ltu.diva-portal.orgltu.diva-portal.org/smash/get/diva2:999529/FULLTEXT01.pdf · Abstract This thesis deals with some generalizations of the discrete Hardy and Carle-man type inequalities

LICENTIATE T H E S I S

Luleå University of TechnologyDepartment of Mathematics

:|: -|: - -- ⁄ --

:

Weight Characterizations of Discrete Hardy and

Carleman Type Inequalities

Christopher Adjei Okpoti

Weight Characterizations ofDiscrete Hardy and Carleman Type

Inequalities

by

Christopher A. Okpoti

Department of Mathematics

Luleå University of Technology

971 87 Luleå, Sweden

August 2005

Examiner

Professor Lars-Erik Persson,Luleå University of Technology, Sweden

Published 2005Printed in Sweden by University Printing Office, Luleå

To Mina and Emelia

Abstract

This thesis deals with some generalizations of the discrete Hardy and Carle-man type inequalities and the relations between them.

In Chapter 1 we give an introduction and overview of the area that servesas a frame for the rest of the thesis. In particular, a fairly complete descrip-tion of the development of discrete Hardy and Carleman type inequalities inone and more dimensions can be found in this chapter.

In Chapter 2 we consider some scales of weight characterizations for theone-dimensional discrete Hardy inequality for the case 1 < p ≤ q < ∞. Thesecharacterizations contain a parameter s and as endpoint results we obtainthe usual characterizations of Muckenhoupt or Bennett type. As limit casessome weight characterizations of Carleman type inequalities are obtained forthe case 0 < p ≤ q < ∞.

In Chapter 3 we present and discuss a new scale of weight characteri-zations for a two-dimensional discrete Hardy type inequality and its limittwo-dimensional Carleman type inequality.

In Chapter 4 we generalize the work done in Chapters 2 and 3 and present,prove and discuss the corresponding general n-dimensional versions.

In Chapter 5 we introduce the study of the general Hardy type inequality

(∗)

(∞∑

n=1

(n∑

k=1

dn,kak

)q

uk

)1/q

≤ C

(∞∑

n=1

apnvn

)1/p

, 1 < p ≤ q < ∞,

with a “discrete kernel” d = {dn,k}∞

n,k=1 involved. For kernels of producttype a weight characterization of (∗) is given, thus generalizing a previousresult of M. Goldman (the case dn,k = bk). A scale of sufficient conditions isproved for the general case.

Finally, in the Appendix some steps in the historical development of thecontinuous Hardy inequality are briefly described.

v

Preface

This thesis is written as a monography and the main content is described inthe abstract on the previous page. It contains an introductory part that ismainly on the history and developments of the discrete Hardy and Carlemantype inequalities and the general frame of the thesis.

In particular, this Licentiate thesis contains the author’s contributionsin the following papers:

• C. A. Okpoti, L.-E. Persson and A. Wedestig, Scales of weight char-acterizations for the discrete Hardy and Carleman inequalities, Proc.Function Spaces, Differential Operators and Nonlinear Analysis (FS-DONA 2004), Math. Institute, Acad. Sci., Czech Republic, Milovy,(2004), 236-258.

• C. A. Okpoti, L. E. Persson and A. Wedestig, Scales of weight char-acterizations for some multidimensional discrete Hardy and Carlemantype inequalities, Proc. A. Razmadze Math. Inst., 2005, to appear.

• C. A. Okpoti, L. E. Persson and A. Wedestig, Weight characterizationsfor the discrete Hardy inequality with kernel, Research report, Depart-ment of Mathematics, Luleå University of Technology, Sweden, 2005(13 pages), submitted.

vii

Acknowledgements

I want to express my deepest and sincere thanks to my main supervisorProfessor Lars-Erik Persson, for his constant support and encouragementduring my stay here in the Mathematics Department and at his home.

My invaluable gratitude also goes to Dr. Anna Wedestig, my co-supervisor,for her patience and thorough support that made me, within a short time,to grab many ideas in my field of study.

I would like to thank my co-authors Professor Lars-Erik Persson, Dr.Anna Wedestig and Ph.D. student Maria Johansson for their cooperationwhich in various ways has been important for my pleasant stay at Luleå Uni-versity of Technology and the results presented in this Licentiate thesis andelsewhere. Moreover, I would like to extend my thanks to Professor Lech Ma-ligranda and the visiting researchers Professor Gord Sinnamon (Universityof Western Ontario, Canada) and Dr. Amiran Gogatishvili (MathematicalInstitute, Academy of Sciences, Czech Republic) for some ideas we shared.

My profound gratitude to all and sundry in the Mathematics Depart-ment, Luleå University of Technology, Sweden for the conducive atmosphereand especially Elin Johansson who assisted me in many ways including trans-lations and Andreas Nilsson for computer support.

Last but not least, I thank the Government of Ghana, the authoritiesof University of Education, Winneba-Ghana, Institute of Mathematical Sci-ences, Ghana who granted me financial support, Professor Francis K. A.Allotey (President of Institute of Mathematical Sciences, Ghana), ProfessorLennart Hasselgren and Dr. Leif Abrahamsson (International Science Pro-gramme, ISP, Uppsala, Sweden), for their immense support and my betterhalf Mrs. Mina Okpoti for her prayers and love.

Luleå, August 2005Christopher A. Okpoti

ix

Contents

Abstract v

Preface vii

Acknowledgements ix

1 Introduction 1

1.1 The standard forms of Hardy’s inequality . . . . . . . . . . . 11.2 The prehistory of Hardy’s original inequalities . . . . . . . . . 21.3 Modern forms of Hardy type inequalities . . . . . . . . . . . . 51.4 On discrete Hardy-type inequalities with kernels . . . . . . . 81.5 On multidimensional discrete Hardy type inequalities . . . . . 101.6 On Carleman’s inequality-the early history and two elemen-

tary proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Modern forms of Carleman type inequalities . . . . . . . . . . 13

2 Scales of weight characterizations for discrete Hardy and

Carleman type inequalities 15

2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 26

3 Scales of weight characterizations for some two-dimensional

discrete Hardy and Carleman type inequalities 29

3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

xi

xii Contents

4 The general multidimensional results 43

4.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Weight characterizations for the discrete Hardy inequality

with kernel 55

5.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3 Concluding remarks and open questions . . . . . . . . . . . . 67

Bibliography 69

A Appendix 75

Chapter 1

Introduction

1.1 The standard forms of Hardy’s inequality

In the literature many authors including G. H. Hardy, J. E. Littlewood andG. Pólya [22] consider the following standard forms of Hardy’s inequality :

The discrete Hardy inequality : if p > 1 and {ak}∞k=1 is an arbitrary

sequence of non-negative real numbers, then

∞∑n=1

(1

n

n∑k=1

ak

)p

≤

(p

p − 1

)p ∞∑n=1

apn; (1.1)

and the continuous Hardy inequality : if p > 1 and f is a non-negative p-integrable function on (0,∞), then f is integrable over the interval (0, x) forall x > 0 and

∞∫0

⎛⎝1

x

x∫0

f(t)dt

⎞⎠

p

dx <

(p

p − 1

)p∞∫0

f(x)pdx. (1.2)

We make the following remarks:

1. The constant(

pp−1

)pin both inequalities (1.1) and (1.2) is sharp in

the sense that it can not be replaced by any smaller number.2. The inequalities (1.1) and (1.2) imply the following information re-

spectively:

1

2 Weight Characterizations of Discrete Hardy and


If∞∑

n=1

apn < ∞, then

∞∑n=1

hpn (a) < ∞, (1.3)

where a = {an} with an ≥ 0 and h (a) = {hn (a)} , hn (a) := 1n

∑nk=1 ak < ∞

is the discrete Hardy operator.

If

∞∫0

fp(x)dx < ∞, then

∞∫0

(Hf (x))p dx < ∞, (1.4)

where f(x) ≥ 0 and Hf (x) := 1x

x∫0

f (x) dx is the continuous Hardy operator.

Note that (1.3) and (1.4) are (see e.g. comments at the end of Hardy’spaper [16]) called the weak forms of (1.1) and (1.2), respectively.

3. The inequalities (1.1) and (1.2) together with remark 1 above implythe important information that the Hardy operators H and h map the spacesLp into Lp and lp into lp respectively (p > 1) and their norms are equal top′ = p

p−1 . Here, as usual, the spaces lp and Lp are the Lebesque spacesconsisting of all sequences a = {an}

∞n=1 of real numbers and all (equivalence

classes modulo equality almost everywhere) measurable functions f = f (x)on (0,∞) , respectively, such that

‖a‖lp:=

(∞∑

n=1

|an|p

)1/p

< ∞ and ‖f‖Lp:=

⎛⎝ ∞∫

0

|f(x)|p dx

⎞⎠

1/p

< ∞.

4. The inequalities (1.1) and (1.2) have been generalized and applied inanalysis and in the theory of differential equations. Some of these develop-ments, generalizations and applications have been described and discussedin the books [22], [35] and [45] and also in the new historical paper [32].

In this introduction and in this thesis we shall focus our interest on thediscrete Hardy inequality (1.1). In the literature much more results concern-ing generalizations of the continuous version (1.2) can be found. For thereader’s convenience we have collected and presented some of these resultsin an Appendix (see also [32]).

1.2 The prehistory of Hardy’s original inequalities

According to the literature the Hardy inequalities (1.1) and (1.2) could betraced back from Hilbert’s inequality, which was first stated and proved by

Introduction 3

D. Hilbert in the early 1900s (see [25] and also [17]). The initial form ofHilbert’s inequality is

∞∑n=1

∞∑m=1

ambn

m + n≤ π

(∞∑

m=1

a2m

) 12(

∞∑n=1

a2n

) 12

,

with π as a sharp constant. The following generalized form with a parameterp is also referred as Hilbert’s inequality:

∞∑n=1

∞∑m=1

ambn

m + n≤

π

sin(

πp

)(

∞∑m=1

apm

) 1p(

∞∑n=1

ap′

n

) 1p′

, (1.5)

where p > 1 and p′ = p/(p−1), even Hilbert himself was not close to considerthis case (the spaces Lp appeared only in 1910). It was M. Riesz and G. H.Hardy who took the first steps towards a proof of (1.5). It was reported in[17] that H. Weyl [61] outlined in his Inaugural-Dissertation that Hilbert’sown proof depends on the theory of Fourier’s series. Another three proofswere given early, one by F. W. Wiener [62] and two by I. Shur [51]. Hardy,however, said that none of these proofs was as simple and elementary asmight be desired. Hardy (see [17] and [16]), therefore, added a fifth proof,which to him was simpler than the earlier four proofs. Hardy’s attempts tosimplify the proofs of Hilbert’s theorem resulted in the discovery of the twoinequalities (1.1) and (1.2).

In Hardy’s studies, he first observed that Hilbert’s theorem (at least inweak form) is an immediate corollary of another theorem, which seems ofinterest in itself. We pronounce that this was Hardy’s original motivation(the beginning of the discovery of (1.1)).

By 1915, Hardy had considered, for an ≥ 0, the following propositions:

(i) If∞∑

n=1a2

n < ∞, then∞∑

n=1

∞∑m=1

aman

m+n < ∞ (a weak version of Hilbert’s

inequality),

(ii) if∞∑

n=1a2

n < ∞, then∞∑

n=1

(a1+a2+...+an

n

)2< ∞ (a weak version of

Hardy’s inequality).Hardy [17] showed that his theorem and Hilbert’s theorem may be de-

duced either from the other by arguments of an entirely simple and elemen-tary kind. Hardy then communicated his theorem to M. Riesz, who foundanother proof that is similar to that of Hardy in terms of simplicity. In

fact, Riesz also left the case p = 2 and proved that if p > 1 and∞∑

n=1ap

n is



convergent, then∞∑

n=1

(1

n

n∑i=1

ai

)p

is convergent and he even obtained that

N∑n=1

(1

n

n∑i=1

ai

)p

≤

(p2

p − 1

)p N∑n=1

apn,

which implies (1.1) but with the constant(

p2

p−1

)pinstead of the sharp one(

pp−1

)p. Moreover, I. Shur pointed out to Hardy that, when p = 2, the

constant(

p2

p−1

)p= 16 can certainly be replaced by

(p

p−1

)p= 4 and that

this fact could almost immediately be deduced from his inequality

N∑n=1

(1 −

2

n + 1

)(1

n

n∑i=1

ai

)2

≤ 4N∑

n=1

a2n.

Therefore in the paper [16] (see also [17]) Hardy claimed that probably the

right value for C =(

pp−1

)p, but without giving a proof.

Finally, in his 1925 note [18] Hardy wrote: “In a letter dated 21 June 1921,Landau communicated to me a direct proof of the theorem, which gives the

correct value for C =(

pp−1

)p” . In 1928 Hardy [20] again confirmed that

it was, in fact, not until the appearance of Landau’s note that the generalvalue of the constant C was definitely fixed. Landau also, in his 1921 letterto Hardy [18], pointed out that, if (1.2) holds, then also (1.1) holds, with the

correct constant C =(

pp−1

)p. This fact can be deduced at once by taking

f(x) = a1 (0 ≤ x < 1) , f(x) = a2 (1 ≤ x < 2) ..., that is step functions.Hence, when Hardy in his famous 1925 paper [18] finally stated and proved(1.2), he also proved (1.1).

Remark 1.1. As can be seen above the prehistory of Hardy’s inequalities ismostly concerned about the discrete version (1.1). But as remarked beforemost of the further developments have been done in the continuous case (seethe books [35], [45] and the review articles [33] and [32]). Therefore theinformation in the remaining part of this introduction and the new results inthis thesis can be of particular interest.

Introduction 5

1.3 Modern forms of Hardy type inequalities

Inspired by the development in the continuous case it is natural to ask:

Question 1: Find necessary and sufficient conditions on the weight se-quences {un}

∞n=1 and {vn}

∞n=1 such that(

∞∑n=1

(n∑

k=1

ak

)q

un

)1/q

≤ C

(∞∑

n=1

apnvn

)1/p

(1.6)

holds for all arbitrary non-negative sequences {an}∞n=1, 0 < p < ∞ and

0 < q < ∞.

The first contribution to answer Question 1 is due to K. F. Andersenand H. P. Heinig ([1], Theorem 4.1) who in 1983 found a result towards aweight characterization of (1.6) for the case p ≤ q. They showed that if1 ≤ p ≤ q < ∞ and

supn∈N

(∞∑

k=n

uk

)1/q ( n∑k=1

v1−p′

k

)1/p′

< ∞,

then the inequality (1.6) holds.Moreover, in 1985 H. P. Heinig ([23], Theorem 3.1) proved that if 1 ≤

q < p < ∞, 1r = 1

q − 1p and

B :=

⎛⎝ ∞∑

n=−∞

(∞∑

k=n

uk

)r/q ( n∑n=−∞

v1−p′

k

)r/q′

v1−p′

n

⎞⎠

1/r

< ∞,

then the inequality (1.6) holds with C ≤ q1/q(p′)1/q′B.In 1987-1991 G. Bennett presented a full characterization of the weighted

inequality (1.6), except for the case 0 < q < 1 < p < ∞, in his papers [5],[6] and [7]. The remaining case 0 < q < 1 < p < ∞ was characterized in1992 (English version was published in 1994) by M. S. Braverman and V. D.Stepanov [10].

Here we state the following important result mainly from the Bennettpaper [7]:

Theorem 1.1. (i) If 1 < p ≤ q < ∞, then (1.6) holds if and only if either

A1 := supN≥1

(N∑

k=1

v1−p′

k

)− 1p(

N∑k=1

uk

(k∑

n=1

v1−p′n

)q) 1q

< ∞, (1.7)



or

A2 := supN≥1

(∞∑

k=N

uk

) 1q(

N∑k=1

v1−p′

k

) 1p′

< ∞, (1.8)

or

A3 := supN≥1

(∞∑

k=N

uk

)−1q′

⎛⎝ ∞∑

k=N

v1−p′

k

(∞∑

n=k

un

)p′⎞⎠

1p′

< ∞. (1.9)

(ii) If 0 < p ≤ 1, p ≤ q < ∞, then (1.6) holds if and only if

A4 := supN≥1

(∞∑

k=N

uk

) 1q

v− 1

p

N < ∞.

(iii) If 1 < p < ∞, q < p, and 1r = 1

q − 1p , then (1.6) holds if and only if

A5 :=∞∑

n=1

⎛⎝un

(∞∑

k=n

uk

) rp(

n∑k=1

v1−p′

k

) r

p′

⎞⎠ < ∞.

(iv) If q < p = 1, then (1.6) holds if and only if

A6 :=∞∑

n=1

⎛⎝un

(∞∑

k=n

uk

) q

(1−q)

max1≤k≤n

vq

(q−1)

k

⎞⎠ < ∞.

(v) If 0 < q < 1 < p and 1r = 1

q − 1p , then (1.6) holds if and only if

A7 :=

⎛⎝ ∞∑

n=1

v1−p′

n

(∞∑

k=n

uk

) rq(

n∑k=1

v1−p′

k

) r

q′

⎞⎠

1r

< ∞.

Here and in the sequel p′ denotes the conjugate number of p, that is1p + 1

p′ = 1 (if p = 1, then p′ = ∞).For the proof of Theorem 1.1(i) see Bennett ([5], Theorem 2), where he

proved that the best C in (1.6) satisfies

A2 ≤ C, A1 ≤ q1/qA2, C ≤ p′A1, C ≤ qA3, A3 ≤(p′)1/p′

. (1.10)

The constants are sharp when q = p. These estimates (1.10) can be rewrittenas

max(q−1/qA1,(p′)−1/p′

A3) ≤ A2 ≤ C ≤ min(p′A1, qA3

)

Introduction 7

≤ min(p′q1/q, q

(p′)1/p′

)A2.

Another proof, with sharp constant, was given by G. Bennett in [7], Theorem9. He proved that for 1 < p ≤ q < ∞, we have

A2 ≤ C ≤(p′ + q)1/p′+1/q

(p′)1/q q1/p′A2.

In [5] and [7] the estimates of the best constant C in (1.6) were describedprecisely with each of the conditions (1.7) - (1.9), that is:

a) if 1 < p < q, then

A1 ≤ C ≤A1

(p − 1)1/q

⎛⎝ q − p

pβ(

pq−p , p(q−1)

q−p

)⎞⎠

q−p

pq

, (1.11)

where β = β(a, b) denotes the usual Beta function, and if q = p, then

A1 ≤ C ≤ p′A1, (1.12)

b) if 1 < p ≤ q, then

A2 ≤ C ≤

(1 +

q

p′

) 1q(

1 +p′

q

) 1p′

A2, (1.13)

c) if 1 < p < q < ∞, then

A3 ≤ C ≤ A3(q − 1)1p′

⎛⎝ q − p

(p − 1)qβ(

qq−p , q(p−1)

q−p

)⎞⎠

q−p

pq

, (1.14)

and if q = p, thenA3 ≤ C ≤ pA3. (1.15)

By comparing with what is known for weighted continuous Hardy-type in-equalities (see e.g. the book [35] and the references given there) we see thatthe correspondence to the conditions (1.8) and (1.7) in the continuous caseare the well-known Muckenhoupt condition (see [41]) and a condition byL.E. Persson and V. D. Stepanov [47] (see also G. Tomaselli [58] for the casep = q). The condition A3 is just the dual of the condition A1.

For the proof of (v) see M. S. Braverman and V. D. Stepanov [10], where

they proved that the best constant C in (1.6) satisfies(

qp′

)1/qA7 ≤ C ≤ A7.



It is worth mentioning that in 1999 L. Miclo [39] (see also [32]) adaptedthe Muckenhoupt proof [41] of the weighted Hardy inequality for integralson (0,∞) to prove the weighted discrete Hardy’s inequality on l2(N). Thenhe applied this weighted Hardy’s inequality in l2 to get an evaluation of thespectral gap for birth and death process on Z as well as the logarithmicSobolev constants on trees.

In Chapter 2 of this thesis we will contribute to the final solution ofQuestion 1 by proving that there are in fact infinite many conditions of thetype (1.7)-(1.9), which for the case 1 < p ≤ q < ∞ characterize (1.6) (seeTheorem 2.1). These scales of conditions are described by some parame-terintervals and in the endpoints of these intervals we have the conditions(1.7)-(1.9). Also in all these cases we obtain estimates of the type (1.11)-(1.15).

1.4 On discrete Hardy-type inequalities with ker-

nels

Inspired by the development in the continuous case (see the book [35] andthe literature therein) we will investigate the following general Hardy typeoperator

H :=

n∑k=1

dn,kak,

described by the kernel d := {dn,k}∞

n,k=1 , dn,k ≥ 0.

Precisely, we want to study the following question:

Question 2: Find necessary and sufficient conditions on the weightsequences {un}

∞n=1 and {vn}

∞n=1 such that

(∞∑

n=1

(n∑

k=1

dn,kak

)q

un

) 1q

≤ C

(∞∑

n=1

apnvn

) 1p

(1.16)


0 < q < ∞.

The first result in this direction was due to K. F. Anderson and H. P.Heinig ([1], Theorem 4.1), who proved a sufficient condition for (1.16) to holdfor the case 1 ≤ p ≤ q < ∞ with special non-negative kernels {dn,k}

∞

n,k=1that was assumed to be non-increasing in k and non-decreasing in n.

Introduction 9

In this thesis, we will contribute to the answer of Question 2 for the case1 < p ≤ q < ∞ by proving some scales of characterizations for the specialcase with product weight kernel, i.e. when dn,k = lnhk, n, k = 1, 2, ... (seeTheorem 5.1). Moreover, we prove a sufficient condition also for a generalkernel (see Theorem 5.2), which at least for a special case is also necessary(see Remark 5.3).

Recently G. Sinnamon [54] proved a remarkable result, which, in particu-lar, means that some Hardy type inequalities for non-increasing sequences infact are equivalent to the corresponding Hardy type inequalities for generalnon-negative sequences. Hence, they can be characterized by the same con-dition(s) (for such conditions see the books [35] and [45] but also some morerecent results e.g. those in [47], [59] and [60]). Partly guided by these resultsby Sinnamon we will prove also the surprising fact that we get the samecharacterizations in some of our results when restricting the set of positivesequences {an}

∞n=1 to the cone of non-increasing sequences {a∗n}

∞n=1 if, in

addition, the weight sequence {vn}∞n=1 is non-increasing (see Theorem 5.3).

The following result by M. Goldman [13] (see also [14]) is an interestinggeneralization of the discrete Hardy inequality in another direction (whichobviously contributes to the answer of Question 2 in a special case):

Theorem 1.2. Let 0 < r ≤ p ≤ q < ∞, σ = prp−r (for p = r, σ = ∞). Then

the inequality

⎛⎝ ∞∑

n=1

(n∑

k=1

(akϕk)r

) q

r

uqn

⎞⎠

1q

≤ C

(∞∑

n=1

(anvn)p) 1

p

(1.17)

for three weight sequences {ϕn} , {un} and {vn} (n = 1, 2, ...) holds if andonly if the (Muckenhoupt type) condition

BG := supn≥1

(n∑

k=1

(ϕkv

−1k

)σ) 1σ(

∞∑k=n

uqk

) 1q

< ∞ (1.18)

holds (with the usual maximum interpretation for the case p = r (σ = ∞)).Moreover, for the best constant C in (1.17) we have that C ≈ BG (but withoutexplicitly specifying the equivalence constants).

In particular, note that if ϕk = 1, k = 1, 2, ... and r = 1, then (1.17) isidentical to the inequality (1.6) and correspondingly (1.18) becomes exactlythe condition (1.8). This fact opens the possibility that also (1.17) can be



characterized by some scales of different conditions and not only by the singlecondition (1.18).

In this thesis we will prove a result (see Corollary 5.1) showing that theGoldman condition (1.18) in fact can be replaced by some scales of conditionsand also the estimate C ≈ BG can be given in a much more precise form.

1.5 On multidimensional discrete Hardy type in-

equalities

In view of Question 1, it is natural to ask the following question concerninga corresponding multidimensional version:

Question 3: Find necessary and sufficient conditions on the weight se-quences {un1n2} , {vn1n2} , n1, n2 = 1, 2, ..., such that

⎛⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∑

k1=1

n2∑k2=1

ak1k2

⎞⎠

q

un1n2

⎞⎠

1/q

≤ C

(∞∑

n1=1

∞∑n2=1

apn1n2

vn1n2

)1/p

(1.19)holds for all arbitrary non-negative sequences {ak1k2} , k1, k2 = 1, 2, ... , 0 <p < ∞ and 0 < q < ∞.

The corresponding question can be asked for the general n-dimensionalcase.

In the literature only a few results can be found on multidimensionaldiscrete Hardy type inequalities. Here we mention that Y. Rakotondratsimba[49] made a contribution by deriving a sufficient condition for the case 1 <p ≤ q < ∞ for (1.19) to hold but with some further restrictions on the weightsequences.

In this thesis we will contribute to the answer of Question 3 by, forthe case 1 < p ≤ q < ∞ and when {vn1n2} is a product weight, i.e.vn1n2 = vn1ωn2, n1, n2 ∈ Z+, proving some scales of necessary and sufficientconditions for (1.19) to hold (see Theorem 3.1 in Chapter 3). Moreover, wesucceeded to prove that the corresponding scales of conditions in n dimen-sions (n ∈ Z+) are necessary and sufficient for the general n-dimensionalHardy type inequality to hold (see Theorem 4.1 in Chapter 4).

The reason why we study only the case when the weights on the righthand side in (1.19) is of product type is that in the general case we needthree conditions to characterize (1.19) like in the continuous case (c.f. theremarkable result by E. Sawyer ([50], Theorem 1)) while in this special case

Introduction 11

we only need one condition, see our Theorem 3.1 in Chapter 3 and also aresult by Wedestig ([60], Theorem 5.2) for the continuous case. Anotherreason to study this special case is that it is sufficient to derive the corre-sponding limit two-dimensional Carleman type inequality also in the general(not product weight) case (see Section 1.7 and our Theorem 3.2 in Chapter3).

1.6 On Carleman’s inequality-the early history and

two elementary proofs

We now discuss the following inequality

∞∑n=1

(a1a2...an)1n < e

∞∑n=1

an, (1.20)

where an, n = 1, 2, ..., are positive numbers and∞∑

n=1an is convergent. In all

these cases in (1.20) the constant e is the best possible and equality can occuronly when all the an’s are zero. The inequality (1.20) is called Carleman’sinequality due to the fact that T. Carleman first presented and proved itin 1922 in [11]. Carleman discovered this inequality during his importantwork on quasi-analytical functions. For the corresponding continuous version(usually called the Pólya-Knopp inequality) see Appendix. Here we presenttwo elementary proofs, which both have influenced the further developmentof (1.20).

Proof 1: In (1.1) replace ak with a1p

k and, using the definition of thederivative (also note that x = elnx), we have that

(1

n

n∑k=1

a1p

k

)p

= exp ln

(1

n

n∑k=1

a1p

k

)p

= exp

⎛⎜⎜⎝[ln

n∑k=1

a1p

k − lnn∑

k=1

a0k

]1p

⎞⎟⎟⎠

= → exp

([D

(ln

n∑k=1

axk

)]x=0

), when p → ∞

= exp

⎛⎜⎜⎝⎡⎢⎢⎣

n∑k=1

axk ln ak

n∑k=1

axk

⎤⎥⎥⎦

x=0

⎞⎟⎟⎠ = exp

1

n

n∑k=1

ln ak =

(n∏

k=1

ak

) 1n

,



which shows that (1.1) leads to the non-strict inequality (1.20) since(

pp−1

)p→

e as p → ∞.

Hardy formulated the inequality (1.1) in 1920 (see [17]) and proved it notlater than 1925 (see [18]) but it looks like Carleman did not know about theinequality (1.1) at that time since he did not refer to the simple connectionthat holds according to the proof above.

It is interesting to note that by using a limit argument we could find that(1.1) implies (1.20). This means that (1.20) may be considered as a limitinequality for the scale (1.1) of Hardy’s inequalities. This was pointed outby Hardy in 1925 in his paper [18], page 156, but he indicated that it wasG. Pólya who made him aware of this interesting fact.

Note that this method does not automatically prove that we have strictinequality in (1.20) and this has to be proved separately (see e.g. our proof2 below).

Proof 2: We use the A-G mean inequality together with the fact that

(k + 1)k

k!=

(1 +

1

1

)(1 +

1

2

)2

...

(1 +

1

k

)k

< ek

and find that

∞∑n=1

an =

∞∑n=1

nan

∞∑k=n

1

k (k + 1)=

∞∑k=1

1

k (k + 1)

k∑n=1

nan

=

∞∑k=1

a1 + 2a2 + ... + kak

k (k + 1)>

∞∑k=1

1

(k + 1)

(k!

k∏n=1

an

)1/k

=∞∑

k=1

(k!

(k + 1)k

)1/k( k∏n=1

an

)1/k

≥1

e

∞∑k=1

(k∏

n=1

an

)1/k

.

The strict inequality holds since we cannot have equality at the same timein all terms of the inequalities. This can only occur if ak = c

k for some c > 0

but this cannot hold since∞∑

k=1

ak is convergent.

Carleman’s original proof is more complicated and it is based on La-grange’s multiplier method. Other proofs of the Carleman inequality can befound in [27], [12], [46], [40] and the recent theses of M. Johansson [26] andA. Wedestig [60] and the related references therein.

Introduction 13

1.7 Modern forms of Carleman type inequalities

As we have seen in our previous section the original Carleman inequality(1.20) may be regarded as a limit case of Hardy’s inequality (1.1) and, similarto Question 1, it is natural to ask the following question:

Question 4: Find necessary and sufficient conditions on the sequences{un}

∞n=1 and {vn}

∞n=1 such that the inequality

⎛⎝ ∞∑

n=1

(n∏

k=1

ak

) q

n

un

⎞⎠

1q

≤ C

(∞∑

n=1

apnvn

) 1p

(1.21)


0 < q < ∞.

Here we note that in 2001 H.P. Heinig, R. Kerman and M. Krbec [[24],Theorem 3.3(i)], made an important contribution by, for the case 0 < p ≤q < ∞, proving a necessary and sufficient condition for (1.21) to hold butwithout some explicit estimates of the best constant C.

In this thesis we will contribute to the answer of Question 4 by provingthat some such weight characterizations can be obtained just as limit re-sults of our weight characterizations of our discrete Hardy type inequalities.Moreover, we give precise lower and upper estimates of the best constant Cin (1.21) (see Theorem 2.2 in Chapter 2).

Concerning multidimensional Carleman type inequalities we raise thefollowing similar question:

Question 5: Find necessary and sufficient conditions on the sequences{un1n2}

∞n1,n2=1 and {vn1n2}

∞n1,n2=1 such that the two-dimensional Carleman

inequality

⎛⎜⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∏

k1=1

n2∏k2=1

ak1,k2

⎞⎠

q

n1n2

un1n2

⎞⎟⎠

1q

≤ C

(∞∑

n1=1

∞∑n2=1

apn1,n2

vn1n2

) 1p

(1.22)holds for all arbitrary non-negative sequences {an1n2}

∞n1,n2=1, 0 < p < ∞

and 0 < q < ∞.

The corresponding question can be asked for the general n-dimensionalcase.



In this thesis we will contribute to the answer of Question 5 by provingsome scales of necessary and sufficient conditions for (1.22) to hold in thecase 0 < p ≤ q < ∞. Moreover, we prove lower and upper estimates of thebest constant C in (1.22) (see Theorem 3.2 in Chapter 3). In fact, we haveeven succeded to prove some scales of necessary and sufficient conditions forthe general n-dimensional version of inequality (1.22) to hold in the case0 < p ≤ q < ∞ (see Theorem 4.2 in Chapter 4).

Note that Theorem 3.2 is a genuine generalization of a result of H.P.Heinig, R. Kerman and M. Krbec [[24], Theorem 3.3]. In fact, for the casep = q = 1 they proved the equivalence between (1.22) and our condition(3.4) but without some explicit estimate of the best constant C in (1.22) asours in (3.6). Moreover, our technique of proof (i.e. to characterize (1.22) byusing a limit argument in the corresponding Hardy inequality) is completelydifferent.

Chapter 2

Scales of weight characterizations

for discrete Hardy and Carleman

type inequalities

The aim of this Chapter is to prove that for the case 1 < p ≤ q < ∞ there areinfinite many conditions characterizing (1.6). In the endpoints of these scalesof conditions we obtain the Bennett conditions presented in Theorem 1.1.Another aim is to prove that some scales of characterizations for Carlemantype inequalities for the case 0 < p ≤ q < ∞ can be obtained as naturallimit results of our Hardy type inequalities. This chapter is organised in thefollowing simple way: The main results are stated and discussed in Section2.1, while the proofs can be found in Section 2.2. Finally, Section 2.3 isreserved for some concluding remarks.

2.1 Main results

Deeply inspired by some recent developments in the theory of continuousweighted Hardy-type inequalities in [36] (see also [59] and [60]) we introducethe following quantities:

A3(s) := supN≥1

(N∑

n=1

v1−p′

n

) s−1p

⎛⎝ ∞∑

n=N

un

(n∑

k=1

v1−p′

k

) q(p−s)p

⎞⎠

1q

, (2.1)

15



for 1 < s ≤ p.

A4(s) := supN≥1

(N∑

k=1

v1−p′

k

)−s⎛⎝ N∑

k=1

uk

(k∑

n=1

v1−p′

n

)q( 1p′

+s)⎞⎠

1q

, (2.2)

for 0 < s ≤ 1p .

A∗3(s) := sup

N≥1

(∞∑

n=N

un

) s−1q′

⎛⎜⎝ N∑

n=1

v1−p′

n

(∞∑

k=n

uk

) p′(q′−s)q′

⎞⎟⎠

1p′

, (2.3)

for 1 < s ≤ q′, and

A∗4(s) = sup

N≥1

(∞∑

n=N

un

)−s⎛⎝ ∞∑

n=N

v1−p′n

(∞∑

k=n

uk

)p′( 1q+s)⎞⎠

1p′

, (2.4)

for 0 < s ≤ 1q′ .

In the sequel, we let {un}∞n=1 and {vn}

∞n=1 denote fixed sequences with

non-negative elements and {an}∞n=1 denotes an arbitrary sequence with non-

negative elements (n ∈ Z+).

Our first main result reads:

Theorem 2.1. Let 1 < p ≤ q < ∞. Then (1.6) is equivalent with each ofthe following conditions (i) − (iv):

(i) A3(s) < ∞ for some s, 1 < s ≤ p,

(ii) A4(s) < ∞ for some s, 0 < s ≤ 1p ,

(iii) A∗3(s) < ∞ for some s, 1 < s ≤ q′,

(iv) A∗4(s) < ∞ for some s, 0 < s ≤ 1

q′ .

Moreover, for the best constant C in (1.6) we have the following esti-mates:

(i)∗

sup1<s<p

(s − 1

s

) 1p

A3(s) ≤ C ≤ inf1<s<p

A3(s)

(p − 1

p − s

) 1p′

,

Scales of weight characterizations for discrete Hardy

and Carleman type inequalities

17

(ii)∗

sup0<s< 1

p

(ps)1p A4(s) ≤ C ≤

1

(p − 1)1/q

⎛⎝ q − p

pB(

pq−p , p(q−1)

q−p

)⎞⎠

q−p

pq

A4(1

p),

if 1 < p < q < ∞, and

A4(1

p) ≤ C ≤ p′A4(

1

p),

if q = p.

(iii)∗

sup1<s<q′

(s − 1

s

) 1q′

A∗3(s) ≤ C ≤ inf

1<s<q′A∗

3(s)

(q′ − 1

q′ − s

) 1q

,

(iv)∗

sup0<s< 1

q′

(q′s)1q′ A∗

4(s) ≤ C ≤ (q − 1)1p′

⎛⎝ q − p

(p − 1)qB(

qq−p , q(p−1)

q−p

)⎞⎠

q−p

pq

A∗4(

1

q′),

if 1 < p < q < ∞, and

A∗4(

1

q′) ≤ C ≤ pA∗

4(1

q′),

if q = p.

Remark 2.1. (i)∗ In the limit case s = p in (2.1) we have A3(p) = A2

so that condition (i) coincides with (1.8). Correspondingly in this case wereplace (i)∗ by (1.13).

(ii)∗ In the limit case s = 1p in (2.2) we have A4(

1p) = A1 i.e. the

conditions (1.7) and (ii) are the same. Moreover, (ii)∗ is identical to theconditions (1.11)-(1.12).

(iii)∗ In the limit case s = q′ in (2.3) it yields that A∗3(q

′) = A2 i.e. theconditions (iii) and (1.8) coincides. Analogously as before, in this case wereplace (iii)∗ by (1.13).

(iv)∗ In the limit case s = 1q′ we see that A∗

4(1q′ ) = A∗

1 and we concludethat the conditions (1.9) and (iv) are equal. Moreover, also (iv)∗ coincideswith the conditions (1.14)-(1.15).



Remark 2.2. There are several relations between the quantities Ai(s), A∗i (s)

and Ai. See e.g. the proof of Theorem 2.1 and Remark 2.7. In fact, this istrue also outside the parameterintervals considered in Theorem 2.1 and thisfact gives further information concerning the results in this chapter.

By making a limit procedure as described in the introduction with theconditions in Theorem 2.1 we can derive some corresponding Carleman typeinequalities of the type (1.21). We point out two examples of this fact. Firstwe state the following improvement of a result of H.P. Heinig, R. Kermanand M. Krbec (see [24], Theorem 3.1(i)).

Theorem 2.2. Let 0 1 and for n = 1, 2, ..., vn > 0

and wn = un

(n∏

k=1

vk

)−q

np

. Then the inequality (1.21) holds for some finite

positive constant C if and only if

B3(s) := supN≥1

Ns−1

p

(∞∑

n=N

n−

sq

p wn

) 1q

< ∞. (2.5)

Moreover, for the best constant C in (1.21) we have the following estimates:

sups>1

((s − 1) es

(s − 1) es + 1

) 1p

B3(s) ≤ C ≤ infs>1

es−1

p B3(s). (2.6)

Remark 2.3. This result was stated and proved in [24] but without explicitestimates of the best constant as in (2.6). Moreover, their proof is differentfrom ours and it does not show that this result is just a limit case of weightcharacterizations of a scale of Hardy type inequalities.

Remark 2.4. The corresponding continuous version of Theorem 2.2 is dueto B. Opic and P. Gurka [44].

Using the condition (2.2) connected to A4(s) in the (Bennett) endpointcase s = 1

p we obtain the following result by A. Wedestig (see [60], Theorem8.1):

Corollary 2.1. Let 0 0. Then theinequality (1.21) holds for some finite constant C > 0 if and only if

B4 := supN≥1

N−1p

⎛⎝ N∑

n=1

un

(n∏

k=1

vk

)−q

np

⎞⎠

1q

< ∞. (2.7)



19

Moreover, for the best constant C in (1.21) we have the following estimate:

B4 ≤ C ≤ e1p B4. (2.8)

Remark 2.5. Corollary 2.1 may be regarded as a discrete version of thecorresponding continuous result by L.-E. Persson and V. D. Stepanov (see[47], Theorem 2). Note also that in the original Carleman inequality case(1.20) (p = 1 and un = vn = 1, n = 1, 2, ...) the estimate (2.8) is goodenough to obtain the best constant e.

2.2 Proofs

We will first state and prove the following elementary technical lemma thatwill be needed in the proofs in this and other chapters:

Lemma 2.1. Let Ak =k∑

n=1an, A0 = 0 and for n = 1, 2, ... let an > 0.

a) If 0 < λ < 1, then, for k = 1, 2, ... ,

λAλ−1k ak ≤ Aλ

k − Aλk−1 ≤ λAλ−1

k−1ak.

b) If λ < 0 or λ > 1 then, for k = 1, 2, ... ,

λAλ−1k−1ak ≤ Aλ

k − Aλk−1 ≤ λAλ−1

k ak.

Proof. For the case k = 1 all inequalities are trivial so we assume thatk ≥ 2. We define the function A(t) as piecewise linear between the points(k − 1, Ak−1) and (k,Ak) , k = 2, 3, ... . As A(t) is continuous on the interval[k − 1, k] and differentiable on the interval (k − 1, k) we can apply the mean-value theorem, and obtain that for some t0 ∈ (k − 1, k) :

Aλk − Aλ

k−1 = λAλ−1(t0)A′(t0). (2.9)

Since A′(t0) = ak and Ak−1 ≤ A(t0) ≤ Ak we find that

Aλ−1k−1 ≤ Aλ−1(t0) ≤ Aλ−1

k , λ > 1,

andAλ−1

k ≤ Aλ−1(t0) ≤ Aλ−1k−1, λ < 1,

and the proof follows by using (2.9).



Proof of Theorem 2.1:

I. (1.6)⇐⇒ (i). According to the Bennett result it is sufficient to considerthe case s < p. Put ap

nvn = bpn in (1.6), and we see that this inequality is

equivalent to

(∞∑

n=1

(n∑

k=1

bkv− 1

p

k

)q

un

) 1q

≤ C

(∞∑

n=1

bpn

) 1p

. (2.10)

Assume that (i) holds, and put Vn =n∑

k=1

v1−p′

k . By applying Hölder’s in-

equality, Lemma 2.1(a) with ak = v1−p′

k and Minkowski’s inequality we findthat

(∞∑

n=1

(n∑

k=1

bkv− 1

p

k

)q

un

) 1q

=

(∞∑

n=1

(n∑

k=1

bkVs−1

p

k V−

s−1p

k v− 1

p

k

)q

un

) 1q

≤

⎛⎝ ∞∑

n=1

(n∑

k=1

bpkV

s−1k

) q

p(

n∑k=1

V−

s−1p−1

k v1−p′

k

) q

p′

un

⎞⎠

1q

≤

(p − 1

p − s

) 1p′

⎛⎝ ∞∑

n=1

(n∑

k=1

bpkV

s−1k

) q

p

V(p−s)q

(p−1)p′

n un

⎞⎠

1q

=

(p − 1

p − s

) 1p′

⎛⎝ ∞∑

n=1

(n∑

k=1

bpkV

s−1k

) q

p

Vq(p−s)

pn un

⎞⎠

1q

≤

(p − 1

p − s

) 1p′

⎛⎝ ∞∑

k=1

bpkV

s−1k

(∞∑

n=k

Vq(p−s)

pn un

) p

q

⎞⎠

1p

≤

(p − 1

p − s

) 1p′

supk≥1

Vs−1

p

k

(∞∑

n=k

Vq(p−s)

pn un

)1q(

∞∑k=1

bpk

) 1p

=

(p − 1

p − s

) 1p′

A3(s)

(∞∑

k=1

bpk

) 1p

.

By taking infimum over s ∈ (1, p) and using the assumption we find that(1.6) holds and also the right hand side estimate in (i)∗ holds.

Now assume that (1.6) holds and thus (2.10) and for fixed N ∈ Z+ apply



21

the following test sequence

bpk =

{V −s

N v1−p′

k for k = 1, ...N

V −sk v1−p′

k for k = N + 1, ...

to (2.10).For the left hand side of (2.10) we have that

(∞∑

n=1

(n∑

k=1

bkv− 1

p

k

)q

un

) 1q

≥

(∞∑

n=N

(n∑

k=1

bkv− 1

p

k

)q

un

) 1q

=

(∞∑

n=N

(N∑

k=1

V− s

p

N v1−p′

k +

n∑k=N+1

V− s

p

k v1−p′

k

)q

un

) 1q

≥

(∞∑

n=N

(V

p−s

p

N + V−sp

n

n∑k=N+1

v1−p′

k

)q

un

) 1q

=

(∞∑

n=N

(V

p−s

p

N + V−sp

n (Vn − VN )

)q

un

) 1q

≥

(∞∑

n=N

(V

p−s

p

N + Vp−s

pn − V

−sp

N VN

)q

un

) 1q

=

(∞∑

n=N

Vq(p−s)

pn un

) 1q

.

For the right hand side of (2.10), by applying Lemma 2.1(b) we find that

(∞∑

n=1

bpn

) 1p

=

(N∑

n=1

V −sN v1−p′

k +∞∑

n=N+1

V −sk v1−p′

k

) 1p

≤

(V 1−s

N +1

s − 1V 1−s

N

) 1p

≤

(1 +

1

s − 1

) 1p

V1−s

p

N

=

(s

s − 1

) 1p

V1−s

p

N .



Hence, (2.10) implies that

(∞∑

n=N

Vq(p−s)

pn un

) 1q

≤ C

(s

s − 1

) 1p

V1−s

p

N ,

i.e. that (s − 1

s

) 1p

Vs−1

p

N

(∞∑

n=N

Vq(p−s)

pn un

) 1q

≤ C.

We conclude that (i) and the left hand estimate of (i)∗ hold.Summing up, we have proved that (1.6) is equivalent to (i) and that (i)∗

holds.II. (1.6)⇔(ii). Assume that (ii) holds. By using the fact that {Vn} is a

non-decreasing sequence we get

A4(s) : = supN≥1

V −sN

(N∑

k=1

ukVq( 1

p′+s)

k

) 1q

= supN≥1

V −sN

(N∑

k=1

ukVq(s− 1

p)

k V qk

) 1q

≥ supN≥1

V− 1

p

N

(N∑

k=1

ukVqk

) 1q

= A1.

Hence, according to the Bennett result, (1.6) holds and also the righthand side inequality in (ii). Now assume that (1.6) holds and for fixedN ∈ Z

+ apply the following test sequence {ak} with

ak =

{V

s− 1p

k v1−p′

k for k = 1, ...N0 for k = N + 1, ...

to the inequality (1.6). For the left hand side we have

(∞∑

n=1

(n∑

k=1

ak

)q

un

) 1q

≥

(N∑

n=1

(n∑

k=1

ak

)q

un

) 1q

=

(N∑

n=1

(n∑

k=1

Vs− 1

p

k v1−p′

k

)q

un

) 1q



23

≥

(N∑

n=1

Vq(s− 1

p

)k

(n∑

k=1

v1−p′

k

)q

un

) 1q

=

(N∑

n=1

Vq(s+ 1

p′)

n un

)1q

.

Moreover, by applying Lemma 2.1(a) with b = ps and an = v1−p′n the sum on

the right hand side of (1.6) can be estimated as follows:

(∞∑

n=1

apnvn

) 1p

=

(N∑

n=1

V ps−1n v1−p′

n

) 1p

≤

(1

ps

) 1p

V sN .

We conclude that(ps)

1p A4(s) ≤ C,

which means that (ii) and also the left hand side inequality in (ii)∗ hold.Hence, we have proved that (1.21) is equivalent to (ii) and also that (ii)∗

holds.III. (1.6)⇔(iii). This equivalence follows by using what we have proved

in I and standard duality in the context of Hardy type inequalities (see [[5],p. 407-408] and [[35], p. 13-14]). The estimates in (iii)∗ also follow from thisargument.

IV. (1.6)⇔(iv)∗. According to the duality principle mentioned in III thisequivalence together with (iv)∗ is more or less equivalent to what we provedin II. The proof is complete.�


We replace an by bnv− 1

pn , n = 1, 2, ... in (1.21) and see that this inequality

is equivalent to

(∞∑

n=1

(n∏

k=1

bk

)q

wn

) 1q

≤ C

(∞∑

n=1

bpn

) 1p

. (2.11)

By using Theorem 2.1(i) with un = wnn−q, and vn = 1, n = 1, 2, ..., for1 < s 0

Ns−1

p

(∞∑

n=N

wnn−

sq

p

) 1q

< ∞ (2.13)

and the best possible constant can be estimated as follows:

sup1<s<p

(s − 1

s

) 1p

B3(s) ≤ C ≤ inf1<s<p

B3(s)

(p − 1

p − s

) 1p′

. (2.14)

Now we replace bk in (2.12) with bαk , 0 < α < p, and after that we replace p

with pα and q with q

α in (2.12)-(2.14) and we find that for 1 < s < pα it yields

that ⎛⎝ ∞∑

n=1

(1

n

n∑k=1

bαk

) q

α

wn

⎞⎠

1q

≤ Cα

(∞∑

n=1

bpn

) 1p

(2.15)

holds if and only if B3(s) < ∞. Moreover, according to the right hand sideestimate in (2.14), we have that the best constant Cα in (2.15) satisfies

Cα ≤ inf1<s< p

α

B3(s)

(p − α

p − αs

) p−α

αp

. (2.16)

We also note that(1

n

n∑k=1

bαk

) 1α

↓

(n∏

k=1

bk

) 1n

, as α −→ 0+

(the scale of power means converges to the geometric mean) and

limα−→0+

(p − α

p − αs

) p−α

αp

= es−1

p .

Thus, by letting α → 0+ in (2.16) we get the right hand side of the estimate(2.6).

The left hand side of the estimate (2.6) can be obtained directly byapplying the test sequence

bpk =

{N

− 1p for k = 1, ...N

e−sN s−1k−s for k = N + 1, ...

to (2.11). The proof follows by combining these two facts.�



25

Proof of Corollary 2.1:

Again we note that (1.21) is equivalent with (2.11), where wn =

un

(n∏

k=1

vk

)−q

np

. By using Theorem 2.1(ii) in the special case s = 1p (the

Bennett case), un = wnn−q and vn = 1, n = 1, 2, ... we find that the inequal-ity (

∞∑n=1

(1

n

n∑k=1

bk

)q

wn

) 1q

≤ C

(∞∑

n=1

bpn

) 1p

(2.17)

holds for some finite constant C > 0 if and only if

B4 := supN>0

N− 1

p

(N∑

n=1

wn

) 1q

< ∞ (2.18)

and the best constant C in (2.17) can be estimated in the following way:

B4 ≤ C ≤ p′B4. (2.19)

Next we replace bn with bαn, 0 < α < p in (2.17) and after that we replace

p with pα and q with q

α in (2.17)-(2.19). We find that

⎛⎝ ∞∑

n=1

(1

n

n∑k=1

bαk

) q

α

wn

⎞⎠

1q

≤ Cα

(∞∑

n=1

bpn

) 1p

(2.20)

holds for all non-negative sequences {bk}∞k=1 if and only if

Bα4 := sup

N>0N−α

p

(N∑

n=1

wn

)αq

< ∞.

We note that the best possible constant Cα in (2.20) can be estimated asfollows:

B4 ≤ Cα ≤

(p

p − α

) 1α

B4.

Moreover, it yields that

(1

n

n∑k=1

bαk

) q

α

↓

(n∏

k=1

bk

) q

n



and (p

p − α

) 1α

−→ e1p as α −→ 0+.

This proves that if B4 < ∞, then (1.21) and the right hand side estimatein (2.8) hold.

For the lower bound in (2.8) we can for fixed N ∈ Z+ use the simple test

sequence

bpn :=

{1, for n = 1, 2, ...N0, for n = N + 1, ....

in (2.17). The proof follows by combining these two facts.�

2.3 Concluding remarks

Remark 2.6. Remark 2.1 shows that the criteria of Bennett all appear inthe endpoints of the scales of criteria we have presented in Theorem 2.1. Butwe have also seen that a part of our proof of Theorem 2.1 is depending on justthese crucial endpoint criteria by Bennett. This way of thinking is related toideas in interpolation theory.

Remark 2.7. There are many relations between the quantities Ai and A∗i defined

in Section 1.2 of Chapter 1 and Section 2.1 of this chapter. For example inthe proof of Theorem 2.1 we proved that if 0 < s ≤ 1

p , then

A1 ≤ A4(s). (2.21)

Note that in a similar way we can prove that (2.21) holds in the reversedirection if s > 1

p . Moreover, the following holds

A2 ≤ A3(s) ≤

(p − 1

s − 1

) 1q

A2, 1 < s < p. (2.22)

In fact, the left hand side inequality easily follows in the following way:

A3(s) = supN≥1

(∞∑

k=N

ukVq(p−s)

p

k

) 1q

Vs−1

p

N

≥ supN≥1

(∞∑

k=N

uk

) 1q

Vp−s

p

N Vs−1

p

N



27

≥ supN≥1

(∞∑

k=N

uk

) 1q

Vp−1

p

N = A2.

Here, as usual, VN =N∑

n=1v1−p′n .

Furthermore, in view of (1.8), we have

A3(s) = supN≥1

(∞∑

k=N

ukVq(p−s)

p

k

) 1q

Vs−1

p

N

≤ A2 supN≥1

⎛⎝ ∞∑

k=N

uk

(∞∑

n=k+1

un

)−p−s

p−1

⎞⎠

1q (

∞∑n=N

un

)− 1q

(s−1p−1

)

≤ A2

(p − 1

s − 1

) 1q

(∞∑

n=N

un

) 1q

(s−1p−1

) (∞∑

n=N

un

)− 1q

(s−1p−1

)

= A2

(p − 1

s − 1

) 1q

.

Here we have used the fact that if B∗N =

∞∑n=N

bn, then by using the mean value

theorem we find that∞∑

k=N

bk

(B∗

k+1

)α≤

(B∗

N)α+1

α+1 . Hence (2.22) is proved.

Remark 2.8. Using Theorem 2.1 we can get better estimates of the bestconstant C (= the operator norm of the corresponding averaging operator)in (1.6) then using just the estimate in the endpoint cases in the Bennettcriteria. We only give one concrete example of this fact.

Example 2.1. In the Hardy inequality (1.6)

(∞∑

n=1

(n∑

k=1

ak

)q

un

) 1q

≤ C

(∞∑

n=1

apnvn

) 1p

,

let un = n1−p if n < 100 and 0 elsewhere, vn = 100 − n if n < 100 and 0elsewhere. In particular, for p = q = 2,we have the following estimates ofthe best constant C in (1.6):

i) With A1 and the estimate (1.12):

C � 1.457287035,



(here � means that the estimate is correct up to 9 digits)ii) With A2 and the estimate (1.13):

C � 1.396344359,

iii) With A3(s) and the estimate (i)∗:

C � 1.133830792.

Remark 2.9. Theorem 2.1 shows that the conditions (i)-(iv) are in factequivalent. It would be of interest to find a direct proof of this equivalence.

Remark 2.10. We point out the fact that Theorem 2.1 can partly be general-ized to two dimensions. More precisely we will consider the two-dimensionalversion in the next chapter.

Chapter 3

Scales of weight characterizations

for some two-dimensional discrete

Hardy and Carleman type

inequalities

The aim of this Chapter is to prove the two-dimensional versions of Theorem2.1 and Theorem 2.2. These results are an improvement of some recentresults in [43]. This chapter is organised in the following simple way: Themain results are stated and discussed in Section 3.1, while the proofs can befound in Section 3.2.

3.1 Main results

Our main results in this chapter read:

Theorem 3.1. Let 1 < p ≤ q < ∞, s1, s2 ∈ (1, p) and let {an1n2} ,n1, n2 = 1, 2, ... be an arbitrary non-negative sequence. Moreover, assumethat {un1,n2} , n1, n2 = 1, 2, ... , {vn1} , n1 = 1, 2, ..., and {ωn2} , n2 = 1, 2, ...,

29



are fixed non-negative sequences. Then the inequality

⎛⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∑

k1=1

n2∑k2=1

ak1k2

⎞⎠

q

un1n2

⎞⎠

1/q

≤ C

(∞∑

n1=1

∞∑n2=1

apn1n2

vn1ωn2

)1/p

(3.1)(3.1) holds for some finite constant C in (3.1) if and only if

A (s1, s2) := supN1,N2≥1

⎛⎝ N1∑

k1=1

v1−p′

k1

⎞⎠

s1−1p⎛⎝ N2∑

k2=1

ω1−p′

k2

⎞⎠

s2−1p

× (3.2)

⎛⎜⎝ ∞∑

n1=N1

∞∑n2=N2

un1,n2

⎛⎝ n1∑

k1=1

v1−p′

k1

⎞⎠

q(p−s1)p

⎛⎝ n2∑

k2=1

ω1−p′

k2

⎞⎠

q(p−s2)p

⎞⎟⎠

1/q

< ∞.

Moreover, if C is the best constant in (3.1), then

sup1<s1,s2<p

(s1 − 1

s1

) 1p(

s2 − 1

s2

) 1p

A (s1, s2) ≤ C (3.3)

≤ inf1<s1,s2 1. Moreover, let {an1,n2} ,n1, n2 = 1, 2, ..., be an arbitrary non-negative sequence and let {un1,n2} and{vn1,n2} be fixed non-negative sequences, where vn1,n2 > 0, n1, n2 = 1, 2, .... Then the Carleman type inequality (1.22) holds for some finite constantC > 0 if and only if

B (s1, s2) := supN1,N2≥1

Ns1−1

p

1 Ns2−1

p

2

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

n−

s1q

p

1 n−

s2q

p

2 wn1,n2

⎞⎠

1q

< ∞,

(3.4)where

wn1,n2 = un1,n2

⎛⎝ n1∏

k1=1

n2∏k2=1

vk1,k2

⎞⎠

−q

n1n2p

. (3.5)

Scales of weight characterizations for some two-

dimensional discrete Hardy and Carleman type inequali-

ties

31

Moreover, for the best constant C in (1.22), we have the following estimates:

sups1,s2>1

(es1 (s1 − 1)

es1 (s1 − 1) + 1

) 1p(

es2 (s2 − 1)

es2 (s2 − 1) + 1

) 1p

B (s1, s2) (3.6)

≤ C ≤ infs1,s2>1

es1+s2−2

p B (s1, s2) .

Remark 3.1. Theorem 3.2 is a generalization of the result of H. P. Heinig,R. Kerman and M. Krbec [24], where they gave a necessary and sufficientcondition for the case p = q = 1. In one dimension, Theorem 3.2 is adiscrete correspondence of a result by B. Opic and P. Gurka [44].

Remark 3.2. Note that Theorem 3.1 is formulated only with a productweight sequence {vn1ωn2} on the right hand side in (3.1). This is crucialbecause as in the continuous case (see the Sawyer result [50]) it seems impos-sible to avoid to have three conditions when characterizing (3.1) with vn1ωn2

replaced by a general weight vn1n2 . However, as we see in Theorem 3.2 inthe limit case one condition is sufficient to characterize (1.22) for a generalweight vn1n2. A similar observation for the continuous case can be found inthe PhD thesis of A. Wedestig [60].

3.2 Proofs

Proof of Theorem 3.1:For simplicity we only present the proof for the case when ωk = vk,

k = 1, 2, but the proof in the general case is completely the same.Put bp

n1,n2 = apn1,n2vn1ωn2 in (3.1). Then (3.1) is equivalent to⎛

⎝ ∞∑n1=1

∞∑n2=1

⎛⎝ n1∑

k1=1

n2∑k2=1

bk1,k2v− 1

p

k1v− 1

p

k2

⎞⎠

q

un1,n2

⎞⎠

1/q

≤ C

(∞∑

n1=1

∞∑n2=1

bpn1,n2

)1/p

.

(3.7)

Assume that (3.2) holds and let Vni=

ni∑k=1

v1−p′

ki, i = 1, 2. Applying

Hölder’s inequality, Lemma 2.1(a) with ak = v1−p′

k and λ = λi = p−si

p−1 ,i = 1, 2 (note that 0 < λ < 1) and Minkowski’s inequality to the left handside of (3.7) we find that

⎛⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∑

k1=1

n2∑k2=1

bk1,k2v− 1

p

k1v− 1

p

k2

⎞⎠

q

un1,n2

⎞⎠

1q



=

⎛⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∑

k1=1

n2∑k2=1

bk1,k2Vs1−1

p

k1V

s2−1p

k2V

−s1−1

p

k1

V−

s2−1p

k2v− 1

p

k1v− 1

p

k2

)q

un1,n2

) 1q

≤

⎛⎜⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∑

k1=1

n2∑k2=1

bpk1,k2

V s1−1k1

V s2−1k2

⎞⎠

q

p

×

⎛⎝ n1∑

k1=1

V−

(s1−1)p′

p

k1v−

p′

p

k1

⎞⎠

q

p′⎛⎝ n2∑

k2=1

V−

(s2−1)p′

p

k2v−

p′

p

k2

⎞⎠

q

p′

un1,n2

⎞⎟⎠

1q

≤

(p − 1

p − s1

) 1p′(

p − 1

p − s2

) 1p′

⎛⎜⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∑

k1=1

n2∑k2=1

bpk1,k2

V s1−1k1

V s2−1k2

⎞⎠

q

p

×

⎛⎝ n1∑

k1=1

V

(p−s1p−1

)k1

− V

(p−s1p−1

)k1−1

⎞⎠

q

p′

×

⎛⎝ n2∑

k2=1

V

(p−s2p−1

)k2

− V

(p−s2p−1

)k2−1

⎞⎠

q

p′

un1,n2

⎞⎟⎠

1q

=

(p − 1

p − s1

) 1p′(

p − 1

p − s2

) 1p′

×⎛⎜⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∑

k1=1

n2∑k2=1

bpk1,k2

V s1−1k1

V s2−1k2

⎞⎠

q

p

×

V

(p−s1

p

)q

n1 V

(p−s2

p

)q

n2 un1,n2

) 1q

≤

(p − 1

p − s1

) 1p′(

p − 1

p − s2

) 1p′

⎛⎝ ∞∑

k1=1

∞∑k2=1

bpk1,k2

V s1−1k1

V s2−1k2



ties

33

×

⎛⎝ ∞∑

n1=k1

∞∑n2=k2

V

(p−s1

p

)q

n1 V

(p−s2

p

)q

n2 un1,n2

⎞⎠

p

q

⎞⎟⎠

1p

≤

(p − 1

p − s1

) 1p′(

p − 1

p − s2

) 1p′

supk1, k2>0

Vs1−1

p

k1V

s2−1p

k2⎛⎝ ∞∑

n1=k1

∞∑n2=k2

V

(p−s1

p

)q

n1 V

(p−s2

p

)q

n2 un1,n2

⎞⎠

1q⎛⎝ ∞∑

k1=1

∞∑k2=1

bpk1,k2

⎞⎠

1p

=

(p − 1

p − s1

) 1p′(

p − 1

p − s2

) 1p′

A3 (s1, s2)

(∞∑

n1=1

∞∑n2=1

bpn1,n2

) 1p

. (3.8)

Hence, by taking infimum over s1, s2 ∈ (1, p) , (3.7) and, thus, (3.1) holdswith a constant C satisfying the right hand side inequality in (3.3).

Next, assume that (3.1) and, thus, (3.7) holds and apply the followingtest sequence to (3.7):

bpk1k2

:=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

V −s1N1

v1−p′

k1V −s2

N2v1−p′

k2if

{k1 = 1, ..., N1,k2 = 1, ..., N2

V −s1N1

v1−p′

k1V −s2

k2v1−p′

k2if

{k1 = 1, ..., N1,k2 = N2 + 1...,

V −s1k1

v1−p′

k1V −s2

N2v1−p′

k2if

{k1 = N1 + 1, ...,k2 = 1, ..., N2

V −s1k1

v1−p′

k1V −s2

k2v1−p′

k2if

{k1 = N1 + 1, ...,k2 = N2 + 1, ... .

(Here N1 and N2 are fixed natural numbers).For the right hand side of (3.7), by applying Lemma 2.1(b) we obtain

that(∞∑

n1=1

∞∑n2=1

bpn1,n2

)1/p

=

(N1∑

n1=1

V −s1N1

v1−p′

n1

N2∑n2=1

V −s2N2

v1−p′

n2

+

N1∑n1=1

V −s1N1

v1−p′n1

∞∑n2=N2+1

V −s2n2

v1−p′n2

+

N2∑n2=1

V −s2N2

v1−p′

n2

∞∑n1=N1+1

V −s1n1

v1−p′

n1



+∞∑

n1=N1+1

V −s1n1

v1−p′

n1

∞∑n2=N2+1

V −s2n2

v1−p′

n2

⎞⎠

1/p

≤

(V 1−s1

N1V 1−s2

N2+

(1

s2 − 1

)V 1−s1

N1V 1−s2

N2(1

s1 − 1

)V 1−s1

N1V 1−s2

N2

+

(1

s1 − 1

)(1

s2 − 1

)V 1−s1

N1V 1−s2

N2

) 1p

=

(1 +

1

s1 − 1+

1

s2 − 1+

(1

s1 − 1

)(1

s2 − 1

)) 1p

×V1−s1

p

N1V

1−s2p

N2.

Hence,(∞∑

n1=1

∞∑n2=1

bpn1,n2

)1/p

≤

(s1

s1 − 1

) 1p(

s2

s2 − 1

) 1p

V1−s1

p

N1V

1−s2p

N2. (3.9)

For the left hand side of (3.7) we have⎛⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∑

k1=1

n2∑k2=1

bk1,k2v− 1

p

k1v− 1

p

k2

⎞⎠

q

un1,n2

⎞⎠

1/q

≥

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

⎛⎝ n1∑

k1=1

n2∑k2=1

bk1,k2v− 1

p

k1v− 1

p

k2

⎞⎠

q

un1,n2

⎞⎠

1/q

=

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

⎛⎝ N1∑

k1=1

V−

s1p

N1v1−p′

k1

N2∑k2=1

V−

s2p

N2v1−p′

k2

+

N1∑k1=1

V−

s1p

N1v1−p′

k1

n2∑k2=N2+1

V−

s2p

k2v1−p′

k2

+

n1∑k1=N1+1

V−

s1p

k1v1−p′

k1

N2∑m2=1

V−

s2p

N2v1−p′

k2

+

n1∑k1=N1+1

V−

s1p

k1v1−p′

k1

n2∑k2=N2+1

V−

s2p

k2v1−p′

k2

⎞⎠

q

un1n2

⎞⎠

1q



ties

35

≥

⎛⎝ ∞∑

n1=k1

∞∑n2=k2

(V

p−s1p

N1V

p−s2p

N2+

+Vp−s1

p

N1V

−s2p

n2

n2∑k2=N2+1

v1−p′

k2+ V

p−s2p

N2V

−s1p

n1

n1∑k1=N1+1

v1−p′

k1

+ V−

s1p

n1 V−

s2p

n2

n1∑k1=N1+1

v1−p′

k1

n2∑k2=N2+1

v1−p′

k2

⎞⎠

q

un1,n2

⎞⎠

1q

=

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

(V

p−s1p

N1V

p−s2p

N2+ V

p−s1p

N1V

−s2p

n2 (Vn2 − VN2)

+Vp−s2

p

N2V

−s1p

n1 (Vn1 − VN1)

+V−

s1p

n1 (Vn1 − VN1)V−

s2p

n2 (Vn2 − VN2)

)q

un1n2

) 1q

=

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

(V

p−s1p

N1V

p−s2p

N2+ V

p−s1p

N1V

p−s2p

n2 − Vp−s1

p

N1V

−s2p

n2 VN2

+Vp−s2

p

N2V

p−s1p

n1 − Vp−s2

p

N2V

−s1p

n1 VN1

+

(V

p−s1p

n1 − V−

s1p

n1 VN1

)(V

p−s2p

n2 − V−

s2p

n2 VN2

))q

un1n2

)1q

≥

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

(V

p−s1p

N1V

p−s2p

N2+ V

p−s1p

N1V

p−s2p

n2 − Vp−s1

p

N1V

−s2p

N2VN2

+Vp−s2

p

N2V

p−s1p

n1 − Vp−s2

p

N2V

−s1p

N1VN1

+

(V

p−s1p

n1 − V−

s1p

N1VN1

)(V

p−s2p

n2 − V−

s2p

N2VN2

))q

un1n2

)1q

=

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

Vq(p−s1)

pn1 V

q(p−s2)p

n2 un1n2

⎞⎠

1q

. (3.10)



Hence, according (3.7), (3.9) and (3.10),⎛⎝ ∞∑

n1=N1

∞∑n2=N2

Vq(p−s1)

pn1 V

q(p−s2)p

n2 un1,n2

⎞⎠

1q

≤ C

(s1

s1 − 1

) 1p(

s2

s2 − 1

) 1p

V1−s1

p

N1V

1−s2p

N2,

so that (s1 − 1

s1

) 1p(

s2 − 1

s2

) 1p

Vs1−1

p

N1V

s2−1p

N2×

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

Vq(p−s1)

pn1 V

q(p−s2)p

n2 un1,n2

⎞⎠

1q

≤ C.

Hence, by taking supremum over N1, N2 ≥ 1 and supremum over s1, s2 ∈(1, p) we conclude that (3.2) and the left hand side of the estimate (3.3) hold.

Summing up, we have proved that (3.1) is equivalent to (3.2) and that(3.3) holds. The proof is complete.�

Proof of Theorem 3.2:Assume that (3.4) holds. Replace ap

n1,n2vn1,n2 with bpn1,n2 in (1.22). Then

(1.22) is equivalent to⎛⎜⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∏

k1=1

n2∏k2=1

bk1,k2

⎞⎠

q

n1n2

wn1,n2

⎞⎟⎠

1q

≤ C

(∞∑

n1=1

∞∑n2=1

bpn1,n2

) 1p

,

(3.11)where wn1,n2 is defined by (3.5).

In Theorem 3.1 put un1,n2 = wn1,n2 n−q1 n−q

2 and vn1 = ωn2 = 1, n1, n2 =1, 2..., and we find that⎛⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ 1

n1n2

n1∑k1=1

n2∑k2=1

ak1,k2

⎞⎠

q

wn1,n2

⎞⎠

1/q

≤ C

(∞∑

n1=1

∞∑n2=1

apn1,n2

)1/p

(3.12)holds if and only if

A (s1, s2) := supN1,N2≥1

Ns1−1

p

1 Ns2−1

p

2

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

n−

s1q

p

1 n−

s2q

p

2 wn1,n2

⎞⎠

1q

< ∞

(3.13)



ties

37

and the best possible constant C in (3.12) can be estimated as follows:

sup1<s1,s2<p

(s1 − 1

s1

) 1p(

s2 − 1

s2

) 1p

A (s1, s2) ≤ C ≤ (3.14)

inf1<s1,s2<p

(p − 1

p − s1

) 1p′(

p − 1

p − s2

) 1p′

A (s1, s2) .

Now we replace ak1,k2 by bαk1,k2

in (3.12) with 0 < α < p and after that wereplace p by p

α and q by qα in (3.12) - (3.14) and since 1 < p

α ≤ qα < ∞, in

view of Theorem 3.1, we find that if 1 < s1, s2 < pα , then the inequality

⎛⎜⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ 1

n1n2

n1∑k1=1

n2∑k2=1

bαk1,k2

⎞⎠

q

α

wn1,n2

⎞⎟⎠

1/q

≤ Cα

(∞∑

n1=1

∞∑n2=1

bpn1,n2

)1/p

(3.15)holds for bk1,k2 > 0 if and only if A (s1, s2) < ∞. Now A

1α (s1, s2) = B (s1, s2)

for all α > 0 and wn1,n2 is defined by (3.5), so according to the right handside of estimate (3.14), we have that the best possible constant Cα in (3.15)satisfies

Cα ≤ inf1<s1,s2< p

α

(p − α

p − αs1

) p−α

pα(

p − α

p − αs2

) p−α

pα

B (s1, s2) .

Moreover, we note that

⎛⎝ 1

n1n2

n1∑k1=1

n2∑k2=1

bαk1k2

⎞⎠

q

α

↓

⎛⎝ n1∏

k1=1

n2∏k2=1

bk1k2

⎞⎠

q

n1n2

, as α −→ 0+


(p − α

p − αs1

)p−α

αp(

p − α

p − αs2

) p−α

αp

−→ es1+s2−2

p as α −→ 0+.

Hence, it follows that (3.11) and, thus, (1.22) hold with a constant satisfyingthe upper estimate in (3.6).

Next we assume that (1.22) holds. To prove the lower bound of the best



constant C in (3.6) we apply the following test sequence to (3.11):

bpk1,k2

:=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

N−11 N−1

2 if

{k1 = 1, ..., N1,k2 = 1, ..., N2

N−11 e−s2N s2−1

2 k−s22 if

{k1 = 1, ..., N1,k2 = N2 + 1...,

N−12 e−s1N s1−1

1 k−s11 if

{k1 = N1 + 1, ...,k2 = 1, ..., N2

e−(s1+s2)N s1−11 k−s1

1 N s2−12 k−s2

2 if

{k1 = N1 + 1, ...,k2 = N2 + 1... .

(Here N1 and N2 are fixed natural numbers).Applying the test sequence to the right hand side of (3.11), we have(

∞∑n1=1

∞∑n2=1

bpn1,n2

)1/p

=

⎛⎝ N1∑

n1=1

N2∑n2=1

N−11 N−1

2 +

N1∑n1=1

∞∑n2=N2+1

N−11 e−s2N s2−1

2 n−s22

+

N2∑n2=1

∞∑n1=N1+1

N−12 e−s1N s1−1

1 n−s11

+∞∑

n1=N1+1

∞∑n2=N2+1

e−(s1+s2)N s1−11 n−s1

1 N s2−12 n−s2

2

⎞⎠

1p

=

⎛⎝1 + e−s2N s2−1

2

∞∑n2=N2+1

n−s22 + e−s1N s1−1

1

∞∑n1=N1+1

n−s11

+ e−(s1+s2)N s1−11 N s2−1

2

∞∑n1=N1+1

n−s11

∞∑n2=N2+1

n−s22

⎞⎠

1p

. (3.16)

For s > 1,

n−s ≤

n∫n−1

x−sdx =

[x1−s

1 − s

]n

n−1

=(n − 1)1−s − n1−s

s − 1.

and∞∑

n=N+1

n−s ≤

∞∫N

x−sdx =

[x1−s

1 − s

]∞N

=N1−s

s − 1, (3.17)



ties

39

where n = ni, N = Ni, s = si, i = 1, 2.By using (3.17) in (3.16) we obtain(∞∑

n1=1

∞∑n2=1

bpn1,n2

)1/p

≤

(1 +

e−s2

s2 − 1+

e−s1

s1 − 1+

e−(s1+s2)

(s1 − 1) (s2 − 1)

) 1p

=

(1 + es1 (s1 − 1)

es1 (s1 − 1)

) 1p(

1 + es2 (s2 − 1)

es2 (s2 − 1)

) 1p

.(3.18)

Moreover, the left hand side of (3.11), can be estimated as follows:⎛⎜⎝ ∞∑

n1=1

∞∑n2=1

⎛⎝ n1∏

k1=1

n2∏k2=1

bpk1,k2

⎞⎠

q

pn1n2

wn1,n2

⎞⎟⎠

1q

≥

⎛⎜⎝ ∞∑

n1=N1

∞∑n2=N2

⎛⎝ n1∏

k1=1

n2∏k2=1

bpk1,k2

⎞⎠

q

pn1n2

wn1,n2

⎞⎟⎠

1q

=

⎛⎜⎝ ∞∑

n1=N1

∞∑n2=N2

⎛⎝exp

n1∑k1=1

n2∑k2=1

ln bpk1,k2

⎞⎠

q

pn1n2

wn1,n2

⎞⎟⎠

1q

. (3.19)

Applying the test sequence to the inner summation of (3.19), we get that

n1∑k1=1

n2∑k2=1

ln bpk1k2

=

N1∑k1=1

N2∑k2=1

ln(N−1

1 N−12

)(3.20)

+

N1∑k1=1

n2∑k2=N2+1

ln(N−1

1 e−s2N s2−12 k−s2

2

)

+

n1∑k1=N1+1

N2∑k2=1

ln(N−1

2 e−s1N s1−11 k−s1

1

)

+

n1∑k1=N1+1

n2∑k2=N2+1

ln(e−(s1+s2)N s1−1

1 N s2−12 k−s1

1 k−s22

).

Using the mean value theorem, we find that

n∑k=N+1

ln k =

n+1∑k=N+2

k ln (k − 1) −

n∑k=N+1

k ln k



=

n∑k=N+1

k [ln (k − 1) − ln k] − (N + 1) ln N + (n + 1) ln n

=n∑

k=N+1

−k

k∫k−1

1

xdx + (n + 1) ln n − (N + 1) ln N

≤

n∑k=N+1

(−1) + (n + 1) ln n − (N + 1) ln N

so that

−

n∑k=N+1

ln k ≥ (n − N) − (n + 1) ln n + (N + 1) ln N. (3.21)

We now consider (3.20) termwise. Below we represent the first, second,third and forth terms of the right hand side of (3.20) by

∑1,∑

2,∑

3 and∑4, respectively. We find that

∑1

=

N1∑k1=1

N2∑k2=1

ln(N−1

1 N−12

)= −N1N2 ln N1 − N1N2 ln N2,

and, in view of (3.21),

∑2

=

N1∑k1=1

n2∑k2=N2+1

ln(N−1

1 e−s2N s2−12 k−s2

2

)= −N1(n2 − N2) ln N1 + s2N1(N2 − n2) +

(s2 − 1)N1(n2 − N2) ln N2 − s2N1

n2∑k2=N2+1

ln k2

≥ −N1(n2 − N2) ln N1 − s2N1(n2 − N2) +

(s2 − 1)N1(n2 − N2) ln N2 + N1s2 (n2 − N2)

−N1s2 (n2 + 1) ln n2 + N1s2 (N2 + 1) ln N2,

= −N1(n2 − N2) ln N1 + (s2 − 1)N1(n2 − N2) ln N2

−N1s2 (n2 + 1) ln n2 + N1s2 (N2 + 1) ln N2.

Similarly, we obtain that

∑3

=

n1∑k1=N1+1

N2∑k2=1

ln(N−1

2 e−s1N s1−11 k−s1

1

)



ties

41

= −N2(n1 − N1) ln N2 + s1N2(N1 − n1) +

(s1 − 1)N2(n1 − N1) ln N1 − s1N2

n1∑k1=N1+1

ln k1

≥ −N2(n1 − N1) ln N2 − s1N2(n1 − N1) +

−N2s1 (n1 + 1) ln n1 + N2s1 (N1 + 1) ln N1

= −N2(n1 − N1) ln N2 + (s1 − 1)N2(n1 − N1) ln N1 +

−N2s1 (n1 + 1) ln n1 + N2s1 (N1 + 1) ln N1,

and

∑4

=

n1∑k1=N1+1

n2∑k2=N2+1

ln(e−(s1+s2)N s1−1

1 N s2−12 k−s1

1 k−s22

)= −(n1 − N1)(n2 − N2) (s1 + s2) + (s1 − 1) (n1 − N1) (n2 − N2) ln N1

+(s2 − 1) (n1 − N1) (n2 − N2) ln N2 − s1(n2 − N2)

n1∑k1=N1+1

ln k1

−s2(n1 − N1)

n2∑k2=N2+1

ln k2

≥ −(n1 − N1)(n2 − N2) (s1 + s2) + (s1 − 1) (n1 − N1) (n2 − N2) ln N1

+(s2 − 1) (n1 − N1) (n2 − N2) ln N2

s1(n2 − N2)(n1 − N1) − s1(n2 − N2)(n1 + 1) ln n1

+s1(n2 − N2)(N1 + 1) ln N1 + s2(n1 − N1)(n2 − N2)

−s2(n1 − N1)(n2 + 1) ln n2 + s2(n1 − N1)(N2 + 1) ln N2

= (s1 − 1) (n1 − N1) (n2 − N2) ln N1 +

(s2 − 1) (n1 − N1) (n2 − N2) ln N2 − s1 (n2 − N2) (n1 + 1) ln n1

+s1 (n2 − N2) (N1 + 1) lnN1 − s2 (n1 − N1) (n2 + 1) ln n2

+s2 (n1 − N1) (N2 + 1) lnN2.

Summing up and simplifying, we have, for any n1 ≥ N1, n2 ≥ N2,

n1∑k1=1

n2∑k2=1

ln bpk1k2

=∑

1+∑

2+∑

3+∑

4

≥ (n1n2 + n2) ln n−s11 + (n1n2 + n1) lnn−s2

2 +

+ (n2s1 + (s1 − 1) n1n2) ln N1

+ (n1s2 + (s2 − 1) n1n2) ln N2



≥ n1n2 ln n−s11 + n1n2 ln n−s2

2 + n1n2 ln N s1−11 +

n1n2 ln N s2−12

= n1n2 ln(n−s1

1 n−s22 N s1−1

1 N s2−12

). (3.22)

Hence⎛⎜⎝ ∞∑

n1=N1

∞∑n2=N2

⎛⎝exp

n1∑k1=1

n2∑k2=1

ln bpk1,k2

⎞⎠

q

pn1n2

wn1,n2

⎞⎟⎠

1q

≥

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

(exp ln

(n−s1

1 n−s22 N s1−1

1 N s2−12

)n1n2) q

pn1n2 wn1,n2

⎞⎠

1q

=

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

n−

qs1p

1 n−

qs2p

2 Nq(s1−1)

p

1 Nq(s2−1)

p

2 wn1,n2

⎞⎠

1q

= Ns1−1

p

1 Ns2−1

p

2

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

n−

qs1p

1 n−

qs2p

2 wn1,n2

⎞⎠

1q

. (3.23)

Combining (3.23), (3.19), (3.18) and (1.22) we get that

Ns1−1

p

1 Ns2−1

p

2

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

n−

qs1p

1 n−

qs2p

2 wn1,n2

⎞⎠

1q

≤ C

(1 + es1 (s1 − 1)

es1 (s1 − 1)

) 1p(

1 + es2 (s2 − 1)

es2 (s2 − 1)

) 1p

,

that is, (es1 (s1 − 1)

1 + es1 (s1 − 1)

) 1p(

es2 (s2 − 1)

1 + es2 (s2 − 1)

) 1p

× (3.24)

Ns1−1

p

1 Ns2−1

p

2

⎛⎝ ∞∑

n1=N1

∞∑n2=N2

n−

qs1p

1 n−

qs2p

2 wn1,n2

⎞⎠

1q

≤ C.

Thus, by taking supremum over N1, N2 ≥ 1 and supremum over s1, s2 ∈(1, p) we find that (3.4) and also the left hand side inequality (3.6) hold.Hence, we have proved that (1.22) is equivalent to (3.4) and that (3.6) holds.The proof is complete.�

Chapter 4

The general multidimensional

results

In this chapter we will follow up the investigations in the previous chapters byproving the corresponding n-dimensional results. This chapter is organisedin the following simple way: The main results are stated and discussed inSection 4.1, while the proofs can be found in Section 4.2.

4.1 Main results

Our main results read:

Theorem 4.1. Let M ∈ Z+, 1 < p ≤ q < ∞, s1, ..., sM ∈ (1, p) andlet {an1,...,nM

} , n1, ..., nM = 1, 2, ..., be an arbitrary non-negative sequence.Moreover, let {un1,...,nM

} , n1, ..., nM = 1, 2, ..., and {vi,nk}∞nk=1 , i = 1, 2, ...,M

be fixed non-negative weight sequences. Then the inequality

⎛⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ n1∑

k1=1

...

nM∑kM =1

ak1,...,kM

⎞⎠

q

un1,...,nM

⎞⎠

1q

(4.1)

≤ C

(∞∑

n1=1

...

∞∑nM =1

apn1,...,nM

v1,n1...vM,nM

) 1p

43




A3 (s1, ..., sn) : = supN1,...,NM≥1

⎛⎝ N1∑

k1=1

v1−p′

1,k1

⎞⎠

s1−1p

...

⎛⎝ NM∑

kM=1

v1−p′

M,kM

⎞⎠

sM−1

p

×

⎛⎜⎝ ∞∑

n1=N1

...∞∑

nM =NM

un1,...,nM

⎛⎝ n1∑

k1=1

v1−p′

1,k1

⎞⎠

q(p−s1)p

×

⎛⎝ n2∑

k2=1

v1−p′

2,k2

⎞⎠

q(p−s2)p

...

⎛⎝ nM∑

kM =1

v1−p′

M,kM

⎞⎠

q(p−sM )p

⎞⎟⎟⎠

1q

< ∞. (4.2)

Moreover, if C is the best constant in (4.1), then

sup1<s1,...,sM<p

(s1 − 1

s1

) 1p

...

(sM − 1

sM

) 1p

A3 (s1, ..., sM ) (4.3)

≤ C ≤ inf1<s1,...,sM 1, and let{an1,...,nM

} , n1, ..., nM = 1, 2, ..., be an arbitrary non-negative sequence. More-over, let {un1,...,nM

} , n1, ..., nM = 1, 2, ..., and {vn1,...,nM} , n1, ..., nM =

1, 2, ..., be fixed non-negative sequences, where vn1,...,nM> 0, n1, ..., nM =

1, 2, ...,. Then the Carleman type inequality⎛⎜⎝ ∞∑

n1=1

...∞∑

nM =1

⎛⎝ n1∏

k1=1

...

nM∏kM =1

ak1,...,kM

⎞⎠

q

n1...nM

un1,...,nM

⎞⎟⎠

1q

(4.4)

≤ C

(∞∑

n1=1

...

∞∑nM =1

apn1,...,nM

vn1,...,nM

) 1p


B (s1, ..., sM ) := supN1,...,NM≥1

Ns1−1

p

1 ...NsM−1

p

M × (4.5)

The general multidimensional results 45

⎛⎝ ∞∑

n1=N1

...

∞∑nM =NM

n−

qs1p

1 ...n−

qsMp

M wn1,...,nM

⎞⎠

1q

< ∞,

where

wn1,...,nM:= un1,...,nM

⎛⎝ n1∏

k1=1

...

nM∏kM =1

vk1,...,kM

⎞⎠

−q

pn1...nM

. (4.6)

Moreover, for the best constant C in (4.4) we have the following estimates:

sups1,..,sM>1

(es1 (s1 − 1)

1 + es1 (s1 − 1)

) 1p

...

(esM (sM − 1)

1 + esM (sM − 1)

) 1p

× (4.7)

B (s1, ..., sM ) ≤ C ≤ infs1,...,sM>1

es1+...+sM−M

p B (s1, ..., sM ) .

4.2 Proofs

Proof of Theorem 4.1:The result is known for M = 1 (see Theorem 2.1, Chapter 2 also [ [43],

Theorem 1]) and M = 2 (see Theorem 3.1) so we assume that M = 3, 4, ... .Put bp

n1,...,nM= ap

n1,...,nMv1,n1...vM,nM

in (4.1). Then (4.1) is equivalent to

⎛⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ n1∑

k1=1

...

nM∑kM =1

bk1,...,kMv− 1

p

1,k1...v

− 1p

M,kM

⎞⎠

q

un1,...,nM

⎞⎠

1q

(4.8)

≤ C

(∞∑

n1=1

...∞∑

nM =1

bpn1,...,nM

) 1p

.

Assume that (4.2) holds and let Vm,nj=

nj∑ki=1

v1−p′

m,kifor i, j,m = 1, 2, ...,M,

Vm,0 = 0, m = 1, 2, ...,M . Applying Hölder’s inequality, Lemma 2.1(a) with

ak = aki= v1−p′

m,ki, and λ = λi = p−si

p−1 , i,m = 1, 2, ...M (note that 0 < λ < 1)and Minkowski’s inequality to the left hand side of (4.8) we have that

⎛⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ n1∑

k1=1

...

nM∑kM =1

bk1,...,kMv− 1

p

1,k1...v

− 1p

M,kM

⎞⎠

q

un1,...,nM

⎞⎠

1q



=

⎛⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ n1∑

k1=1

...

nM∑kM =1

bk1,...,kMV

s1−1p

1,k1...V

sM−1

p

M,kM×

× V−

s1−1p

1,k1...V

−sM−1

p

M,kMv− 1

p

1,k1...v

− 1p

M,kM

)q

un1,...,nM

) 1q

≤

⎛⎜⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ n1∑

k1=1

...

nM∑kM =1

bpk1,...,kM

V s1−11,k1

...V sM−1M,kM

⎞⎠

q

p

⎛⎝ n1∑

k1=1

V−

(s1−1)p′

p

1,k1v1−p′

1,k1

⎞⎠

q

p′

...

⎛⎝ nM∑

kM=1

V−

(sM−1)p′

p

M,kMv1−p′

M,kM

⎞⎠

q

p′

un1,...,nM

⎞⎟⎠

1q

≤

(p − 1

p − s1

) 1p′

...

(p − 1

p − sM

) 1p′

×⎛⎜⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ n1∑

k1=1

...

nM∑kM =1

bpk1,...,kM

V s1−11,k1

...V sM−1M,kM

⎞⎠

q

p

×

⎛⎝ n1∑

k1=1

(V

(p−s1p−1

)1,k1

− V

(p−s1p−1

)1,k1−1

)⎞⎠

q

p′⎛⎝ n2∑

k2=1

(V

(p−s2p−1

)2,k2

− V

(p−s2p−1

)2,k2−1

)⎞⎠

q

p′

×

... ×

⎛⎝ nM∑

kM=1

(V

(p−sMp−1

)M,kM

− V

(p−sMp−1

)M,kM−1

)⎞⎠q

p′

un1,...,nM

⎞⎟⎠

1q

=

(p − 1

p − s1

) 1p′

...

(p − 1

p − sM

) 1p′

⎛⎜⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ n1∑

k1=1

...

nM∑kM =1

bpk1,...,kM

V s1−11,k1

...V sM−1M,kM

⎞⎠

q

p

×

V

(p−s1p−1

)q

p′

1,n1...V

(p−sMp−1

)q

p′

M,nMun1,...,nM

) 1q

≤

(p − 1

p − s1

) 1p′

...

(p − 1

p − sM

) 1p′

⎛⎝ ∞∑

k1=1

...∞∑

kM =1

bpk1,...,kM

V s1−11,k1

...V sM−1M,kM

×


⎛⎝ ∞∑

n1=k1

...

∞∑nM =kM

Vq(p−s1)

p

1,n1...V

q(p−sM )p

M,nMun1...nM

⎞⎠

p

q

⎞⎟⎠

1p

≤

(p − 1

p − s1

) 1p′

...

(p − 1

p − sM

) 1p′

×

supk1,...,kM>0

Vs1−1

p

1,k1...V

sM−1

p

M,kM

⎛⎝ ∞∑

n1=k1

...∞∑

nM =kM

Vq(p−s1)

p

1,n1...V

q(p−sM )p

M,nMun1...nM

⎞⎠

1q

×

⎛⎝ ∞∑

k1=1

...

∞∑kM =1

bpk1,...,kM

⎞⎠

1p

=

(p − 1

p − s1

) 1p′

...

(p − 1

p − sM

) 1p′

A3 (s1, ..., sM )

(∞∑

n1=1

...

∞∑nM =1

bpn1...nM

) 1p

.

Hence, by taking infimum over s1, ..., sM ∈ (1, p) , (4.8) and, thus, (4.1)holds with a constant C satisfying the right hand side inequality in (4.3).

Now assume (4.1) and, thus, (4.8) holds. Similar to the two-dimensionalcase we consider the following test sequence:

bpk1,...,kM

:=

{M∏i=1

(V −si

m,Niv1−p′

m,kifor ki = 1, 2, ...Ni

V −si

m,kiv1−p′

m,kifor ki = Ni + 1, ...

, m = 1, ...,M

).

(4.9)We claim that

n1∑k1=1

..

nM∑kM=1

bk1,...,kMv− 1

p

1,k1...v

− 1p

M,kM≥ V

p−s1p

1,n1...V

p−sMp

M,nM. (4.10)

Using the calculations in (3.10) we see that (4.10) holds for M = 2.We assume that (4.10) holds with M replaced by M − 1, M ≥ 3. By

using this assumption, the relation −1p + 1−p′

p = 1 − p′ and the fact thatVM,n is nondecreasing, we get

n1∑k1=1

...

nM∑kM =1

bk1,...,kMv− 1

p

1,k1...v

− 1p

M,kM

≥ Vp−s1

p

1,n1...V

p−sM−1p

M−1,nM−1

⎛⎝ NM∑

kM=1

V−

sMp

M,NMv1−p′

M,kM+

nM∑kM=NM+1

V−

sMp

M,kMv1−p′

M,kM

⎞⎠



≥ Vp−s1

p

1,n1...V

p−sM−1p

M−1,nM−1

(V

p−sMp

M,NM+ V

−sMp

M,nM(VM,nM

− VM,NM)

)

= Vp−s1

p

1,n1...V

p−sM−1p

M−1,nM−1

(V

p−sMp

M,nM+ V

p−sMp

M,NM− V

−sMp

M,nMVM,NM

)

≥ Vp−s1

p

1,n1...V

p−sM−1p

M−1,nM−1

(V

p−sMp

M,nM+ V

p−sMp

M,NM− V

−sMp

M,NMVM,NM

)

= Vp−s1

p

1,n1...V

p−sM−1p

M−1,nM−1V

p−sMp

M,nM.

Hence, by the induction axiom (4.10) holds for each M, M ≥ 2. We concludethat

⎛⎝ ∞∑

n1=N1

...

∞∑nM =NM

⎛⎝ n1∑

k1=1

...

nM∑kM =1

bk1,...,kMv−

1p

1,k1...v

−1p

M,kM

⎞⎠

q

un1,...,nM

⎞⎠

1q

=

⎛⎝ ∞∑

n1=N1

...

∞∑nM =NM

(V

q(p−s1)p

1,n1...V

q(p−sn)p

M,nM

)un1 ,...,nM

⎞⎠

1q

. (4.11)

Moreover, applying the test sequence (4.9) to the right hand side of (4.8)and using Lemma 2.1(b) with λ = λi = 1− si < 0, i = 1, 2, ...,M, we obtainthat

(∞∑

n1=1

∞∑n2=1

...∞∑

nM =1

bpn1,...,nM

) 1p

=

⎛⎝⎛⎝ N1∑

n1=1

V −s11,N1

v1−p′

1,n1+

∞∑n1=N1+1

V −s11,n1

v1−p′

1,n1

⎞⎠ ×

... ×

⎛⎝ NM∑

nM=1

V −sn

M,nMv1−p′

M,nM+

∞∑nM=NM +1

V −sn

M,nMv1−p′

M,nM

⎞⎠⎞⎠

1p

≤

((V 1−s1

1,N1+

1

s1 − 1V 1−s1

1,N1

)...

(V 1−sM

M,NM+

1

sM − 1V 1−sM

M,NM

)) 1p

=

(s1

s1 − 1

) 1p

...

(sM

sM − 1

) 1p

V1−s1

p

1,N1...V

1−sMp

M,NM. (4.12)


Hence, according (4.11), (4.12) and (4.8),

⎛⎝ ∞∑

n1=N1

...

∞∑nM =NM

(V

q(p−s1)p

1,n1...V

q(p−sM )p

M,nM

)un1 ,...,nM

⎞⎠

1q

≤ C

(s1

s1 − 1

) 1p

...

(sM

sM − 1

) 1p

V1−s1

p

1,N1...V

1−sMp

M,NM

so that (s1 − 1

s1

) 1p

...

(sM − 1

sM

) 1p

Vs1−1

p

1,N1...V

sM−1

p

M,NM×

⎛⎝ ∞∑

n1=N1

...∞∑

nM =NM

(V

q(p−s1)p

1,n1...V

q(p−sM )p

M,nM

)un1 ,...,nM

⎞⎠

1q

≤ C.

Thus, by taking supremum over N1, ..., NM ≥ 1 and supremum overs1, ..., sM ∈ (1, p) we find that (4.2) and the left hand side of the estimate(4.3) hold. Hence, we have proved that (4.1) is equivalent to (4.2) and that(4.3) holds. The proof is complete.�

Proof of Theorem 4.2:Assume that (4.5) holds. The result is known for M = 1 (see Proposition

2.2 in Chapter 2 and also [43]) and M = 2 (see Theorem 3.2 in Chapter 3)so we assume that M = 3, 4, ... . Put bp

n1,...,nM= ap

n1,...,nMvn1,...,nM

in (4.4).Then (4.4) is equivalent to

⎛⎜⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ n1∏

k1=1

...

nM∏kM =1

bk1,...,kM

⎞⎠

q

n1...nM

wn1,...,nM

⎞⎟⎠

1q

(4.13)

≤ C

(∞∑

n1=1

...

∞∑nM =1

bpn1,...,nM

) 1p

,

where wn1,...,nMis defined in (4.6).

Now we use Theorem 4.1 with un1,...,nM= wn1,...,nM

n−q1 ...n−q

M and v1,n1 =v2,n2 = ... = vM,nM

= 1, and obtain

⎛⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ 1

n1...nM

n1∑k1=1

...

nM∑kM =1

ak1,...,kM

⎞⎠

q

wn1,...,nM

⎞⎠

1q

(4.14)



≤ C

(∞∑

n1=1

...

∞∑nM =1

apn1,...,nM

) 1p

holds for some finite constant C if and only if

A (s1, ..., sM ) := supN1,...,NM>0

Ns1−1

p

1 ...NsM−1

p

M × (4.15)

⎛⎝ ∞∑

n1=N1

...

∞∑nM =NM

(n−

qs1p

1 ...n−

qsMp

M

)wn1,...,nM

⎞⎠

1q

< ∞

and the best possible constant C in (4.14) can be estimated as follows:

sup1<s1,...,sM<p

(s1 − 1

s1

) 1p

...

(sM − 1

sM

) 1p

A (s1, ..., sM ) (4.16)

≤ C ≤ inf1<s1,...,sM<p

A3 (s1, ..., sM )

(p − 1

p − s1

) 1p′

...

(p − 1

p − sM

) 1p′

.

Now we replace ak1,...,kMby bα

k1,...,kMin (4.14) with 0 < α < p and after

that we replace p by pα and q by q

α in (4.14)-(4.16) and we find that for1 < s1, ..., sM < p

α we have that

⎛⎜⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ 1

n1...nM

n1∑k1=1

...

nM∑kM =1

bαk1,...,kM

⎞⎠

q

α

wn1,...,nM

⎞⎟⎠

1q

(4.17)

≤ Cα

(∞∑

n1=1

...

∞∑nM =1

bpn1,...,nM

) 1p

holds for bk1,...,kM> 0 if and only if

Aα(s1, ..., sM ) := supN1,...,NM>0

Nα(

s1−1p

)1 , ..., N

α(

sM−1

p

)M ×

⎛⎝ ∞∑

n1=N1

...

∞∑nM =NM

(n

−qs1p

1 ...n−qsM

p

M

)wn1,...,nM

⎞⎠

αq

< ∞.


Since now A1αα = B for all α > 0 and wn1,...,nM

is defined in (4.6), theupper estimate of the best possible constant Cα in (4.17) can be estimatedas follows:

Cα ≤ inf1<s1,...,sM<p

B (s1, ..., sM )

(p − α

p − αs1

) p−α

pα

...

(p − α

p − αsM

) p−α

pα

.

Moreover, we note that⎛⎝ 1

n1...nM

n1∑k1=1

...

nM∑kM =1

bαk1,...,kM

⎞⎠

q

α

↓

⎛⎝ n1∏

k1=1

...

nM∏kM =1

bk1...kM

⎞⎠

q

n1...nM


(p − α

p − αs1

) p−α

pα

...

(p − α

p − αsM

) p−α

pα

−→ es1+...+sM−M

p as α −→ 0+.

Thus, we get

C ≤ infs1,...,sM>1

B (s1, ..., sM ) es1+...+sM−M

p . (4.18)

Hence, it follows that (4.4) holds with a constant satisfying the upper esti-mate in (4.7).

Next, we assume that (4.4) and, thus, (4.13) holds and apply the followingtest sequence to (4.13):

bpk1,...,kM

:=

{M∏i=1

(N−1

i for ki = 1, ..., Ni

e−siN si−1i k−si

i for ki = Ni + 1, ...

). (4.19)

We apply the test sequence (4.19) to the right hand side of (4.13) anduse estimate (3.17) in the calculation. Then we have, since bp

k1,...,kMis of

product type, that(∞∑

n1=1

...

∞∑nM =1

bpn1,...,nM

) 1p

=

⎛⎝⎛⎝ N1∑

n1=1

N−11 +

∞∑n1=N1+1

e−s1N s1−11 n−s1

1

⎞⎠× ...

... ×

⎛⎝ NM∑

nM =1

N−1M +

∞∑nM =NM+1

e−sM N sM−1M n−sM

M

⎞⎠⎞⎠

1p



=

⎛⎝⎛⎝1 + e−s1N s1−1

1

∞∑n1=N1+1

n−s11

⎞⎠×

... ×

⎛⎝1 + e−sM N sM−1

M

∞∑nM =NM+1

n−sM

M

⎞⎠⎞⎠

1p

≤

(1 +

e−s1

s1 − 1

) 1p

...

(1 +

e−sM

sM − 1

) 1p

=

(1 + es1 (s1 − 1)

es1 (s1 − 1)

) 1p

....

(1 + esM (sM − 1)

esM (sM − 1)

) 1p

. (4.20)

Moreover, the left hand side of (4.13) can be estimated as follows:

⎛⎜⎝ ∞∑

n1=1

...

∞∑nM =1

⎛⎝ n1∏

k1=1

...

nM∏kM =1

bpk1,...,kM

⎞⎠

q

pn1...nM

wn1,...,nM

⎞⎟⎠

1q

≥

⎛⎜⎝ ∞∑

n1=N1

...

∞∑nM =NM

⎛⎝ n1∏

k1=1

...

nM∏kM =1

bpk1,...,kM

⎞⎠

q

pn1...nM

wn1,...,nM

⎞⎟⎠

1q

=

⎛⎜⎝ ∞∑

n1=N1

...

∞∑nM =NM

⎛⎝exp

n1∑k1=1

...

nM∑kM =1

ln bpk1,...,kM

⎞⎠

q

pn1...nM

wn1,...,nM

⎞⎟⎠

1q

.(4.21)

By following the argumentation in the proof of Theorem 3.2 we see that itis sufficient to prove that

n1∑k1=1

...

nM∑kM =1

ln bpk1,...,kM

≥ n1...nM ln(n−s1

1 ...n−sM

M N s1−11 ...N sM−1

M

). (4.22)

We have already proved that this formula holds for M = 2 (see (3.22)).Moreover, by mathematical induction as in the proof of Theorem 3.2, weeasily find that it is true in general.

Using (4.21) and (4.22), we find that the left hand side of (4.13) can beestimated as follows:⎛

⎜⎝ ∞∑n1=1

...

∞∑nM =1

⎛⎝ n1∏

k1=1

...

nM∏kM =1

bpk1,...,kM

⎞⎠

q

pn1...nM

wn1,...,nM

⎞⎟⎠

1q


≥ Ns1−1

p

1 ...NsM−1

p

M

⎛⎝ ∞∑

n1=N1

...

∞∑nM =NM

n−

qs1p

1 ...n−

qsMp

M wn1,...,nM

⎞⎠

1q

(4.23)

Now, by combining (4.23), (4.20) and (4.4), we obtain that

Ns1−1

p

1 ...NsM−1

p

M

⎛⎝ ∞∑

n1=N1

...∞∑

nM =NM

n−

qs1p

1 ...n−

qsMp

M wn1,...,nM

⎞⎠

1q

≤ C

(1 + es1 (s1 − 1)

es1 (s1 − 1)

) 1p

....

(1 + esM (sM − 1)

esM (sM − 1)

) 1p

,

that is (es1 (s1 − 1)

1 + es1 (s1 − 1)

) 1p

....

(esM (sM − 1)

1 + esM (sM − 1)

) 1p

×

Ns1−1

p

1 ...NsM−1

p

M

⎛⎝ ∞∑

n1=N1

...

∞∑nM =NM

n−

qs1p

1 ...n−

qsMp

M wn1,...,nM

⎞⎠

1q

≤ C.

Hence, by taking supremum over N1, ..., NM ≥ 1 and supremum overs1, ..., sM ∈ (1, p) we find that (4.5) and the left hand side of (4.7) hold. Theproof is complete.�



Chapter 5

Weight characterizations for the

discrete Hardy inequality with

kernel

In this chapter we consider (1.16) and also make a new proof of a somewhatmore precise version of Theorem 1.2. For kernels of product type a weightcharacterization of (1.16) is given, thus generalizing a previous result of M.Goldman (the case dn,k = bk). A scale of sufficient conditions is provedfor the general case. This chapter is organised as follows: In Section 5.1we present the main results together with some related remarks and theproofs are given in Section 5.2. Finally, some concluding remarks and openquestions can be found in Section 5.3.

5.1 Main results

As before, in this chapter {an}∞n=1 denotes an arbitrary (weight) sequence of

non-negative numbers. Moreover, {un}∞n=1 , {vn}

∞n=1 , {ln}

∞n=1 and {hk}

∞k=1

denote fixed weight sequences and d = {dn,k}∞

n,k=1 is a non-negative discretekernel, i.e. a sequence of non-negative numbers.

First we state the following generalization and unification of Theorem1.2 and Theorem 2.1:

55



Theorem 5.1. Let 1 < p ≤ q < ∞ and consider the kernel d = {dn,k}∞

n,k=1 ,where dn,k = lnhk, n, k = 1, 2, ... . Then the inequality

(∞∑

n=1

(n∑

k=1

dn,kak

)q

un

) 1q

≤ C

(∞∑

n=1

apnvn

) 1p

(5.1)

holds if and only if

D1(s) := supN≥1

(N∑

n=1

hp′

n v1−p′

n

)s⎛⎜⎝ ∞∑

n=N

lqnun

(n∑

k=1

hp′

k v1−p′

k

)q(

1p′−s)⎞⎟⎠

1q

< ∞,

(5.2)for some s, 0 < s ≤ 1

p′ , or

D2(s) := supN≥1

(N∑

n=1

hp′n v1−p′

n

)−s⎛⎝ N∑

n=1

lqnun

(n∑

k=1

hp′

k v1−p′

k

)q( 1p′

+s)⎞⎠

1q

< ∞,

(5.3)for some s, 0 < s < 1

p , or

D3(s) := supN≥1

(∞∑

n=N

lqnun

)s⎛⎜⎝ N∑

n=1

hp′

n v1−p′

n

(∞∑

k=n

lqkuk

)p′(

1q−s)⎞⎟⎠

1p′

< ∞,

(5.4)for some s, 1 < s ≤ 1

q , or

D4(s) := supN≥1

(∞∑

n=N

lqnun

)−s⎛⎝ ∞∑

n=N

hp′n v1−p′

n

(∞∑

k=n

lqkuk

)p′( 1q+s)⎞⎠

1p′

< ∞,

(5.5)for some s, 0 < s ≤ 1

q′ .


sup0<s< 1

p′

(ps

ps + 1

) 1p

D1(s) ≤ C ≤ inf0<s< 1

p′

D1(s)

(p − 1

p (1 − s) − 1

) 1p′

, (5.6)

Weight characterizations for the discrete Hardy inequal-

ity with kernel

57

sup0<s< 1

p

(ps)1p D2(s) ≤ C ≤

1

(p − 1)1/q

⎛⎝ q − p

pβ(

pq−p , p(q−1)

q−p

)⎞⎠

q−p

pq

D2(1

p) (5.7)

if p < q and

D2(1

p) ≤ C ≤ p′D2(

1

p)

if p = q,

sup0<s< 1

q

(q′s

q′s + 1

) 1q′

D3(s) ≤ C ≤ inf0<s< 1

q

D3(s)

(q′ − 1

q′ (1 − s) − 1

) 1q

, (5.8)

and

sup0<s< 1

q′

(q′s)1q′ D4(s) ≤ C ≤ (q − 1)

1p′

⎛⎝ q − p

(p − 1)qβ(

qq−p , q(p−1)

q−p

)⎞⎠

q−p

pq

D4(1

q′)

(5.9)if p < q and

D4(1

q′) ≤ C ≤ pD4(

1

q′)

if p = q.

In particular, we have the following more precise version of Theorem 1.2:

Corollary 5.1. Let 0 < r ≤ p ≤ q < ∞, σ = prp−r and (for p = r, σ = ∞).

Then the inequality (1.17) holds if and only if

B1(s) = supN≥1

(N∑

n=1

(ϕnv−1

n

)σ) sr

⎛⎝ ∞∑

n=N

uqn

(n∑

k=1

(ϕkv−1

k

)σ) q

r (rσ−s)⎞⎠

1q

< ∞

(5.10)for some s, 0 < s ≤ r

σ or

B2(s) = supN≥1

(N∑

n=1

(ϕnv−1

n

)σ)− sr

⎛⎝ ∞∑

n=N

uqn

(n∑

k=1

(ϕkv

−1k

)σ) q

r (rσ

+s)⎞⎠

1q

< ∞

(5.11)



for some s, 0 < s ≤ rp or

B3(s) := supN≥1

(∞∑

n=N

uqn

) sr

⎛⎜⎝ N∑

n=1

(ϕkv

−1k

)σ ( ∞∑k=n

uqk

)( 1q− s

r

)σ⎞⎟⎠

1σ

< ∞

(5.12)for some s, 0 < s ≤ r

q or

B4(s) := supN≥1

(∞∑

n=N

uqn

)− sr

⎛⎜⎝ ∞∑

n=N

(ϕkv

−1k

)σ ( ∞∑k=n

uqn

)( 1q+ s

r

)σ⎞⎟⎠

1σ

< ∞

(5.13)for some s, 0 < s ≤ 1 − r

q .


sup0<s< r

σ

(ps

ps + r

) 1p

B1(s) ≤ C ≤ inf0<s< r

σ

(p − r

p (1 − s) − r

) 1σ

B1(s), (5.14)

sup0<s< r

p

(p

rs) 1

pB2(s) ≤ C ≤

(σ

p

) 1q

⎛⎝ q − p

pβ(

pq−p , p(q−r)

r(q−p)

)⎞⎠

q−p

pq

B2(r

p) (5.15)

if p < q and

B2(r

p) ≤ C ≤

(p

p − r

)B2(

r

p)

if p = q,

sup0<s< r

q

(qs

q (s + 1) − r

) q−r

qr

B3(s) ≤ C ≤ inf0<s< r

q

B3(s)

(r

r − qs

) 1q

, (5.16)

and

sup0<s<1− r

q

(qs

q − r)

q−r

rq B4(s)

≤ C ≤ (q − r

r)

1σ

⎛⎝ r (q − p)

(p − r)qβ(

qq−p , q(p−r)

r(q−p)

)⎞⎠

q−p

pq

B4(q − r

q). (5.17)


ity with kernel

59

if p < q and

B4(q − r

q) ≤ C ≤

p

rB4(

q − r

q)

if p = q.

Remark 5.1. If s = rσ in (5.10), then we have

B1(r

σ) = sup

n≥1

(n∑

k=1

(ϕkv−1

k

)σ) 1σ(

∞∑k=n

uqk

)1q

< ∞,

which coincides with (1.18) (i.e. B1(rσ ) = BG) and the statement in Theorem

1.2 follows.

Remark 5.2. When r = 1 and ϕk = 1, k = 1, 2..., in Corollary 5.1, then

the inequality (1.17) with vn replaced by v1pn and un replaced by u

1qn coincides

with (1.6). In particular, for the case s = 1p′ in (5.10), we have

B1(1

p′) = sup

n≥1

(n∑

k=1

v1−p′

k

) 1p′(

∞∑k=n

uk

) 1q

< ∞,

which coincides with Muckenhoupt’s condition A1(1p′ ) < ∞ (c.f. (5.21) and

also Bennett [5]).

Next we state the following result for the case with a general kernel:

Theorem 5.2. Let 1 < p ≤ q < ∞ and s ∈ (1, p) . If

E(s) := supN≥1

(N∑

n=1

v1−p′

n

)s⎛⎜⎝ ∞∑

n=N

dqn,kun

(n∑

m=1

v1−p′

m

)q(

1p′−s)⎞⎟⎠

1q

< ∞,

(5.18)

holds for some s ∈(0, 1

p′

), then the inequality (1.16) holds with

C ≤ inf0<s< 1

p′

(p − 1

p − sp − 1

) 1p′

E(s). (5.19)

Remark 5.3. For the case dn,k = 1, n, k = 1, 2, ... the condition (5.18)coincides with the condition (5.21) and thus (c.f. Lemma 5.1) in this casethe condition (5.18) is both necessary and sufficient for (1.16) to hold.



Inspired by a recent result of G. Sinnamon [54] we also state the following:

Theorem 5.3. Let 1 < p ≤ q < ∞. Then the inequality

(∞∑

n=1

(n∑

k=1

a∗k

)q

un

) 1q

≤ C

(∞∑

n=1

(a∗n)p vn

) 1p

(5.20)

holds for all non-increasing sequences {a∗n}∞n=1 with the additional condi-

tion that {vn}∞n=1 is non-increasing if and only if the condition (5.21) holds.

Moreover, for the best constant C in (5.20), the estimate (5.25) holds.

Remark 5.4. For the case vn = 1, n = 1, 2, ... the statement in Theorem5.3 is a special case of a recent remarkable result of G. Sinnamon ([54], p.300-301).

5.2 Proofs

In order to make all proofs in this chapter selfcontained and to clarify ourarguments we first rewrite a special case of Theorem 2.1 in the followingform:

Lemma 5.1. Let 1 < p ≤ q < ∞. Then the inequality (1.6) holds if andonly if

A1(s) := supN≥1

(N∑

n=1

v1−p′n

)s⎛⎜⎝ ∞∑

n=N

un

(n∑

k=1

v1−p′

k

)q(

1p′−s)⎞⎟⎠

1q

< ∞, (5.21)

for some s, 0 < s ≤ 1p′ , or

A2(s) := supN≥1

(N∑

n=1

v1−p′

n

)−s⎛⎝ N∑

n=1

uk

(n∑

k=1

v1−p′

k

)q( 1p′

+s)⎞⎠

1q

< ∞, (5.22)

for some s, 0 < s ≤ 1p , or

A3(s) := supN≥1

(∞∑

n=N

un

)s⎛⎜⎝ N∑

n=1

v1−p′n

(∞∑

k=n

uk

)p′(

1q−s)⎞⎟⎠

1p′

< ∞, (5.23)


ity with kernel

61

for some s, 0 < s ≤ 1q , or

A4(s) := supN≥1

(∞∑

n=N

un

)−s⎛⎝ ∞∑

n=N

v1−p′n

(∞∑

k=n

uk

)p′( 1q+s)⎞⎠

1p′

< ∞, (5.24)

for some s, 0 < s ≤ 1q′ .


sup0<s< 1

p′

(ps

ps + 1

) 1p

A1(s) ≤ C ≤ inf0<s< 1

p′

A1(s)

(p − 1

p (1 − s) − 1

) 1p′

, (5.25)

sup0<s< 1

p

(ps)1p A2(s) ≤ C ≤

1

(p − 1)1/q

⎛⎝ q − p

pβ(

pq−p , p(q−1)

q−p

)⎞⎠

q−p

pq

A2(1

p) (5.26)

if p < q and

A2(1

p) ≤ C ≤ p′A2(

1

p)

if p = q,

sup0<s< 1

q

(q′s

q′s + 1

) 1q′

A3(s) ≤ C ≤ inf0<s< 1

q

A3(s)

(q′ − 1

q′ (1 − s) − 1

) 1q

, (5.27)

and

sup0<s< 1

q′

(q′s)1q′ A4(s) ≤ C ≤ (q − 1)

1p′

⎛⎝ q − p

(p − 1)qβ(

qq−p , q(p−1)

q−p

)⎞⎠

q−p

pq

A4(1

q′)

(5.28)if p < q and

A4(1

q′) ≤ C ≤ pA4(

1

q′)

if p = q.

Remark 5.5. (a) The conditions A3(s) < ∞ and A4(s) < ∞ are just thenatural duals of the conditions A1(s) < ∞ and A2(s) < ∞, respectively (c.f.the book [35]).



(b) It is pointed out in [43] that as end point cases of some of the con-ditions above we just obtain some well-known conditions by G. Bennett (see[5] and [7]).


With the kernel {dn,k} = {lnhk} the inequality (5.1) becomes

(∞∑

n=1

(n∑

k=1

lnhkak

)q

un

) 1q

≤ C

(∞∑

n=1

apnvn

) 1p

,

that is,

(∞∑

n=1

(n∑

k=1

hkak

)q

lqnun

) 1q

≤ C

(∞∑

n=1

apnvn

) 1p

. (5.29)

We now put bk = hkak in the inequality (5.29) and note that (5.29) isequivalent to

(∞∑

n=1

(n∑

k=1

bk

)q

lqnun

) 1q

≤ C

(∞∑

n=1

bpnh−p

n vn

) 1p

.

Considering lqnun = un and h−pn vn = vn to be our new fixed non-negative

weight sequences, we have that the inequality

(∞∑

n=1

(n∑

k=1

bk

)q

un

) 1q

≤ C

(∞∑

n=1

bpnvn

) 1p

(5.30)

is equivalent to the Hardy type inequality (1.6). Thus, by replacing un bylqnun and vn by h−p

n vn in the conditions (5.21)-(5.24) ( i.e. those described byA1(s)-A4(s)) and using Lemma 5.1 we obtain that the conditions (5.2)-(5.5)(i.e. those described by D1(s)-D4(s)) are necessary and sufficient conditionsfor (5.30), and, thus, (5.1) to hold. Subsequently, by replacing Ai(s) withDi(s), i = 1, ..., 4 respectively, in the estimates (5.25)-(5.28) we obtain theestimates for the best constant C in (5.1) to be those described in (5.6)-(5.9).The proof is complete.�


ity with kernel

63

Proof of Corollary 5.1:

In the inequality (5.1) we let hk = ϕrk and ln = u

qr−1q

n and replace an

with arn and vn with vpr

n :

(∞∑

n=1

(n∑

k=1

ϕrka

rk

)q

uqrn

) 1q

≤ C

(∞∑

n=1

aprn vpr

n

) 1p

.

Moreover, relace p with pr and q with q

r and we obtain

⎛⎝ ∞∑

n=1

(n∑

k=1

ϕrka

rk

) q

r

uqn

⎞⎠

1q

≤ Co

(∞∑

n=1

apnvp

n

) 1p

,

with Co = C1r which is equivalent to the inequality (1.17).

This means that for the case 0 < r < p ≤ q < ∞ we can characterize theinequality (1.17) by using Theorem 5.1. Thus, in conditions (5.2)-(5.5) we

first let ln = uqr−1

qn , hn = ϕr

n, vn = vprn , after replace p by p

r and q by qr , and

finally raise each estimate to the power 1r .

Hence, by Theorem 5.1, we conclude that each of the conditions (5.10)-(5.13) (i.e those described by B1(s)-B4(s)) characterizes (1.17). Moreover,the estimates (5.14)-(5.17) follow in a similar way from (5.6)-(5.9). The proofis complete.�


Put bpn = ap

nvn in (1.16). Then (1.16) is equivalent to

(∞∑

n=1

(n∑

k=1

dn,kbkv−

1p

k

)q

un

) 1q

≤ C

(∞∑

n=1

bpn

) 1p

. (5.31)

Assume that the condition (5.18) holds and let

Vn =

n∑k=1

v1−p′

k . (5.32)

Applying Hölder’s inequality, Lemma 2.1(a) with ak = v1−p′

k (0 < λ =p−sp−1

p−1 < 1) and Minkowski’s inequality to the left hand side of (5.31), wefind that (

∞∑n=1

(n∑

k=1

dn,kbkv−

1p

k

)q

un

) 1q



=

(∞∑

n=1

(n∑

k=1

dn,kbkVsk V −s

k v− 1

p

k

)q

un

) 1q

≤

⎛⎝ ∞∑

n=1

(n∑

k=1

dpn,kb

pkV

spk

) q

p(

n∑k=1

V −sp′

k v−

p′

p

k

) q

p′

un

⎞⎠

1q

=

⎛⎝ ∞∑

n=1

(n∑

k=1

dpn,kb

pkV

spk

) q

p(

n∑k=1

V−

sp

p−1

k v1−p′

k

) q

p′

un

⎞⎠

1q

≤

(p − 1

p − sp − 1

) 1p′

⎛⎝ ∞∑

n=1

(n∑

k=1

dpn,kb

pkV

spk

) q

p

Vq(

p−sp−1p

)n un

⎞⎠

1q

≤

(p − 1

p − sp − 1

) 1p′

⎛⎝ ∞∑

k=1

bpkV

spk

(∞∑

n=k

dqn,kV

q(

1p′−s)

n un

) p

q

⎞⎠

1p

≤

(p − 1

p − sp − 1

) 1p′

supk>0

V sk

(∞∑

n=k

dqn,kV

q(

1p′−s)

n un

) 1q(

∞∑k=1

bpk

) 1p

.

Hence, (5.31), and, thus, (1.16) holds. By taking infimum over s ∈(0, 1

p′

)we find that also (5.19) holds. The proof is complete.�

Proof of Theorem 5.3:Sufficiency: The proof follows by just using Lemma 5.1 in the present

situation and also the upper estimate in (5.25) is obtained. However, herewe make the following independent proof:

Assume that the condition (5.2), and thus (5.21) holds and let {a∗n}∞n=1

be an arbitrary non-increasing sequence and define a∗n =

(∞∑

m=ntm

) 1p

, n =

1, 2, ... . The inequality (5.20) can equivalently be rewritten as

⎛⎝ ∞∑

n=1

⎛⎝ n∑

k=1

(∞∑

m=k

tm

) 1p

⎞⎠

q

un

⎞⎠

1q

≤ C

(∞∑

n=1

(∞∑

m=n

tm

)vn

) 1p

(5.33)

= C

(∞∑

m=1

tm

m∑n=1

vn

) 1p

.


ity with kernel

65

Taking Vn as it is defined in (5.32) and applying Hölder’s inequality, Lemma2.1(a) (with λ = p−sp−1

p−1 ), Minkowski’s inequality and changing the order ofthe summation to the left hand side of (5.33) we have that

⎛⎝ ∞∑

n=1

⎛⎝ n∑

k=1

(∞∑

m=k

tm

) 1p

⎞⎠

q

un

⎞⎠

1q

=

⎛⎝ ∞∑

n=1

⎛⎝ n∑

k=1

(∞∑

m=k

tm

) 1p

V sk V −s

k v1p

k v− 1

p

k

⎞⎠

q

un

⎞⎠

1q

≤

⎛⎝ ∞∑

n=1

(n∑

k=1

(∞∑

m=k

tm

)V sp

k vk

) q

p(

n∑k=1

V−

sp

p−1

k v1−p′

k

) q

p′

un

⎞⎠

1q

≤

(p − 1

p − sp − 1

) 1p′

⎛⎝ ∞∑

n=1

(n∑

k=1

(∞∑

m=k

tm

)V sp

k vk

) q

p

Vq(p−sp−1)

pn un

⎞⎠

1q

≤

(p − 1

p − sp − 1

) 1p′

⎛⎝ ∞∑

k=1

(∞∑

m=k

tm

)V sp

k vk

(∞∑

n=k

Vq(p−sp−1)

pn un

) p

q

⎞⎠

1p

=

(p − 1

p − sp − 1

) 1p′

⎛⎝ ∞∑

m=1

tm

m∑k=1

vkVspk

(∞∑

n=k

Vq(

1p′−s)

n un

) p

q

⎞⎠

1p

≤

(p − 1

p − sp − 1

) 1p′

supk≥1

V sk

(∞∑

n=k

Vq(

1p′−s)

n un

) 1q(

∞∑m=1

tm

m∑k=1

vk

) 1p

.

Hence, by taking infimum over s ∈(0, 1

p′

), (5.33), and, thus, (5.20) holds

with a constant C satisfying the right hand inequality in (5.25).Necessity: Assume that (5.20) holds and for fixed N ∈ Z+ apply the

following test sequence:

a∗k =

⎧⎪⎨⎪⎩

V−

(1+ps

p

)N v1−p′

k for k = 1, ...N

V−

(1+ps

p

)k v1−p′

k for k = N + 1, ...

to (5.20). (Note that with our assumptions {a∗k}∞

k=1 is a non-increasingsequence).



For the left hand side of (5.20) we have that(∞∑

n=1

(n∑

k=1

a∗k

)q

un

) 1q

=

(∞∑

n=1

(N∑

k=1

V−

(1+ps

p

)N v1−p′

k +

n∑k=N+1

V−

(1+ps

p

)k v1−p′

k

)q

un

) 1q

≥

(∞∑

n=N

(N∑

k=1

V−

(1+ps

p

)N v1−p′

k +

n∑k=N+1

V−

(1+ps

p

)k v1−p′

k

)q

un

) 1q

≥

(∞∑

n=N

(V

1p′−s

N + V−

1+ps

pn

n∑k=N+1

v1−p′

k

)q

un

) 1q

=

(∞∑

n=N

(V

1p′−s

N + V−

1+ps

pn (Vn − VN )

)q

un

) 1q

≥

(∞∑

n=N

(V

1p′−s

N + V1p′−s

n − V−

1+ps

p

N VN

)q

un

) 1q

=

(∞∑

n=N

Vq(

1p′−s)

n un

)1q

. (5.34)

For the right hand side of (5.20), by applying Lemma 2.1(b) we find that(∞∑

n=1

(a∗n)p vn

) 1p

=

(N∑

n=1

V−(1+ps)N v1−p′

k +

∞∑n=N+1

V−(1+ps)k v1−p′

k

) 1p

≤

(V −ps

N +1

psV −ps

N

) 1p

≤

(1 +

1

ps

) 1p

V −sN

=

(ps + 1

ps

) 1p

V −sN . (5.35)

Combining (5.35), (5.34) and (5.20) we have that(∞∑

n=N

Vq(

1p′−s)

n un

)1q

≤ C

(ps + 1

ps

) 1p

V −sN ,


ity with kernel

67

that is, (ps

ps + 1

) 1p

V sN

(∞∑

n=N

Vq(

1p′−s)

n un

) 1q

≤ C < ∞.

Hence, by taking supremum over N ≥ 1 and supremum over s ∈(0, 1

p′

), we

conclude that (5.21) and the left hand side of the estimate (5.25) hold.Summing up, we have proved that (5.20) is equivalent to (5.21) and that

(5.25) holds. The proof is complete.�

5.3 Concluding remarks and open questions

By comparing the statements in Theorem 5.2 and Remark 5.3 and the corre-sponding knowledge from the continuous case (see the book [35]) it is naturalto raise the following question:

Open question 1: Find necessary and sufficient conditions for (1.16) tohold for all non-negative sequences {an}

∞n=1 for as general kernels as

possible.

Remark 5.6. For the case when vn = 1, n = 1, 2, ..., Lemma 5.1 holdsalso if the set of all weight sequences {an}

∞n=1 is restricted to the cone of all

non-increasing weight sequences {a∗n}∞n=1 (c.f. our more general statement in

Theorem 5.3) (c.f. also Remark 5.4). This fact follows from a recent resultof G. Sinnamon [54].

Remark 5.7. The result of G. Sinnamon [54] was recently generalized byL.-E. Persson, V. D. Stepanov and E. P. Ushakova [47] to a more generalcase involving kernels and a general measure. However, these kernels stillhave some restrictions (monotonicity in the second variable). These resultsmake it natural to also raise the following question:

Open question 2: Find necessary and sufficient conditions for (1.16) tohold on the cone of non-increasing sequences for as general kernels aspossible.



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[32] A. Kufner, L. Maligranda, and L.-E. Persson. The prehistory of theHardy inequality. Amer. Math. Monthly,(18 pages), to appear.

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Appendix A

Appendix

On the history and developments of Hardy type inequalities in the

continuous case:

Many investigations and generalizations on the original form of the con-tinuous Hardy inequality (1.2) can be found in the literature. Let us firstmention here some books which are important to these developments. Thebooks are: Hardy-Littlewood-Pólya [22] on classical inequalities, B. Opicand A. Kufner [45] concerning Hardy’s inequalities and A. Kufner and L.-E.Persson [35], which in particular contains most of the newest developmentsof Hardy-type inequalities.

A detailed description of the prehistory of Hardy’s inequality was recentlypublished by A. Kufner, L. Maligranda and L.-E. Persson [33]. A briefdescription of the most important steps in the development was presentedby A. Kufner and L.-E. Persson [34] and this paper has now in [32] beenextended almost to a book manuscript by these authors and L. Maligrandato a fairly complete version including many surprising and not so knownhistorical and mathematical facts. Below we have just selected a few factsfrom this development.

In 1928 G. H. Hardy [19] proved the estimate for some integral operators,from which the first “weighted” modification of inequality (1.2) appeared,namely the famous inequality

∞∫0

⎛⎝1

x

x∫0

f(t)dt

⎞⎠

p

xεdx <

(p

p − ε − 1

)p∞∫0

f(x)pxεdx, (A.1)

75



valid with p > 1 and ε < p − 1, for all measurable nonnegative functions f

(see [22], Theorem 330), where the constant(

pp−ε−1

)pis the best possible.

We mention the following “dual” inequality, which can be derived from (A.1):

∞∫0

⎛⎝1

x

∞∫x

f(t)dt

⎞⎠

p

xεdx <

(p

ε + 1 − p

)p∞∫0

f(x)pxεdx; (A.2)

it holds with p > 1 and ε > p − 1, for all measurable nonnegative functions

f and the constant(

pε+1−p

)pis the best possible (see again [22], Theorem

330).During the last decades the inequality (A.1) has been developed to the

form

⎛⎝ b∫

a

⎛⎝ x∫

a

f(t)dt

⎞⎠

q

u(x)dx

⎞⎠

1q

< C

⎛⎝ b∫

a

f(x)pv(x)dx

⎞⎠

1p

(A.3)

with- a, b real number satisfying −∞ ≤ a < b ≤ ∞,

- u, v weight functions (measurable functions everywhere in (a, b)),- p q real parameters, satisfying 0 < q ≤ ∞ and 1 ≤ p ≤ ∞.

In 1930, the first extension to the case 1 < p < ∞ appeared and was dueHardy and Littlewood [21] and Bliss [8]. They considered the interval (a, b)= (0,∞) and the weight functions v(x) ≡ 1, u(x) = xr−q with r = q−p

p , andderived inequality (A.3) with the constant

C =

(1

q − r − 1

)1/q⎛⎝ rΓ

(qr

)Γ(

1r

)Γ(

q−1r

)⎞⎠

r/q

,

which is the sharp constant.In the late fifties and early sixties of the last century, P. R. Beesack

initiated a systematic investigation of the Hardy inequality. He dealt with thecase p = q but considered also the case p < 0, and even the case 0 < p < 1,where the inequality sign in (A.3) must be reversed (see [32]). Beesack’sapproach was extended to a class of inequalities containing Hardy’s inequalityas a special case, but the value ε = p − 1 in inequality (A.2) was excluded;see [3]. In 1965, Kadlec and Kufner [29] suceeded to “remove” this gap

Appendix 77

considering, instead of weight functions of the power type, xε, weights of theform xε |log x|η.

The problem of finding necessary and sufficient conditions for (A.1) tohold, again for (a, b) = (0,∞) and p = q, was investigated in 1966, 1967 and1969 by G. Talensi [57] and in 1969 by G. Tomaselli [58] and they obtainedthe necessary and sufficient condition for the estimate

b∫0

⎛⎝ x∫

0

f(t)dt

⎞⎠

p

u(x)dx ≤ C

b∫0

f(x)pv(x)dx, (A.4)

f ≥ 0, with 0 < b ≤ ∞ and 1 < p < ∞ as

A := supr∈(0,b)

⎛⎝ b∫

r

u(x)dx

⎞⎠⎛⎝ r∫

0

v(x)1−p′dx

⎞⎠

p−1

< ∞. (A.5)

Moreover, for the sharp constant in (A.4) it yields that

A ≤ C ≤pp

(p − 1)p−1 A.

The condition (A.5) with b = ∞ and written in a different form

AM := supr>0

⎛⎝ ∞∫

r

u(x)dx

⎞⎠

1p⎛⎝ r∫

0

v(x)1−p′dx

⎞⎠

1p′

< ∞ (A.6)

is often called the Muckenhoupt condition since B. Muckenhoupt [41] hasgiven in 1972 a direct proof of his result and also extended it to the case ofthe more general inequality

⎛⎝ ∞∫

0

⎛⎝ x∫

0

f(t)dt

⎞⎠

p

dμ

⎞⎠

1p

< C

⎛⎝ ∞∫

0

f(x)pdν

⎞⎠

1p

, (A.7)

f ≥ 0, with some Borel measures μ and ν. In this case, the necessary andsufficient condition for (A.7) to hold for some finite constant C is

A := supr∈(0,b)

[μ (r,∞)]1/p

⎛⎝ r∫

0

(dν

dx

)1−p′

dx

⎞⎠

1p′

< ∞. (A.8)



For the special measures dμ(t) = u(t)dt, dν(t) = v(t)dt the condition (A.8)reduces to (A.6).

Moreover, the best (= least) constant C in (A.7) satisfies

A ≤ C ≤ p1/p(p′)1/p′

A (A.9)

for 1 < p < ∞ and C = A for p = 1 (and p′ = ∞).Besides condition (A.6), G. Tomaselli derived also some other (equiv-

alent) conditions of the validity of Hardy’s inequality (A.3) (with (a, b) =(0,∞) , p = q), namely

B∗ := supr∈(0,b)

(∫ r

0u(x)

(∫ x

0v(t)1−p′dt

)p

dx

)1/p(∫ r

0v(x)1−p′dx

)−1/p

< ∞

and

B∗∗ := inff

supx∈(0,b)

1

f(x)

∫ x

0u(t)

[f(t) +

∫ t

0v(s)1−p′ds

]p

dt < ∞,

where the infimum is taken over all positive measurable functions f (see[58]).

At this stage, the case 1 ≤ p = q < ∞ was solved, since the step from(0,∞) to an arbitrary interval (a, b) is only a matter of routine (for thecorresponding substitutions, see e.g. A. Kufner and L. E. Persson [35]).

The investigation of the case of different parameters p and q,

1 ≤ p, q ≤ ∞

started (probably) in 1978 by a paper of J. S. Bradley [9]. He showed thatthe condition

Am := supr>0

(∫ ∞

ru(x)dx

)1/q (∫ r

0v1−p′(x)dx

)1/p′

< ∞ (A.10)

is necessary for⎛⎝ ∞∫

0

⎛⎝ x∫

0

f(t)dt

⎞⎠

q

u(x)dx

⎞⎠

1q

≤ C

⎛⎝ ∞∫

0

f(x)pv(x)dx

⎞⎠

1p

(A.11)

to hold if 1 ≤ p, q ≤ ∞ and, moreover, it is sufficient if 1 ≤ p ≤ q ≤ ∞. Inthe second case, he derived for the best constant C in (A.11) the estimates{

Am ≤ C ≤ p1/q(p′)1/p′Am if 1 < p ≤ q < ∞,C = Am if p = 1 or q = ∞.

(A.12)

Appendix 79

See also V. Kokilashvili [31], V. G. Maz’ja [38], K. F. Andersen and B.Muckenhoupt [2].

Moreover, for the case 1 0

(∫ r

0v(x)1−p′dx

)−1/p(∫ r

0u(x)

(∫ x

0v(t)1−p′dt

)q

dx

)1/q

< ∞,

is necessary and sufficient for (A.11) to hold. Moreover, the best constant in(A.11) with (a, b) = (0,∞) can be estimated as follows

APS ≤ C ≤ p′APS .

The estimate (A.12) for the best constant C was improved by severalauthors. If we rewrite it as

Am ≤ C ≤ k(p, q)Am,

then in (A.12) we have

k1(p, q) = p1/q(p′)1/p′ ,

which is due to J. S. Bradley [9] and V. Kokilashvili [31] and by anothermethod by P. Gurka [15]. Moreover, V. G. Maz’ja (see [38]) received

k2(p, q) = p1/q(q′)1/p′

and R. K. Juberg [28] (see also [32]) has minimum of both k1(p, q) andk2(p, q);

k3(p, q) = min(p1/q(p′)1/p′

, p1/q(q′)1/p′).

In 1987 V. D. Stepanov [56] showed that

k4(p, q) = q1/q(p′)(p−1)/q

which was not better than the R. K. Juberg result. In 1990, independentlyB. Opic (see Opic and Kufner [45], Theorem 1.14) improved these estimatesto

k5(p, q) =

(1 +

q

p′

) 1q(

1 +p′

q

) 1p′

.

Remark A.1. For p = q all the five estimates coincide with

kj(p) = p1/p(p′)1/p′ , j = 1, 2, ..., 5.



In 1991, G. Bennett ([7], Theorem 9) and in 1992, V. M. Manakov [37]derived the following sharp constant:

k6(p, q) =

⎛⎝ Γ

(pq

q−p

)Γ(

qq−p

)Γ(

p(q−1)q−p

)⎞⎠

q−p

pq

.

For the case 1 < q < p < ∞ a necessary and sufficient condition wasderived by Maz’ja and A. L. Rozin in the late seventies (see [38]). It has adifferent form, namely⎛

⎜⎝∞∫0

⎛⎝ ∞∫

t

u(x)dx

⎞⎠

rq⎛⎝ t∫

0

v1−p′(x)dx

⎞⎠

r

q′

v1−p′(t)dt

⎞⎟⎠

1r

< ∞ (A.13)

with 1r = 1

q − 1p for inequality (A.11). It was later proved by L.-E. Persson

and V. D. Stepavov that (A.13) can be replaced by the condition

∞∫0

⎛⎝ ∞∫

t

u(x)

⎛⎝ x∫

0

v1−p′(s)ds

⎞⎠

q

dx

⎞⎠

q

r

v1−p′(t)dt < ∞ (A.14)

for the case q < p.The remaining case,

0 < q < 1, p > 1

is due G. Sinnamon [52] who showed in 1987 (published in 1991 in his paper[53]) that again condition (A.13) or (A.14) is necessary and sufficient for(A.11) to hold.

The case 0 < q < 1, p = 1, was, in 1996, described by G. Sinnamon andV. D. Stepanov [55].

There are many papers written on the Hardy inequalities. See the refer-ences in the books [22], [45], [35], [4], [40] and the review paper [33] for moreinformation. We can also see some more contributions from the recent thesiswritten by A. Wedestig [60], D. V. Prokhorov [48], and M. Nassyrova [42].

Finally we recall that by using a limit argument, we found that thediscrete Hardy inequality (1.1) implies the Carleman inequality (1.20). In asimilar way we find that the continuous Hardy inequality (1.2) implies thefollowing inequality called the Pólya-Knopp inequality

∞∫0

exp

⎛⎝1

x

x∫0

ln f(t)dt

⎞⎠

p

dx ≤ C

∞∫0

f(x)pdx. (A.15)

Appendix 81

Aslo the weighted Pólya-Knopp inequality

⎛⎝ ∞∫

0

exp

⎛⎝1

x

x∫0

ln f(t)dt

⎞⎠

q

u(x)dx

⎞⎠

1q

≤ C

⎛⎝ ∞∫

0

f(x)pv(x)dx

⎞⎠

1p

, (A.16)

p, q > 0, has been studied in the literature. We can find more information onthe inequalities (A.15) and (A.16) in [30] and in the recent theses [60] and [26]and the related references therein. In particular, for 0 < p ≤ q < ∞, a scaleof characterizations of (A.16) can be obtained as a natural limit result of thecorresponding scales of description of the Hardy-type inequality proved byA. Wedestig (see [60], Theorem 3.1).

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ltu.diva-portal.orgltu.diva-portal.org/smash/get/diva2:999529/FULLTEXT01.pdf · Abstract This thesis deals with some generalizations of the discrete Hardy and Carle-man type inequalities