists a constant ( )3 3 , , , ,C C k pα β σ= such that

( ), ,3,m k pp

M f C fσ α β ≤ .

54

Proof: Note that

( )( )( ) ( ) ( ) ( ) ( ) ( )( )( )2 1, ,,

0

1mk p k m m

m kM V f x x t x t V f t dtα σσβ σβσ α β σ σ −

∞− −= +∫

( ) ( ) ( ) ( ) ( ) ( )2 2 1

0

1 1mk p k p m mx t x t f t dt

α σσβ σβ σ σ −∞

− − −= +∫ .

Substitute 1y t= . Then

( )( )( ), ,,m kM V f xσ α β

( ) ( ) ( ) ( ) ( ) ( )2 2 1

0

mk p p k m mx y x y f y dyα σσβ α σβ σ σ −

∞− + − −= +∫ .

Suppose that ( ) ( ) ( ) ( ) ( ) ( )2 2 1 ,mk p p k m mx y x y x y

α σσβ α σβ σ σψ−− + − −= + .

Then ( ) ( )1 , ,hx hy h x yψ ψ−= .

Since ( ),x yψ is a homogeneous function of degree 1− , then by Theorem

4.1.3(a(i)), there exists a constant ( )3 3 , , , ,C C k pα β σ= such that

( )( ) ( ), ,3 3,m k ppp

M V f C V f C fσ α β ≤ =

where

( )13

0

1,pC y y dyψ∞

−= ∫

( ) ( ) ( )1 1

0

1mp k my y dy

α σα σβ σ −∞

+ − −= +∫

( ) ( ) ( )1

1 1

0

1mp k my y dy

α σα σβ σ −+ − −= +∫

( ) ( ) ( )1 1

1

1 ,mp k my y dy

α σα σβ σ∞ −+ − −+ + < ∞∫

if ( ) ( ) ( ) 1, 1 , 0 1, 0, 0p k p k kσβ α σβ α σ> − < < < < > > .

Thus, ( ), ,, : p p

m kM f L Lσ α β → .

Also, since ( ) : p pV f L L→ and ( ) ( )( )( ), , , , 2, ,m k m kM f M V fσ α β σ α β= , therefore

( ) ( )( )( ) ( ), , , , 23 3, ,m k m k ppp p

M f M V f C V f C fσ α β σ α β= ≤ = .

55

Hence, ( ), ,, : p p

m kM f L Lσ α β → and there exists a constant ( )3 3 , , , ,C C k pα β σ= such that

( ), ,3,m k pp

M f C fσ α β ≤ .

Lemma 6.1.3: Let

( )( ) ( ) ( ) ( ) ( ) ( ) ( )1 2* , ,,

0

1 , 0mk k p m m

m kM g x x t x t g t dt xα σσβ σβσ α β σ σ −

∞− −= + >∫ .

If ( ) ( ) ( ) ( ) ( )1, 1 1 1, 1 ,0 1p p p k p kσβ α σβ α′ ′ ′> + = − < < < < and

0, 0k σ> > , then ( )* , ,, : p p

m kM g L Lσ α β ′ ′→ and there exists a constant

( )3 3 , , , ,C C k pα β σ′ ′ ′= such that

( )* , ,3,m k pp

M g C gσ α β′′

′≤ .

Proof: For proof, take p′ in place of p , ( ) ( )1 1 1p p′+ = , in the last theorem to get the result. Now, we prove the following product formula. Theorem 6.1.4: Let

( )( ) ( ) ( ) ( ) ( ) ( ) ( )2 1, ,,

0

1 , 0mk p k m m

m kM f x x t x t f t dt xα σσβ σβσ α β σ σ −

∞− −= + >∫

and

( )( ) ( ) ( ) ( ) ( ) ( ) ( )1 2* , ,,

0

1 , 0mk k p m m

m kM g x x t x t g t dt xα σσβ σβσ α β σ σ −

∞− −= + >∫ .

If ( )pf L R+∈ and ( ) pg L R′ +∈ , then

( ) ( )( ) ( ) ( )( ), , * , ,, ,

0 0m k m kg t M f t dt f t M g t dtσ α β σ α β

∞ ∞

=∫ ∫ .

Proof: Use of Fubini’s theorem will prove the result. Theorem 6.1.5: For 0, 0k σ> > , let

( ) ( ) ( )( ) ( )1 2 1 for 0 ;k p

x t x t t t xγ βσ σ σφ− − −= − < < < ∞

0 = for t x≥ . Then

56

( )( )* , ,, xm kM tσ α β σφ ( ) ( ) ( ) ( )

( )

1

=k k

k k

k

k x tσγ σ σββ γ βσ γ

− −Γ Γ −Γ

( )

( )1,1 , , , , ,..., ,

1 , 1, , , ,..., ,

; .

m kkk k k

m m m m m mm m k m kk

k k km m m

F x t

βα β βσσ σ σ

γγ γ

+ − +

+ + − +

× −

Proof: Consider the following integral representation

( )

( )1,1 , , , , ,..., ,

1 , 1, , , ,..., ,

;

m kkk k k

m m m mm m k m kk

k k km m m

F x

βα β βσσ

γγ γ

+ − +

+ + − +

( )

( ) ( )( ) ( ) ( )( ) ( )

111

0

1 1 .mkk mk

k ku u xu du

k

α σγ ββγβ γ β

−− −−Γ= − −

Γ Γ − ∫

Putting u wσ= , we get

( )

( )1,1 , , , , ,..., ,

1 , 1, , , ,..., ,

;

m kkk k k

m m m mm m k m kk

k k km m m

F x

βα β βσσ

γγ γ

+ − +

+ + − +

( )

( ) ( )( ) ( ) ( )( ) ( )

111

0

= 1 1 .mkk mk

k kw w xw dw

k

α σγ βσβ σ σσ γβ γ β

−− −−Γ− −

Γ Γ − ∫

Replacing by m mx x tσ σ− , we have

( )

( )1,1 , , , , ,..., ,

1 , 1, , , ,..., ,

;

m kkk k k

m m m m m mm m k m kk

k k km m m

F x t

βα β βσσ σ σ

γγ γ

+ − +

+ + − +

−

( )

( ) ( ) = k

k kkσ γβ γ βΓ

Γ Γ −

( ) ( ) ( )( ) ( )1

11

0

1 1 .mkk m m mw w x t w dw

α σγ βσβ σ σ σ σ −− −−× − +∫

Put xw y= to get

( )

( )1,1 , , , , ,..., ,

1 , 1, , , ,..., ,

;

m kkk k k

m m m m m mm m k m kk

k k km m m

F x t

βα β βσσ σ σ

γγ γ

+ − +

+ + − +

−

( ) ( )

( ) ( )=

kk

k k

xk

σ σγσ γβ γ β

−ΓΓ Γ −

57

( ) ( ) ( )( ) ( )11

0

1 .m

xkk m my x y t y dy

α σγ βσβ σ σ σ σ −− −−× − +∫

This implies that

( ) ( ) ( )( ) ( )11

0

1m

xkk m my x y t y dy

α σγ βσβ σ σ σ σ −− −− − +∫

( ) ( ) ( )

( )=

kk k

k

k x σγ σβ γ βσ γ

−Γ Γ −Γ

( )

( )1,1 , , , , ,..., ,

1 , 1, , , ,..., ,

; .

k mkk k k

m m m m m mm m k k mk

k k km m m

F x t

βα β βσσ σ σ

γγ γ

+ − +

+ + − +

× −

(6.1.1)

Now, by considering * , ,,m kM σ α β as in Theorem 6.1.3, we obtain the following.

( )( ) ( ) ( ) ( ) ( ) ( ) ( )1 2* , ,,

0

1 , 0mk k p m m

m kM g t t y t y g y dy tα σσβ σβσ α β σ σ −

∞− −= + >∫ .

Then taking ( ) ( ) xg t tσφ= , we get

( )( ) ( ) 1* , ,,

kxm kM t t σβσ α β σφ −=

( ) ( ) ( ) ( ) ( )( ) ( )2 2

0

1 11m kk p pm my t y dyx y y

α σ γ βσβ σ σ σ σ−∞ −− − − × +

−∫

( ) ( ) ( ) ( )( ) ( )1 1

0

11

mkk k m mt y t y dyx yα σγ βσβ σβ σ σσ σ −

∞ −− − −= +−∫ .

By using Equation (6.1.1), we obtain

( )( ) ( ) ( ) ( ) ( )

( )

1* , ,, =

k kk k

xm kk

k x tM t

σγ σ σβσ α β σ β γ β

φσ γ

− −Γ Γ −Γ

( )

( )1,1 , , , , ,..., ,

1 , 1, , , ,..., ,

; .

k mkk k k

m m m m m mm m k k mk

k k km m m

F x t

βα β βσσ σ σ

γγ γ

+ − +

+ + − +

× −

6.2 BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING

HYPERGEOMETRIC FUNCTIONS 1 , , 1m m k mFσ+ ≥

We now formulate integral operators involving hypergeometric functions of the type 1 , , 1m m k mFσ

+ ≥ . We use the results proved in Section 6.1 to prove

58

the pL − boundedness of these integral operators involving hypergeometric functions 1 , , 1m m k mFσ

+ ≥ .

Theorem 6.2.1: Let

( )( ) ( ) ( )( ) 12 1, , ,,

0

kpm kS f x x xt σβσ α β γ

∞−− += ∫

( )

( )

( )1

,1 , , , , ,..., ,

1 , 1, , , ,..., ,

; .

m kkk k k

m m m m m mm m k m kk

k k km m m

F x t f t dt

βα β βσσ σ σ

γγ γ

+ − +

+ + − +

× −

If ( ) ( ) ( )1, 0, 1 , 0 1p k p kγ β σβ α σβ α> − > − < < < < and 0, 0k σ> > , then

a) ( ) ( ) ( ){ }, , , , ,, ,, , 2km k m kk pS f I MA fσ α β γ γ β σ α β

σ−= where

( )( )

kk

kA

γβ

Γ=Γ

,

b) ( ), , ,, : p p

m kS f L Lσ α β γ → and explicitly

c) there exists a constant kCσ such that ( ), , ,

, km k ppS f C fσ α β γ σ≤ .

Proof: We employ the Theorem 6.1.1 to get

( ) ( ){ }( ), ,,, , 2 m kk pI M f xγ β σ α β

σ−

( ) ( )( )

( )

2 p k

k

xk

σ σ γ βσγ β

− + − −

=Γ −

( ) ( )( ) ( ) ( )( )1 2 1 , ,

,0

.x k p

m kx t t M f t dtγ βσ σ σ α β− − −× −∫

An application of Theorem 6.1.4 and Theorem 6.1.5 then yields

( ) ( ){ }( ), ,,, , 2 m kk pI M f xγ β σ α β

σ−

( ) ( )( )

( )

2 p k

k

xk

σ σ γ βσγ β

− + − −

=Γ −

( ) ( ) ( )( ) ( )1 2 1* , ,,

0

k p

m kf t M x t t dtγ βσ α β σ σ

∞ − − − × −

∫

( ) ( )( )

( ) ( ) ( )( )2

* , ,,

0

p k

xm kk

xf t M t dt

k

σ σ γ βσ α β σσ φ

γ β

∞− + − −

=Γ − ∫

59

( ) ( )( )

( ) ( ) ( ) ( )( )

( )2

0

p kk kk k

k k

kxf t x t

k

σ σ γ βσγ σ σββ γ βσ

γ β σ γ

∞− + − −−Γ Γ −

=Γ − Γ∫

( )

( )1,1 , , , , ,..., ,

1 , 1, , , ,..., ,

;

m kkk k k

m m m m m mm m k m kk

k k km m m

F x t dt

βα β βσσ σ σ

γγ γ

+ − +

+ + − +

× −

( ) ( )

( ) ( )( )2 1

1

0

pkk

k

xxt σββ

γ

∞− +−Γ

=Γ ∫

( )

( )

( )1

,1 , , , , ,..., ,

1 , 1, , , ,..., ,

;

m kkk k k

m m m m m mm m k m kk

k k km m m

F x t f t dt

βα β βσσ σ σ

γγ γ

+ − +

+ + − +

× −

( )( ) ( )( ), , ,

,k

m kk

S f xσ α β γβγ

Γ=Γ

.

Hence, ( ) ( ) ( ){ }, , , , ,, ,, , 2km k m kk pS f I MA fσ α β γ γ β σ α β

σ−=

where ( )( )

kk

kA

γβ

Γ=Γ

.

Since ( ) ( ), , 2 : p pk pI f L Lγ βσ− → by Theorem 6.1.1 and it, therefore, follows from

Theorem 6.1.2 that if ( ) ( ) ( )1, 0, 1 , 0 1p k p kγ β σβ α σβ α> − > − < < < < and

0, 0k σ> > , then

( ) ( ) ( ){ } ( ), , , , , , ,, , ,, , 2

3 .

k km k m k m kk pp pp

k kp p

S f A I M f A C M f

A C C f C f

σ α β γ γ β σ α β σ α βσ

σ

− ′′= ≤

′′≤ =

Hence, ( ), , ,, : p p

m kS f L Lσ α β γ → and there exists a constant 3k kC A C Cσ ′′= such that

( ), , ,, km k pp

S f C fσ α β γ σ≤ .

60

CHAPTER 7

pL −BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING GENERALIZED HYPERGEOMETRIC

FUNCTIONS , , 1m m k mFσ ≥ AS KERNEL In this chapter, we consider integral operators involving generalized

k −hypergeometric functions , , 1m m k mFσ ≥ as kernel and discuss the pL − boundedness of these integral operators by using properties of generalized

k −hypergeometric functions , , 1m m k mFσ ≥ .

7.1 BOUNDEDNESS OF INTEGRALS ( ),, m kH fσ β INVOLVING

EXPONENTIAL FUNCTIONS AS KERNEL

In this section, we first consider integral operators of the type

( )( ) ( ) ( ) ( ) ( ) ( )2 1,,

0

, 0m mk p k x t

m kH f x x t e f t dt xσ σσβ σβσ β

∞− − −= >∫ .

We prove their boundedness in pL and other relations that are useful in our study of integral operators involving generalized k −hypergeometric functions , , 1m m k mFσ ≥ . We begin with the following result. Theorem 7.1.1: Let

( )( ) ( ) ( ) ( ) ( ) ( )2 1,,

0

, 0m mk p k x t

m kH f x x t e f t dt xσ σσβ σβσ β

∞− − −= >∫ .

If ( ) ( )1, 1 , 0, 0p k p kσβ σ> > > > , ( ),,then : p p

m kH f L Lσ β → and there

exists a constant ( )4 4 , , ,C C k pβ σ= such that

( ),4,m k pp

H f C fσ β ≤ .

Proof: Note that

( )( )( ) ( ) ( ) ( ) ( )( )( )2 1,,

0

m mk p k x tm kH V f x x t e V f t dt

σ σσβ σβσ β∞

− − −= ∫

61

( ) ( ) ( ) ( ) ( )2 2 1

0

1m mk p k p x tx t e f t dtσ σσβ σβ

∞− − − −= ∫ .

Substitute 1y t= .Then

( )( )( ) ( ) ( ) ( ) ( ) ( )2 2 1,,

0

m mk p p k x ym kH V f x x y e f y dy

σ σσβ σβσ β∞

− − − −= ∫ .

Suppose that ( ) ( ) ( ) ( ) ( )2 2 1 ,m mk p p k x yx y x y eσ σσβ σβψ − − − −= .

Then ( ) ( )1, ,hx hy h x yψ ψ−= .

Since ( ),x yψ is a homogeneous function of degree 1− , then by Theorem

4.1.3(a(i)), there exists a constant ( )4 4 , , ,C C k pβ σ= such that

( )( ) ( ),4 4,m k ppp

H V f C V f C fσ β ≤ =

where

( )14

0

1,pC y y dyψ∞

−= ∫

( ) ( )1 1 1

0

mp k yy e dyσσβ

∞− − −= < ∞∫ ,

if ( ) ( )1, 1 , 0, 0p k p kσβ σ> > > > .

Thus, ( ),, : p p

m kH f L Lσ β → .

Also since ( ) : p pV f L L→ and ( ) ( )( )( ), , 2, , m k m kH f H V fσ β σ β= , therefore

( ) ( )( )( ) ( ), , 24 4, ,m k m k ppp p

H f H V f C V f C fσ β σ β= ≤ = .

Hence, ( ),, : p p

m kH f L Lσ β → and there exists a constant ( )4 4 , , ,C C k pβ σ= such that

( ),4,m k pp

H f C fσ β ≤ .

Lemma 7.1.2: Let

( )( ) ( ) ( ) ( ) ( ) ( )1 2* ,,

0

, 0m mk k p t

m kxH g x x t g t dt xe

σ σσβ σβσ β∞

− − −= >∫ .

If ( ) ( ) ( ) ( ) 1, 1 , 1 1 1, 0, 0p k p p p kσβ σ′ ′ ′> > + = > > , then

( )* ,, : p p

m kH g L Lσ β ′ ′→ and there exists a constant ( )4 4 , , ,C C k pβ σ′ ′ ′= such that

( )* ,4,m k pp

H g C gσ β′′

′≤ .

62

Proof: The proof is similar to that as given in the last theorem. Now, we consider the following product formula. Theorem 7.1.3: Let

( )( ) ( ) ( ) ( ) ( ) ( )2 1,,

0

, 0m mk p k x t

m kH f x x t e f t dt xσ σσβ σβσ β

∞− − −= >∫

and

( )( ) ( ) ( ) ( ) ( ) ( )1 2* ,,

0

, 0m mk k p t

m kxH g x x t g t dt xe

σ σσβ σβσ β∞

− − −= >∫ .

If ( )pf L R+∈ and ( ) pg L R′ +∈ , then

( ) ( )( ) ( ) ( )( ), * ,, ,

0 0m k m kg t H f t dt f t H g t dtσ β σ β

∞ ∞

=∫ ∫ .

Proof: Fubini’s theorem allows the change of variables which leads to the proof. Theorem 7.1.4: For 0, 0k σ> > , let

( ) ( ) ( )( ) ( )1 2 1 for 0 ; k p

x t x t t t xγ βσ σ σφ− − −= − < < < ∞

0 = for t x≥ . Then

( )( ) ( ) ( ) ( ) ( )

( )

1* ,, =

k kk k

xm kk

k x tH t

σγ σ σβσ β σ β γ β

φσ γ

− −Γ Γ −Γ

( )

( )1, , , ,..., ,

, 1, , , ,..., ,

; .

m kkk k k

m m m m mm m k m kk

k k km m m

F x t

ββ βσ σ σ

γγ γ

+ − +

+ − +

× −

Proof: Consider the following integral representation

( )

( )1, , , ,..., ,

, 1, , , ,..., ,

;

m kkk k k

m m mm m k m kk

k k km m m

F x

ββ βσ

γγ γ

+ − +

+ − +

( )

( ) ( )( ) ( ) ( )( )

111

0

1 .mkk xuk

k ku u e du

kγ ββγ

β γ β− −−Γ

= −Γ Γ − ∫

63

Substitute u wσ= to get

( )

( )1, , , ,..., ,

, 1, , , ,..., ,

;

m kkk k k

m m mm m k m kk

k k km m m

F x

ββ βσ

γγ γ

+ − +

+ − +

( )

( ) ( )( ) ( ) ( )( )1

11

0

= 1 .mkk xwk

k kw w e dw

k

σγ βσβ σσ γβ γ β

− −−Γ−

Γ Γ − ∫

Replacing by m mx x tσ σ− , we have

( )

( )1, , , ,..., ,

, 1, , , ,..., ,

;

m kkk k k

m m m m mm m k m kk

k k km m m

F x t

ββ βσ σ σ

γγ γ

+ − +

+ − +

−

( )

( ) ( )( ) ( ) ( )( )1

11

0

= 1 .m m mkk x t wk

k kw w e dw

k

σ σ σγ βσβ σσ γβ γ β

− −− −Γ−

Γ Γ − ∫

Put xw y= to get

( )

( )1, , , ,..., ,

, 1, , , ,..., ,

;

m kkk k k

m m m m mm m k m kk

k k km m m

F x t

ββ βσ σ σ

γγ γ

+ − +

+ − +

−

( ) ( )

( ) ( )( ) ( ) ( )( ) 11

0

= .m m

xk kk t yk

k k

xy x y e dy

k

σ σσ σγ γ βσβ σ σσ γ

β γ β

− − −− −Γ−

Γ Γ − ∫

This implies that

( ) ( ) ( )( ) 11

0

m mx

kk t yy x y e dyσ σγ βσβ σ σ − −− −−∫

( ) ( ) ( )

( )=

kk k

k

k x σγ σβ γ βσ γ

−Γ Γ −Γ

( )

( )1, , , ,..., ,

, 1, , , ,..., ,

; .

m kkk k k

m m m m mm m k m kk

k k km m m

F x t

ββ βσ σ σ

γγ γ

+ − +

+ − +

× −

(7.1.1)

Now, consider * ,,m kH σ β as in Theorem 7.1.2, we obtain the following.

( )( ) ( ) ( ) ( ) ( ) ( )1* ,,

0

, 0m mk k p y

m ktH g t t y g y dy teσ σσβ σβσ β

∞− − −= >∫ .

64

Then taking ( ) ( ) xg t tσφ= , we get

( )( ) ( ) 1* ,,

kxm kH t t σβσ β σφ −=

( ) ( ) ( ) ( )( ) ( )2 2

0

1 1m m kk p pyty dye x y yσ σ γ βσβ σ σ

∞ −− − −− ×

−∫

( ) ( ) ( ) ( )( ) 11 1

0

m mx kk k t yt y x y e dy

σ σγ βσβ σβ σ σ − −− − −= −∫ .

By using Equation (7.1.1), we obtain

( )( ) ( ) ( ) ( ) ( )

( )

1* ,, =

k kk k

xm kk

k x tH t

σγ σ σβσ β σ β γ β

φσ γ

− −Γ Γ −Γ

( )

( )1, , , ,..., ,

, 1, , , ,..., ,

; .

k mkk k k

m m m m mm m k k mk

k k km m m

F x t

ββ βσ σ σ

γγ γ

+ − +

+ − +

× −

7.2 BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING HYPERGEOMETRIC FUNCTIONS , , 1m m k mFσ ≥

We now formulate integral operators involving generalized k −hypergeometric functions of the type , , 1m m k mFσ ≥ . We use the results

proved in Section 7.1 to prove the pL − boundedness of these integral operators involving generalized k −hypergeometric functions , , 1m m k mFσ ≥ .

Theorem 7.2.1: Let

( )( ) ( ) ( )( ) 12 1, ,,

0

kpm kG f x x xt σβσ β γ

∞−− += ∫

( )

( )

( )1

, , , ,..., ,

, 1, , , ,..., ,

; .

m kkk k k

m m m m mm m k m kk

k k km m m

F x t f t dt

ββ βσ σ σ

γγ γ

+ − +

+ − +

× −

( ) ( )If 1, 1 , 0, 0, 0p k p kσβ γ β σ> < − > > > , then

a) ( ) ( ) ( ){ }, , ,, ,, , 2km k m kk pG f A I H fσ β γ γ β σ β

σ−= where

( )( )

kk

kA

γβ

Γ=Γ

,

65

b) ( ), ,, :m k

p pG f L Lσ β γ → and explicitly

c) there exists a constant kCσ′ such that

( ), ,, km k pp

G f C fσ β γ σ′≤ .

Proof: We employ the Theorem 6.1.1 to get

( ) ( ){ }( ),,, , 2 m kk pI H f xγ β σ β

σ−

( ) ( )( )

( )

2 p k

k

xk

σ σ γ βσγ β

− + − −

=Γ −

( ) ( )( ) ( ) ( )( )1 2 1 ,

,0

.x k p

m kx t t H f t dtγ βσ σ σ β− − −× −∫

Now, use of Theorem 7.1.3 and Theorem 7.1.4 then yields

( ) ( ){ }( ),,, , 2 m kk pI H f xγ β σ β

σ−

( ) ( )( )

( )

2 p k

k

xk

σ σ γ βσγ β

− + − −

=Γ −

( ) ( ) ( )( ) ( )1 2 1* ,,

0

k p

m kf t H x t t dtγ βσ β σ σ

∞ − − − × −

∫

( ) ( )( )

( ) ( ) ( )( )2

* ,,

0

p k

xm kk

xf t H t dt

k

σ σ γ βσ β σσ φ

γ β

∞− + − −

=Γ − ∫ .

( ) ( )( )

( )

2 p k

k

xk

σ σ γ βσγ β

− + − −

=Γ −

( ) ( ) ( )( )

( )

0

k kk k

k

kf t x tσγ σ σββ γ β

σ γ

∞−Γ Γ −

×Γ∫

( )

( )1, , , ,..., ,

, 1, , , ,..., ,

;

m kkk k k

m m m m mm m k m kk

k k km m m

F x t dt

ββ βσ σ σ

γγ γ

+ − +

+ − +

× −

( ) ( )

( ) ( )( )2 1

1

0

pkk

k

xxt σββ

γ

∞− +−Γ

=Γ ∫

( )

( )

( )1

, , , ,..., ,

, 1, , , ,..., ,

;

k mkk k k

m m m m mm m k k mk

k k km m m

F x t f t dt

ββ βσ σ σ

γγ γ

+ − +

+ − +

× −

66

where( )( )

kk

kA

γβ

Γ=Γ

.

Since ( ) ( ), , 2 : p pk pI f L Lγ βσ− → by Theorem 6.1.1 and it, therefore, follows from

Theorem 7.1.1 that if ( ) ( ) 1, 1 , 0, 0, 0p k p kσβ γ β σ> < − > > > , then

( ) ( )( ) ( ), , , , , ,, , ,

4 .

k km k k m k m kp pp

k kp p

G f A I H f A C H f

A C C f C f

σ β γ σ σ β γ σ β σ σ β

σ σ

′′= ≤

′′ ′≤ =

Hence, ( ), ,, : p p

m kG f L Lσ β γ → and there exists a constant 4k kC A C Cσ σ′ ′′= such that

( ), ,, km k pp

G f C fσ β γ σ′≤ .

67

CHAPTER 8

pL −BOUNDEDNESS OF INTEGRAL OPERATORS WITH WEIGHT FUNCTIONS INVOLVING GENERALIZED

HYPERGEOMETRIC FUNCTION 1 ,m m kFσ+ AND , , 1m m k mFσ ≥

AS KERNEL

In this chapter, we consider integral operators involving generalized k −hypergeometric functions 1 , , 1m m k mFσ

+ ≥ as kernel and discuss the pL -

boundedness of these integral operators with weight functions xν by using properties of generalized k −hypergeometric functions 1 , , 1m m k mFσ

+ ≥ .

8.1 BOUNDEDNESS OF INTEGRALS ( ), ,

,m kM fσ α β WITH WEIGHT

FUNCTION

In this section, we first consider integral operators of the type

( )( ) ( ) ( ) ( ) ( ) ( ) ( )2 1, ,,

0

1 , 0mk p k m m

m kM f x x t x t f t dt xα σσβ σβσ α β σ σ −

∞− −= + >∫ .

We prove their boundedness in pL and other relations that are useful in our study of integral operators involving generalized k −hypergeometric functions 1 , , 1m m k mFσ

+ ≥ . We begin with the following result.

Theorem 8.1.1: Let

( )( ) ( ) ( ) ( ) ( ) ( )2 1, ,,

0

1mk p k m m

m kM f x x t x t f t dtα σσβ σβσ α β σ σ −

∞− −= +∫ .

If ( ) ( ) ( ) ( ) 1, 1 , 0 1, 0, 0p k p p k kσβ α ν σβ α σ> − < − < < < > > ,

then ( ), ,, : p p

m kM f L Lσ α β → and there exists a constant ( )5 5 , , , , ,C C k pα β σ ν= such that

( )( ) ( )1 1

, ,5,

0 0

p pp p

m kx M f x dx C x f x dxν σ α β ν∞ ∞

≤ ∫ ∫ .

Proof: From Theorem 6.1.2, note that

68

( )( )( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 1, ,,

0

mk p p k m mm kM V f x x y x y f y dy

α σσβ α σβσ α β σ σ −∞

− + − −= +∫ .

Suppose that ( ) ( ) ( ) ( ) ( ) ( )2 2 1 ,mk p p k m mx y x y x y

α σσβ α σβ σ σψ−− + − −= + .

( ) ( )1, ,hx hy h x yψ ψ−= .

Since ( ),x yψ is a homogeneous function of degree 1− , by Theorem 4.1.3(a(ii)),

there exists a constant ( )5 5 , , , , ,C C k pα β σ ν= such that

( )( ) ( )( ) ( )1 1 1

, ,5 5,

0 0 0

p p pp p p

m kx M f x dx C x V f x dx C x f x dxν σ α β ν ν∞ ∞ ∞

≤ = ∫ ∫ ∫

where

( ) ( ) ( )15

0

1,p pC y y dyν ψ∞

− −= ∫

( ) ( ) ( ) ( )1 1

0

1mp p k my y dy

α σν α σβ σ −∞

− + − −= +∫

( ) ( ) ( ) ( )1

1 1

0

1mp p k my y dy

α σν α σβ σ −− + − −= +∫

( ) ( ) ( ) ( )1 1

1

1mp p k my y dy

α σν α σβ σ −∞

− + − −+ + < ∞∫ ,

if ( ) ( ) ( ) ( )1, 1 , 0 1, 0, 0p k p p k kσβ α ν σβ α σ> − < − < < < > > .

Hence, ( ), ,, : p p

m kM f L Lσ α β → and there exists a constant

( )5 5 , , , , ,C C k pα β σ ν= such that

( )( ) ( )1 1

, ,5,

0 0

p pp p

m kx M f x dx C x f x dxν σ α β ν∞ ∞

≤ ∫ ∫ .

69

Theorem 8.1.2: Let 1, 0, 1, 0, 0p kγ β ν σ> − > > > > . If

( ) ( )( )( ) ( )( )

( )

2

, , 2

p k

k pk

xI f x

k

σ σ γ βγ βσ γ β

− + − −− =

Γ −

( ) ( )( ) ( ) ( )1 2 1

0

( ) , 0 ,x k px t t f t dt x

γ βσ σ − − −× − >∫

then ( ) ( ), , 2 : p pk pI f L Lγ βσ− → and there exists a constant ( ), , , , ,C C k pβ γ σ ν′′′ ′′′=

such that

( ) ( )( ) ( )1 1

, , 20 0

p pp p

k px I f x dx C x f x dxν γ β νσ

∞ ∞−

′′′≤

∫ ∫ .

Proof: Consider

( ) ( )( )( ) ( )( )

( )

2

, , 2

p k

k pk

xI f x

k

σ σ γ βγ βσ

σγ β

− + − −− =

Γ −

( ) ( )( ) ( )1 2 1

0

( ) .x k px t t f t dt

γ βσ σ − − −× −∫

( ) ( ) ( )( ) ( ) ( )( ) ( )12 2 1Suppose that ,kp k px t x x t t

γ βσ σ γ β σ σψ− −− + − − −= − .

Then ( ) ( )1 , ,hx ht h x tψ ψ−= .

Since ( ),x yψ is a homogeneous function of degree 1− , by Theorem 4.1.3

(a(ii)), there exists a constant ( ), , , , ,C C k pβ γ σ ν′′′ ′′′= such that

( ) ( )( ) ( )1 1

, , 20 0

p pp p

k px I f x dx C x f x dxν γ β νσ

∞ ∞−

′′′≤

∫ ∫

where

( ) ( ) ( )1

1

0

1,p pC t t dtν ψ− −′′ = ∫

( ) ( ) ( ) ( )( )1 11 1

0

1kp pt t dt

γ βν σ − −− −= − < ∞∫ ,

if 1, 0, 1, 0, 0p kγ β ν σ> − > > > > .

Hence, ( ) ( ), , 2 : p pk pI f L Lγ βσ− → and there exists a constant

( ), , , , ,C C k pβ γ σ ν′′′ ′′′= such that

70

( ) ( )( ) ( )1 1

, , 20 0

p pp p

k px I f x dx C x f x dxν γ β νσ

∞ ∞−

′′′≤

∫ ∫ .

8.2 BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING

HYPERGEOMETRIC FUNCTIONS 1 , , 1m m k mFσ+ ≥ WITH

WEIGHT FUNCTION xν We now formulate new integral operators involving generalized k −hypergeometric functions of the type 1 , , 1m m k mFσ

+ ≥ . We use the results on * , ,,m kM σ α β proved in Section 8.1 to establish the pL − boundedness of these

integral operators involving generalized k −hypergeometric functions

1 , , 1m m k mFσ+ ≥ with weight function xν .

Theorem 8.2.1: Let

( )( ) ( ) ( )( ) 12 1, , ,,

0

kpm kS f x x xt σβσ α β γ

∞−− += ∫

( )

( )

( )1

,1 , , , , ,..., ,

1 , 1, , , ,..., ,

; .

m kkk k k

m m m m m mm m k m kk

k k km m m

F x t f t dt

βα β βσσ σ σ

γγ γ

+ − +

+ + − +

× −

If ( ) ( ) ( ) ( )1, 0, 1, 1 ,p k p p kγ β ν σβ α ν σβ> − > > − < − < and

0 1, 0, 0kα σ< < > > there exists a constant ,kCσ ν such that

( )( ) ( )1 1

, , , ,,

0 0

p pp p

km kx S f x dx C x f x dxν σ α β γ σ ν ν∞ ∞

≤ ∫ ∫ .

Proof: We employ the Theorem 8.1.1 and 8.1.2 to get

( )( ) ( ) ( ){ }( )1 1

, , , , ,, ,, , 2

0 0

p ppp

m k m kk px S f x dx x I M f x dxν σ α β γ ν γ β σ α βσ

∞ ∞−

=

∫ ∫

( )( ) ( )1 1

, , ,,

0 0

p pp p

km kx M f x dx C x f x dxν σ α β σ ν ν∞ ∞

≤ ≤ ∫ ∫

71

Hence, ( ), , ,, : p p

m kS f L Lσ α β γ → and there exists a constant ,kCσ ν such that

( )( ) ( )1 1

, , , ,,

0 0

p pp p

km kx S f x dx C x f x dxν σ α β γ σ ν ν∞ ∞

≤ ∫ ∫ .

8.3 BOUNDEDNESS OF INTEGRALS ,

,m kHσ β WITH WEIGHT

FUNCTION In this section, we take up integral operators involving generalized

k −hypergeometric functions , , 1m m k mFσ ≥ as kernel and discuss the pL − boundedness of these integral operators with weight function xν by using properties of generalized k −hypergeometric functions , , 1m m k mFσ ≥ .

We first consider integral operators of the type

( )( ) ( ) ( ) ( ) ( ) ( )2 1,,

0

, 0m mk p k x t

m kH f x x t e f t dt xσ σσβ σβσ β

∞− − −= >∫ .

We prove their boundedness in pL and other relations that are useful in our study of integral operators involving generalized k −hypergeometric functions , , 1m m k mFσ ≥ with weight function xν . We begin with the following result. Theorem 8.3.1: Let

( )( ) ( ) ( ) ( ) ( ) ( )2 1,,

0

, 0m mk p k x t

m kH f x x t e f t dt xσ σσβ σβσ β

∞− − −= >∫ .

If ( ) ( ) ( ) 1, 1 , 0, 0p k p p kσβ ν σ> > − > > , then

( ),, : p p

m kH f L Lσ β → and there exists a constant ( )6 6 , , , ,C C k pβ σ ν= such that

( )( ) ( )1 1

,6,

0 0

p pp p

m kx H f x dx C x f x dxν σ β ν∞ ∞

≤ ∫ ∫ .

Proof: From Theorem 7.1.1

( )( )( ) ( ) ( ) ( ) ( ) ( )2 2 1,,

0

m mk p p k x ym kH V f x x y e f y dy

σ σσβ σβσ β∞

− − − −= ∫ .

Suppose that ( ) ( ) ( ) ( ) ( )2 2 1 ,m mk p p k x yx y x y eσ σσβ σβψ − − − −= .

72

Then ( ) ( )1 , ,hx hy h x yψ ψ−= .

Since ( ),x yψ is a homogeneous function of degree 1− , by Theorem

4.1.3(a(ii)), there exists a constant ( )6 6 , , , ,C C k pβ σ ν= such that

( )( ) ( )( ) ( )1 1 1

,6 6,

0 0 0

p p pp p p

m kx H f x dx C x V f x dx C x f x dxν σ β ν ν∞ ∞ ∞

≤ = ∫ ∫ ∫where

( ) ( ) ( )16

0

1,p pC y y dyν ψ∞

− −= ∫

( ) ( ) ( )1 1 1

0

mp p k yy e dyσν σβ

∞− − − −= < ∞∫ ,

if ( ) ( ) ( ) 1, 1 , 0, 0p k p p kσβ ν σ> > − > > .

Hence, ( ),, : p p

m kH f L Lσ β → and there exists a constant ( )6 6 , , , ,C C k pβ ν σ= such that

( )( ) ( )1 1

,6,

0 0

p pp p

m kx H f x dx C x f x dxν σ β ν∞ ∞

≤ ∫ ∫ .

8.4 BOUNDEDNESS OF INTEGRAL OPERATORS INVOLVING

HYPERGEOMETRIC FUNCTIONS , , 1m m k mFσ ≥ WITH WEIGHT

FUNCTION xν We now formulate integral operators involving generalized k −hypergeometric functions of the type , , 1m m k mFσ ≥ . We establish the pL − boundedness of these integral operators involving generalized

k −hypergeometric functions , , 1m m k mFσ ≥ with weight function xν .

Theorem 8.4.1: Let

( )( ) ( ) ( )( ) 12 1, ,,

0

kpm kG f x x xt σβσ β γ

∞−− += ∫

( )

( )

( )1

, , , ,..., ,

, 1, , , ,..., ,

; .

m kkk k k

m m m m mm m k m kk

k k km m m

F x t f t dt

ββ βσ σ σ

γγ γ

+ − +

+ − +

× −

73

If ( ) ( ) ( ) 1, 0, 1, 1p k p pγ β ν σβ ν> − > > > − , 0, 0k σ> > , then there

exists a constant ,kCσ ν′ such that

( )( ) ( )1 1

, , ,,

0 0

p pp p

km kx G f x dx C x f x dxν σ β γ σ ν ν∞ ∞

′≤ ∫ ∫ .

Proof: We employ Theorem 8.1.2 and 8.3.1 to get

( )( ) ( ) ( ){ }( )1 1

, , ,, ,, , 2

0 0

p ppp

m k m kk px G f x dx x I H f x dxν σ β γ ν γ β σ βσ

∞ ∞−

=

∫ ∫

( )( ) ( )1 1

, ,,

0 0

p pp p

km kx H f x dx C x f x dxν σ β σ ν ν∞ ∞

′≤ ≤ ∫ ∫

Hence, ( ), ,, : p p

m kG f L Lσ β γ → and there exists a constant ,kCσ ν′ such that

( )( ) ( )1 1

, , ,,

0 0

p pp p

km kx G f x dx C x f x dxν σ β γ σ ν ν∞ ∞

′≤ ∫ ∫ .

74

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