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Morrey Spaces

Hendra GUNAWAN*

*http://personal.fmipa.itb.ac.id/hgunawan/

Analysis and Geometry GroupBandung Institute of TechnologyBandung 40132, INDONESIA

Functional Analysis & Operator Algebras Seminar 2019UPI Bandung, 24 September 2019

HG* (*ITB Bandung) Morrey Spaces 24 September 2019 1 / 29

Outline

1 Introduction: Geometric Constants for Banach Spaces

2 Morrey Spaces

3 Small Morrey Spaces

4 Acknowledgement


Outline


2 Morrey Spaces


4 Acknowledgement


Outline


2 Morrey Spaces


4 Acknowledgement


Outline


2 Morrey Spaces


4 Acknowledgement


Abstract

In this talk, we shall discuss about Morrey spaces. In particular, weshall be interested in their geometric properties.

Three geometric constants will be computed, namely VonNeumann-Jordan constant, James constant, and Dunkl-Williamsconstant.

These constants measure the uniformly non-squareness of the spaces.We found that Morrey spaces are not uniformly non-square, whichconfirms the fact that they are not reflexive spaces.


Introduction: Geometric Constants for Banach Spaces

Von Neumann-Jordan and James Constants

Let (X , ‖ · ‖X ) be a Banach space. The von Neumann-Jordanconstant1 CNJ(X ) and the James constant2 CJ(X ) for X are given by

CNJ(X ) := sup

{‖x + y‖2X + ‖x − y‖2X

2(‖x‖2X + ‖y‖2X ): x , y ∈ X \ {0}

}and

CJ(X ) := sup {min{‖x + y‖X , ‖x − y‖X} : x , y ∈ SX} ,

respectively. Here SX := {x ∈ X : ‖x‖X = 1}.

1P. Jordan and J. von Neumann, “On inner products in linear, metricspaces”, Ann. Math. (2) 36 (1935), 719–723

2R.C. James, “Uniformly non-square Banach spaces”, Ann. Math. 80(1964), 542–550



Parallelogram’s Diagonals



Dunkl-Williams Constant

There is a third constant, namely the Dunkl-Williams constant3

CDW(X ) for X , given by

CDW(X ) := sup

{‖x‖X + ‖y‖X‖x − y‖X

∥∥∥∥ x

‖x‖X− y

‖y‖X

∥∥∥∥X

: x 6= y ∈ X \ {0}}.

3C. F. Dunkl and K.S. Williams, “A simple norm inequality”, Amer.Math. Monthly 71 (1964), 53–54



It is well known that 1 ≤ CNJ(X ) ≤ 2 for every Banach space X , andthat CNJ(X ) = 1 if and only if X is a Hilbert space. 4

Meanwhile,√

2 ≤ CJ(X ) ≤ 2 holds for every Banach space X , andCJ(X ) =

√2 if (but not only if) X is a Hilbert space. 5

As for the Dunkl-Williams constant, we have 2 ≤ CDW(X ) ≤ 4 andCDW(X ) = 2 if and only if X is a Hilbert space.6

4E. Casini, “About some parameters of normed linear spaces”, AttiAccad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. – Ser. VIII 80(1986), no. 1-2, 11–15

5J. Gao and K.-S. Lau, “On the geometry of spheres in normed linearspaces”, J. Aust. Math. Soc. Ser. A 48 (1990), 101–112

6C.F. Dunkl and K.S. Williams, “A simple norm inequality”, Amer.Math. Monthly 71 (1964), 53–54



For Lebesgue spaces Lp = Lp(Rd) where 1 ≤ p ≤ ∞, we have

CNJ(Lp) = max{22/p−1, 21−2/p}

andCJ(Lp) = max{21/p, 21−1/p}.7

Meanwhile, we know that CDW(L1) = CDW(L∞) = 4.8

7M. Kato and L. Maligandra, “On James and Jordan-von Neumannconstants of Lorentz sequence spaces”, J. Math. Anal. Appl. 258 (2001),457–465

8A. Jimenez-Melado, E. Liorens-Fuster, and E.M. Mazcunan-Navarro,“The Dunkl–Williams constant, convexity, smoothness and normalstructure”, J. Math. Anal. Appl. 342, Issue 1 (2008), 298–310



Theorem. For every Banach spaces X , we have

C 2J (X )

2≤ CNJ(X ) ≤ CJ(X ).9

Theorem (James). A Banach space X is reflexive iff X is uniformlynon-square, that is, there exists δ > 0 s.t.

min{‖x + y‖X , ‖x − y‖X} ≤ 2(1− δ)

for all x , y ∈ SX .10

Corollary. A Banach space X is reflexive iff CJ(X ) < 2.

9M. Kato, L. Maligandra, and Y. Takahashi, Studia Math. 144 (2001)& M. Kato and Y. Takahashi, JMAA 359 (2009)

10R.C. James, Ann. Math. 80 (1964)HG* (*ITB Bandung) Morrey Spaces 24 September 2019 9 / 29

Morrey Spaces

Morrey Spaces

Let 1 ≤ p ≤ q <∞. The Morrey space Mpq =Mp

q(Rd) is the set ofall the measurable functions f on Rd for which

‖f ‖Mpq

:= supB=B(a,r)

|B |1q− 1

p

(∫B

|f (y)|p dy) 1

p

<∞,

where B(a, r) denotes the ball centered at a ∈ Rd having radiusr > 0 and Lebesgue measure |B |.11

Example. For 1 ≤ p < q <∞, | · |−d/q ∈Mpq \ Lq.

Note. In general, Lq ⊆Mpq, and the inclusion is proper. Moreover,

Mp1q ⊆Mp2

q

whenever 1 ≤ p2 ≤ p1 ≤ q.11D.R. Adams and J. Xiao, “Morrey spaces in harmonic analysis”, Ark.

Mat. 50 (2012), 201–230HG* (*ITB Bandung) Morrey Spaces 24 September 2019 10 / 29

Morrey Spaces

Since Mpq is a Banach space, it follows that

CNJ(Mpq), CJ(Mp

q) ≤ 2 and CDW(Mpq) ≤ 4.

The question is: how large are the constants?

Note. The larger the constant, the lesser round the unit ball.


Morrey Spaces

The following theorem12 tells us that the unit ball in Morrey spaces isvery far from round: it contains a square!

Theorem 2.1

If 1 ≤ p < q <∞, then CNJ(Mpq) = CJ(Mp

q) = 2 andCDW(Mp

q) = 4.

Note. For p = q, Mpq = Lq and their norms are identical. The

above theorem tells us that the case where q > p and the case wherep = q are substantially different. When q > p, the three constantsCJ(Mp

q), CNJ(Mpq), and CDW(Mp

q) take the extreme values, whichare the same values as those for L1 and L∞ spaces.

12G., E. Kikianty, Y. Sawano, and C. Schwanke, “Three geometricconstants for Morrey spaces”, to appear in Bull. Korean Math. Soc.


Morrey Spaces

We prove the theorem constructively, that is, by finding two elementsin the space such that the associated expressions are equal to 2, 2,and 4, respectively.

Proof. Let 1 ≤ p < q <∞, and let f (x) := |x |−d/q(x ∈ Rd), where|x | denotes the Euclidean norm of x . Then f ∈Mp

q(Rd).

Next, defineg := χ(0,1)(| · |)f ,

h := f − g ,

andk := −f + 2g .

Note. Drawing their graphs might help.


Morrey Spaces

By a change of variables, we see that

‖td/qg(t·)‖Mpq

= ‖g‖Mpq

and‖td/qh(t·)‖Mp

q= ‖h‖Mp

q

for all t > 0.

Sincetd/qg(tx) = χ(0,1)(t|x |)f (x)

andtd/qh(tx) = χ(0,1)(t|x |)f (x)− χ[1,∞)(t|x |)f (x)

for t > 0 and x ∈ Rd , by the Monotone Convergence Property wehave

‖f ‖Mpq

= ‖g‖Mpq

= ‖h‖Mpq

= ‖k‖Mpq∈ (0,∞).


Morrey Spaces

This implies that

‖f + k‖2Mpq

+ ‖f − k‖2Mpq

= 4(‖f ‖2Mpq

+ ‖k‖2Mpq)

and

min{‖f + k‖Mpq, ‖f − k‖Mp

q} = min{‖2g‖Mp

q, ‖2h‖Mp

q}

= 2‖f ‖Mpq

= 2‖k‖Mpq.

By definition and the fact that both CNJ(Mpq), CJ(Mp

q) ≤ 2, weconclude that

CNJ(Mpq) = CJ(Mp

q) = 2,

as desired.


Morrey Spaces

We now move to the Dunkl–Williams constant. We use the sametechniques as in Jimenez-Melado et al.13

Consider the previous function f and the function (1 + r)g + (1− r)hfor r ∈ (0, 1). We have

‖f ‖Mpq

+ ‖(1 + r)g + (1− r)h‖Mpq

‖f − (1 + r)g − (1− r)h‖Mpq

∥∥∥∥∥ f

‖f ‖Mpq

− (1 + r)g + (1− r)h

‖(1 + r)g + (1− r)h‖Mpq

∥∥∥∥∥Mp

q

=‖f ‖Mp

q+ (1 + r)‖f ‖Mp

q

r‖f ‖Mpq

∥∥∥∥∥ f

‖f ‖Mpq

− (1 + r)g + (1− r)h

(1 + r)‖f ‖Mpq

∥∥∥∥∥Mp

q

=‖f ‖Mp

q+ (1 + r)‖f ‖Mp

q

r‖f ‖Mpq

∥∥∥∥∥ 2rh

(1 + r)‖f ‖Mpq

∥∥∥∥∥Mp

q

=4 + 2r

1 + r.

Letting r ↓ 0, we obtain CDW(Mpq) = 4, as desired.

13A. Jimenez-Melado, E. Liorens-Fuster, and E.M. Mazcunan-Navarro,“The Dunkl–Williams constant, convexity, smoothness and normalstructure”, J. Math. Anal. Appl. 342, Issue 1 (2008), 298–310


Morrey Spaces

Remarks

Let 1 ≤ p ≤ q <∞. The central Morrey space cMpq = cMp

q(Rd) isthe set of all the measurable functions f on Rd for which

‖f ‖cMpq

:= supB=B(0,r)

|B |1q− 1

p

(∫B

|f (y)|p dy) 1

p

<∞.

Using the same arguments, we see that CNJ(cMpq) = CJ(cMp

q) = 2and CDW(cMp

q) = 4 whenever 1 ≤ p < q <∞.

With the values of these constants, we can confirm that Morreyspaces and central Morrey spaces are not reflexive.


Small Morrey Spaces

Small Morrey Spaces

The small Morrey space mpq = mp

q(Rd) is defined to be the set of allmeasurable functions f such that

‖f ‖mpq

:= supa∈Rd ,R∈(0,1)

|B(a,R)|1q− 1

p

( ∫B(a,R)

|f (y)|pdy) 1

p

<∞.

Note:(1) Small Morrey spaces are also Banach spaces.(2) For each p and q, the small Morrey mp

q space properly containsthe Morrey space Mp

q.(3) For p = q, the space mq

q is identical with the Lquloc space.14

14Y. Sawano, “A thought on generalized Morrey spaces”,arXiv:1812.08394 [math.FA]


Small Morrey Spaces

By using a similar idea (but not quite the same) as in the previouspart, we obtain the following theorem.15

Theorem 3.1

Let 1 ≤ p < q <∞. Then CNJ(mpq) = CJ(mp

q) = 2 andCDW(mp

q) = 4.

15A. Mu’tazili and G., “Von Neumann-Jordan constant, Jamesconstant, and Dunkl-Williams constant for small Morrey spaces”,submitted


Small Morrey Spaces

Define f := | · |−dq , g := χ(0,ε)(| · |)f , h = f − g , k = −f + 2g and

l = (1 + δ)g + (1− δ)h with 0 < ε, δ < 1.

Here g , h and k depend on ε, while l depends on ε and δ.

Note that all functions are radial functions, f , g , l are radiallydecreasing, h ≤ f , and |k | = f . Hence, we have

‖f ‖mpq

= supR∈(0,1)

|B(0,R)|1q− 1

p

( ∫B(0,R)

|y |−dpq dy

) 1p

=(Cd

d

) 1q(

1−p

q

)− 1p,


Small Morrey Spaces

Next, we compute

‖g‖mpq

= supR∈(0,1)

|B(0,R)|1q− 1

p

( ∫B(0,R)

|f (y)|pχ(0,ε)(|y |)dy) 1

p

= supR∈(0,ε)

|B(0,R)|1q− 1

p

( ∫B(0,R)

|y |−dpq dy

) 1p

=(Cd

d

) 1q(

1− p

q

)− 1p

= ‖f ‖mpq;


Small Morrey Spaces

‖h‖mpq≥ sup

R∈(0,1)|B(0,R)|

1q− 1

p

( ∫B(0,R)

|f (y)(1− χ(0,ε)(|y |))|pdy) 1

p

= supR∈(0,1)

|B(0,R)|1q− 1

p

( ∫B(0,R)

|y |−dpq (1− χ(0,ε)(|y |))dy

) 1p

= supR∈(ε,1)

(Cd

d

) 1q− 1

pR

dq− d

p

(Cd

R∫ε

r−dpq+d−1dr

) 1p

= supR∈(ε,1)

(Cd

d

) 1q(

1− p

q

)− 1p(1− R

dpq−dε−

dpq+d)

1p

= ‖f ‖mpq(1− ε−

dpq+d)

1p ;


Small Morrey Spaces

and . . .

‖k‖mpq

= supR∈(0,1)

|B(0,R)|1q− 1

p

( ∫B(0,R)

|f (y)(2χ(0,ε)(|y |)− 1)|pdy) 1

p

= supR∈(0,1)

|B(0,R)|1q− 1

p

( ∫B(0,R)

|f (y)|pdy) 1

p

= ‖f ‖mpq,

where Cd denotes the ‘area’ of the unit sphere in Rd .


Small Morrey Spaces

We observe that

CNJ(mpq) ≥

‖f + k‖2mp

q+ ‖f − k‖2

mpq

2(‖f ‖2mp

q+ ‖k‖2

mpq)

=‖2g‖2

mpq

+ ‖2h‖2mp

q

2(‖f ‖2mp

q+ ‖k‖2

mpq)

=4‖g‖2

mpq

+ 4‖h‖2mp

q

2(‖f ‖2mp

q+ ‖k‖2

mpq)

≥2(‖f ‖2

mpq

+ ‖f ‖2mp

q(1− ε−

dpq+d)

2p)

‖f ‖2mp

q+ ‖f ‖2

mpq

= 1 + (1− ε−dpq+d)

2p ,

which holds for any 0 < ε < 1.

Since ε can be arbitrarily small, we obtain CNJ(mpq) ≥ 2. But

CNJ(mpq) ≤ 2, and we conclude that CNJ(mp

q) = 2.


Small Morrey Spaces

Next, for James constant, we observe that

min{‖f + k‖mpq, ‖f − k‖mp

q} = min{‖2g‖mp

q, ‖2h‖mp

q}

≥ 2 min{‖f ‖mpq, ‖f ‖mp

q(1− ε−

dpq+d)

1p }

= 2‖f ‖mpq(1− ε−

dpq+d)

1p .

Dividing both sides by ‖f ‖mpq

(= ‖k‖mpq), we get

CJ(mpq) ≥ min

{∥∥∥∥ f

‖f ‖mpq

+k

‖k‖mpq

∥∥∥∥mp

q

,

∥∥∥∥ f

‖f ‖mpq

− k

‖k‖mpq

∥∥∥∥mp

q

}= 2(1− ε−

dpq+d)

1p ,

and this also holds for any 0 < ε < 1.

Using similar arguments as before, we conclude that CJ(mpq) = 2.


Small Morrey Spaces

Finally, we compute the Dunkl-Williams constant. Observe that

‖l‖mpq

= supR∈(0,1)

|B(0,R)|1q− 1

p

( ∫B(0,R)

|(1 + δ)g(y) + (1− δ)h(y)|pdy) 1

p

= supR∈(0,1)

|B(0,R)|1q− 1

p

( ∫B(0,R)

|f (y)(1 + δ(2χ(0,ε)(|y |)− 1))|pdy) 1

p

= supR∈(0,1)

(Cd

d

) 1q− 1

pR

dq− d

p

(Cd

R∫0

r−dpq+d−1(1 + δ(2χ(0,ε)(r)− 1))pdr

) 1p

= supR∈(ε,1)

(Cd

d

) 1qd

1pR

dq− d

p

((1 + δ)p

ε∫0

r−dpq+d−1dr+

+ (1− δ)pR∫ε

r−dpq+d−1dr

) 1p

(1 + δ)‖f ‖mpq.


Small Morrey Spaces

It follows that

CDW(mpq) ≥

‖f ‖mpq

+ ‖l‖mpq

‖f − l‖mpq

∥∥∥∥ f

‖f ‖mpq

− l

‖l‖mpq

∥∥∥∥mp

q

=‖f ‖mp

q+ (1 + δ)‖f ‖mp

q

‖f − ((1 + δ)g + (1− δ)h)‖mpq

∥∥∥∥ f

‖f ‖mpq

−

(1 + δ)g + (1− δ)h

(1 + δ)‖f ‖mpq

∥∥∥∥mp

q

=(2 + δ)‖f ‖mp

q

‖δk‖mpq

∥∥∥∥ 2δh

(1 + δ)‖f ‖mpq

∥∥∥∥mp

q

≥(2 + δ)‖f ‖mp

q

δ‖f ‖mpq

2δ‖f ‖mpq(1− ε−

dpq+d)

1p

(1 + δ)‖f ‖mpq

=(4 + 2δ)

1 + δ(1− ε−

dpq+d)

1p ,

which holds for any 0 < δ, ε < 1.HG* (*ITB Bandung) Morrey Spaces 24 September 2019 27 / 29

Small Morrey Spaces

As δ and ε can be arbitrarily small, we must have CDW(mpq) ≥ 4.

Using the same arguments as before, we conclude that CDW(mpq) = 4.

Remarks. With the values of these constants, we can also tell thatsmall Morrey spaces are not reflexive.


Acknowledgement

Acknowledgement. The work on Morrey spaces is joint with E.Kikianty, Y. Sawano, and C. Schwanke; while that on small Morreyspaces is joint with A. Mu’tazili.

We acknowledge the financial support from ITB through the 2019Research and Innovation Program.

THANK YOU FOR YOUR ATTENTION.


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