Boundedness of some linear operators in various function spaces

DOCTORA L T H E S I S

Evgeniy a Bur tse va Boundedness of Some Linear Operator s in Var ious Function Spaces

Department of Engineering Sciences and Mathematics Division of Mathematical Sciences

ISSN 1402-1544 ISBN 978-91-7790-687-2 (print)

ISBN 978-91-7790-688-9 (pdf) Luleå University of Technology 2020

Boundedness of Some Linear Operators in Various Function Spaces

Evgeniya Burtseva

Applied Mathematics

131781-LTU-Evgeniya.indd Alla sidor

131781-LTU-Evgeniya.indd Alla sidor 2020-11-10 10:252020-11-10 10:25

Boundedness of Some Linear Operators in Various Function Spaces

Evgeniya Burtseva

Department of Engineering Sciences and Mathematics Lule˚ a University of Technology

SE-971 87 Lule˚ a, Sweden

2020 Mathematics Subject Classification: 46E30, 46B26, 42B35, 26A33.

Keywords and phrases: Morrey–Orlicz spaces, central Morrey–Orlicz

spaces, Orlicz functions, Riesz potential, fractional integral operator, Hardy

operators, H¨older spaces.

Abstract

This PhD thesis is devoted to boundedness of some classical linear oper- ators in various function spaces. We prove boundedness of weighted Hardy type operators and the weighted Riesz potential in Morrey–Orlicz spaces.

Furthermore, we consider central Morrey–Orlicz spaces and prove bounded- ness of the Riesz potential in these spaces. We also present results concerning boundedness of Hardy type operators in H¨older type spaces. The thesis con- sists of four papers (Papers A–D), two complementary appendices (A

, B

) and an introduction.

The introduction is divided into three parts. In the first part we give main definitions and properties of Morrey spaces, Orlicz spaces and Morrey–Orlicz spaces. In the second part we consider boundedness of the Riesz potential and Hardy type operators in various Banach ideal spaces. These operators have lately been studied in Lebesgue spaces, Morrey spaces and Orlicz spaces by many authors. We briefly describe this development and thereafter we present how these results have been extended to Morrey–Orlicz spaces (Paper A) and central Morrey–Orlicz spaces (Paper B). Finally, in the third part, we introduce H¨older type spaces and present our main results from Paper C and Paper D, which concern boundedness of Hardy type operators in H¨older type spaces.

In Paper A we prove boundedness of the Riesz fractional integral opera- tor between distinct Morrey–Orlicz spaces, which is a generalization of the Adams type result. Moreover, we investigate boundedness of some weighted Hardy type operators and weighted Riesz fractional integral operator between distinct Morrey–Orlicz spaces. The Appendix A

contains detailed calcula- tions of some examples, which illustrate one of our main results presented in Paper A.

In Paper B we prove strong and weak boundedness of the Riesz potential in central Morrey–Orlicz spaces. We also give some examples, which illustrate the main theorem. Detailed calculations connected to one of the examples are described in the Appendix B

.

In Paper C we consider n-dimensional Hardy type operators and prove that these operators are bounded in H¨older spaces.

In Paper D we develop the results from paper C and derive necessary and

sufficient conditions for the boundedness of n-dimensional weighted Hardy

type operators in H¨older type spaces. We also obtain necessary and sufficient

conditions for the boundedness of weighted Hardy operators in H¨older spaces

on compactification of R

.

Preface

This PhD thesis consists of four papers (Papers A–D), two complementary appendices (A

, B

) and introduction.

[A] E. Burtseva, Weighted fractional and Hardy type operators in Orlicz–

Morrey spaces, to appear.

[A

] E. Burtseva, Appendix to Paper A.

[B] E. Burtseva, L. Maligranda and K. Matsuoka, Boundedness of the Riesz potential in central Morrey–Orlicz spaces, submitted.

[B

] E. Burtseva, L. Maligranda and K. Matsuoka, Appendix to Paper B.

[C] E. Burtseva, S. Lundberg, L.-E. Persson and N. Samko, Multi-dimensional Hardy type inequalities in H¨older spaces, J. Math.

Inequal. 12 (2018), no. 3, 719–729.

[D] E. Burtseva, L.-E. Persson and N. Samko, Necessary and sufficient

conditions for the boundedness of weighted Hardy operators in H¨older

spaces, Math. Nachr. 291 (2018), no. 11–12, 1655–1665.

Acknowledgements

First of all, I would like to thank my main supervisor Professor Lech Maligranda, Professor and Head of Department Elisabet Kassfeldt and Chair Professor Peter Wall. All three of them have in different ways been incredibly important to me during my doctoral studies.

My studies at Lule˚ a University of Technology (LTU) could have started better. In short, my original supervisors made my study conditions unsus- tainable and I was close to drop out of my studies. However, as a gift from heaven Head of Department Elisabet Kassfeldt became aware of my situa- tion. She contacted me and I felt at once that she saw me as a person, which gave me confidence to describe my situation. I will remember for life how she then appointed new supervisors and arranged everything. I am so grateful for all she did for me, both on a personal and professional level, despite that it created a lot of problems for her. Without her I would have never been able to complete my doctoral studies.

Professor Lech Maligranda gave me the best supervision one can ever dream of. It is well-known that Lech is an authority within his research field and that he has an extraordinary knowledge about the literature and the historical background. I am very grateful for all our discussions, where he with enthusiasm shared this knowledge with me. I have enjoyed every moment of our work. Lech really made me grow both as a mathematician and a person. I will remember for life his generosity and unbelievable support in the most crucial moment in my PhD studies.

Both Elisabet and Lech have made a deep impression on me. I hope that in the future I can support someone like they have supported me.

I also want to express my gratitude to my co-supervisors Professor Inge S¨oderkvist and Dr. Stefan Ericsson for helping me in various ways.

During my time as doctoral student, I also got the opportunity to take part in some very interesting research in mathematical modelling of lubri- cation. In particular, I want to thank Professor Andreas Almqvist for this possibility. It was a valuable experience for me.

I am also grateful to Dr. Johan Bystr¨om for his constant moral support.

It means a lot.

I would like to thank the Department of Engineering Sciences and Math- ematics at Lule˚ a University of Technology for offering me so good conditions and posibilities during my studies.

Finally, I am very grateful to my family and close friends for that they

are always there for me.

Introduction

In this PhD thesis we deal with Morrey–Orlicz spaces and H¨older type spaces, and we investigate the boundedness of some classical linear operators.

The thesis consists of an introduction to the research field and four papers, Papers A–D, containing the main results. The introduction is divided into three sections.

In Section 1 definitions and properties of Morrey spaces, Orlicz spaces and Morrey–Orlicz spaces are presented.

In Section 2 we consider boundedness of Hardy type operators and the Riesz potential in various Banach ideal spaces. These operators have been studied in Lebesgue spaces, Morrey spaces and Orlicz spaces by many au- thors. First we present a short overview of the development within this field.

Thereafter we describe the main results from Papers A and B. In Paper A boundedness of Hardy type operators and the Riesz potential in Morrey–

Orlicz spaces is proved. In Paper B strong and weak boundedness of the Riesz potential in central Morrey–Orlicz spaces is investigated.

In Section 3 we introduce H¨older type spaces and consider the bounded- ness of some Hardy type operators in those spaces. This section contains our main results from Papers C and D.

1 Morrey spaces, Orlicz spaces and Morrey–

Orlicz spaces

In this section we introduce Morrey spaces, Orlicz spaces and Morrey–Orlicz spaces. All these spaces are Banach ideal spaces. Therefore to describe some properties of those spaces we first give a short overview about Banach ideal spaces.

For two Banach spaces X and Y the symbol X ֒ − → Y means that the

embedding X ⊂ Y is continuous with norm at most C, i.e., kfk

≤ Ckfk

for all f ∈ X. When X ֒ − → Y holds with some (unknown) constant C > 0,

we write X ֒ − → Y.

Banach ideal spaces

Let X = (X, k · k

) be Banach ideal space on a measure space (Ω, µ).

The so-called ideal property means that if |f| ≤ |g| almost everywhere on Ω and g ∈ X, then f ∈ X and kfk

≤ kgk

. For details we refer, for instance, to [3] and [43].

1

A Banach ideal space X on a measure space (Ω, µ) is separable if and only if the measure µ is separable and the norm in X is absolutely continuous. The latter means that for any f ∈ X and any decreasing sequence of measurable sets E

with empty intersection we have kχ

f k

→ 0 as n → 0 (see, for example, [3], [39] and [43]).

For a Banach ideal space X = (X, k · k

) on (Ω, µ) the associated space X

is the collection of all measurable functions g on Ω for which

kgk

= sup

 

 Z

Ω

|f(x)g(x)| dµ: f ∈ X, kfk

≤ 1

 

 < ∞.

The associated space X

is itself a Banach ideal space and X

⊂ X

, where X

is the dual space to X. The space X

coincides with the whole dual space if and only if the norm of X is absolutely continuous. Moreover, X is continuously imbedded in X

= (X

)

, i.e. X ֒ → X

and we have X = X

with equality of the norms if and only if the norm of X has the Fatou property, that is, if 0 ≤ f

ր f a.e. on Ω and sup

n∈N

kf

k

< ∞, then f ∈ X and kf

k

ր kfk

. For more details about Banach ideal spaces we refer, for instance, to [3], [39], [43] and references therein.

1.1 Morrey spaces

Morrey spaces are known as Banach ideal spaces well suited for applications to harmonic analysis and partial differential equations (PDE). These spaces were introduced in 1938 by C. Morrey [54] in his work on systems of second order elliptic PDE. But Morrey himself was only interested in related integral inequalities in connection with smoothness properties (H¨older continuity) of solutions of nonlinear elliptic and parabolic equations. The reformulation in terms of function spaces goes back to Yu. A. Brudnyi, S. Campanato and J. Peetre in the 1960s, see, for instance, [4], [16], [17], [64], [65]. Classical and generalized Morrey spaces are presented in various sources, for example in the books [2], [28], [44], [78], [79] and the papers [5], [59], [80].

In the sequel B(a, r) denotes the open ball with a center at a ∈ R

and radius r > 0, that is {x ∈ R

: |x − a| < r}. Let |B(a, r)| be the Lebesgue measure of the ball B(a, r) which is |B(a, r)| = v

r

with v

= |B(0, 1)|. We will also use the following notation B

: = B(0, r).

2

1.1.1 Classical Morrey spaces

Let 1 ≤ p < ∞ and λ ∈ R. The Morrey space M

( R

) contains all classes of Lebesgue measurable real functions f on R

such that f ∈ L

( R

) and

Z

B(a,r)

|f(x)|

dx ≤ c|B(a, r)|

(1)

for all a ∈ R

and for all r > 0. This is Banach ideal space on R

with respect to the norm

kfk

:= sup

a∈Rⁿ,r>0

1

|B(a, r)|

Z

B(a,r)

|f(x)|

dx

!

. (2)

When λ > 1 or λ < 0, the space M

( R

) is trivial, that is, M

( R

) = {0}, where {0} is the set of all functions equivalent to 0 on R

(see [8, Lemma 1]), M

( R

) = L

( R

) and M

( R

) = L

( R

) (for the proof we refer to [44, Theorem 4.3.6]). Thus, the admissible range of the parameter λ is 0 ≤ λ ≤ 1. Moreover, the space M

( R

) does not coincide with the Lebesgue space L

( R

) or L

( R

) if and only if 0 < λ < 1.

The norm (2) in case 0 ≤ λ ≤ 1 has the Fatou property and therefore M

= M

with the equality of the norms. However, for 0 < λ ≤ 1 we have proper imbedding M

⊂ M

, since the norm (2) is not absolutely continuous. It is also important to mention that Morrey spaces M

for λ ∈ (0, 1] are not separable spaces (for the proof we refer to [80]).

One can also describe imbedding property of Morrey spaces:

if 1 ≤ p < q < ∞, 0 ≤ µ < λ < 1 and

=

, then

L

( R

) = M

( R

) M

( R

) ֒ − → M

( R

). (3) In fact, by the H¨older–Rogers inequality with

> 1 we have for any a ∈ R

Z

B(a,r)

|f(x)|

dx ≤ Z

B(a,r)

|f(x)|

dx

!

|B(a, r)|

= 1

|B(a, r)|

Z

B(a,r)

|f(x)|

dx

!

|B(a, r)|

= 1

|B(a, r)|

Z

B(a,r)

|f(x)|

dx

!

|B(a, r)|

3

,

since 1 −

+ µ

=

(µ − 1) + 1 = −(1 − λ) + 1 = λ, which implies 1

|B(a, r)|

Z

B(a,r)

|f(x)|

dx ≤ 1

|B(a, r)|

Z

B(a,r)

|f(x)|

dx

!

,

and (3) follows. Observe that both inclusions in (3) are proper (examples showing strict inclusions can be found, for instance in [36]).

One can also define the corresponding Morrey space over a domain Ω contained in R

by integrating in (1) over the sets B(x, r) ∩ Ω and taking supremum over 0 < r ≤ diam Ω. But mostly we will deal with Morrey spaces over the entire R

.

1.1.2 Generalized Morrey spaces

Let G be the set of all measurable functions ϕ: [0, ∞) → [0, ∞) satisfying the following assumptions:

r

lim

ϕ(r) = ϕ(0) = 0, ϕ(r) = 0 ⇔ r = 0, ϕ(r) ≥ Cr

for some constant C > 0 and all 0 < r ≤ 1.

Replacing |B(a, r)|

by such a function ϕ(r) ∈ G in the definition of Morrey spaces M

( R

) we obtain generalized Morrey spaces

M

( R

) = (

f ∈ L

( R

) : sup

a∈Rⁿ, r>0

1 ϕ(r)

Z

B(a,r)

|f(x)|

dx < ∞ )

,

with the norm defined by

kfk

:= sup

a∈Rⁿ, r>0

1 ϕ(r)

Z

B(a,r)

|f(x)|

dx

!

.

Generalized Morrey spaces appeared already in 1969 in the paper [65] of J. Peetre, but were first investigated by C. T. Zorko in the paper [83] from 1986. For the historical development of Morrey type spaces and their different properties we refer to the survey paper by V. I. Burenkov [5].

1.1.3 Central Morrey spaces

Let 1 ≤ p < ∞ and λ ∈ R. The central Morrey space M

(0) is defined as the space of all classes of Lebesgue measurable real functions f on R

, such

4

that f ∈ L

( R

) and

kfk

:= sup

r>0



 1

|B

|

Z

Br

|f(x)|

dx





1 p

< ∞. (4)

The space M

(0) is non-trivial, that is, M

(0) 6= {0} if and only if λ ≥ 0.

For the proof we refer to [8] and [10], see also the survey paper [5].

Obviously, for 0 ≤ λ ≤ 1

M

( R

) ֒ − → M

(0)

and the inclusion is proper for 0 < λ ≤ 1. Moreover, M

( R

) = M

(0) = L

( R

) and M

( R

) = L

( R

). However, L

( R

) ֒ − → M

(0) and the inclusion is strict.

Note, that in the case of central Morrey spaces the admissible range of the parameter λ is λ ≥ 0, which is essentially wider than in the case of classical Morrey spaces, when we only have 0 ≤ λ ≤ 1.

1.2 Orlicz spaces

Let us recall the definition of Orlicz spaces on R

and some of their properties which will be used later on (see [42] and [49] for details). These spaces were introduced by W. Orlicz in [61] and [62] as a generalization of L

-spaces.

A function Φ : [0, ∞) → [0, ∞) is called an Orlicz function if it is an in- creasing, continuous and convex function with Φ(0) = 0. Each such a function Φ has an integral representation Φ(u) =

R

p(s) ds, where p is a nondecreasing right continuous function. Here, Φ

(u) = p(u) almost everywhere on (0, ∞).

If we want to include in the Orlicz spaces, for example, spaces L

( R

) and L

( R

) ∩ L

( R

), then we need to consider the so-called Young functions. A function Φ : [0, ∞) → [0, ∞] is called a Young function if it is a nondecreasing and convex function with lim

u→0+

Φ(u) = Φ(0) = 0, and not identically 0 or ∞ in (0, ∞). It may have jump up to ∞ at some point u > 0, but then it should be left continuous at u.

To each Young function Φ one can associate a complementary function Φ

, defined for v ≥ 0 by

Φ

(v) = sup

u>0

[uv − Φ(u)].

5

We say that a Young function Φ satisfies the ∆

−condition, and we write Φ ∈ ∆

, if 0 < Φ(u) < ∞ for u > 0 and there exists a constant C ≥ 1 such that Φ(2u) ≤ CΦ(u) for all u > 0.

For any Young function Φ the Orlicz space L

( R

) contains all classes of Lebesgue measurable real functions f on R

such that

Z

Rⁿ

Φ 

k |f(x)| 

dx < ∞ for some k = k(f) > 0.

This space with the Luxemburg–Nakano norm kfk

= inf

( ε > 0 :

Z

Rⁿ

Φ |f(x)|

ε

! dx ≤ 1

)

is a Banach ideal space. For further properties of Orlicz spaces we refer, for instance, to the books [42], [49] and [68].

1.3 Morrey–Orlicz spaces

Next we define Morrey–Orlicz (or Orlicz–Morrey) spaces. The Morrey–Orlicz spaces M

( R

), introduced by E. Nakai [58] in 2004, unify Morrey and Orlicz spaces.

For a Young function Φ, λ ∈ R and r > 0, let kfk

denote the λ-central mean Luxemburg–Nakano norm of f on B(a, r), defined by

kfk

= inf (

ε > 0 : 1

|B(a, r)|

Z

B(a,r)

Φ |f(x)|

ε

! dx ≤ 1

) .

The corresponding (smaller) weak λ-central mean Luxemburg–Nakano norm kfk

is

kfk

= inf (

ε > 0 : sup

u>0

Φ u ε

 1

|B(a, r)|

d(f χ

, u) ≤ 1 )

,

where d(f, u) = |{x ∈ R

: |f(x)| > u}|.

In particular,

kfk

= inf (

ε > 0 : 1

|B

|

Z

Br

Φ |f(x)|

ε

! dx ≤ 1

)

6

and

kfk

= inf (

ε > 0 : sup

u>0

Φ u ε

 1

|B

|

d(f χ

, u) ≤ 1 )

.

Then using these notions we can define Morrey–Orlicz spaces M

( R

) and weak Morrey–Orlicz spaces W M

( R

) in the following way:

M

( R

) = (

f ∈ L

( R

) : kfk

= sup

B=B(a,r)

kfk

< ∞ )

,

W M

( R

) = (

f ∈ L

( R

) : kfk

= sup

B=B(a,r)

kfk

< ∞ )

.

Similarly, we can define central Morrey–Orlicz spaces M

(0) and weak cen- tral Morrey–Orlicz spaces W M

(0):

M

(0) = (

f ∈ L

( R

) : kfk

= sup

r>0

kfk

< ∞ )

,

W M

(0) = (

f ∈ L

( R

) : kfk

= sup

r>0

kfk

< ∞ )

.

The last spaces were defined by L. Maligranda and K. Matsuoka in [50], where the boundedness of maximal function was investigated. There are also other ways to define these spaces (we refer for example to [22] and [71]) but in this thesis we follow the definition given in [58].

In particular cases we obtain the following spaces (see [50] for more de- tails):

(a) Orlicz and weak Orlicz spaces: M

( R

) = M

(0) = L

( R

) and W M

( R

) = W M

(0) = W L

( R

).

(b) Classical Morrey, weak Morrey, central Morrey and weak central Mor- rey spaces: If Φ(u) = u

, 1 ≤ p < ∞ and λ ≥ 0, then M

( R

) = M

( R

), W M

( R

) = W M

( R

) and M

(0) = M

(0), W M

(0) = W M

(0).

(c) Beurling–Orlicz and weak Beurling–Orlicz spaces: M

( R

) = B

( R

) and W M

( R

) = W B

( R

).

7

Let Φ be a Young function and ϕ ∈ G. We define generalized Morrey–

Orlicz spaces M

( R

) in the following way:

M

( R

) = (

f ∈ L

( R

) : kfk

= sup

B=B(a,r)

kfk

< ∞ )

, (5)

where r > 0, a ∈ R

and kfk

is defined by

kfk

= inf (

λ > 0 : 1 ϕ(r)

Z

B(a,r)

Φ |f(x)|

λ

! dx ≤ 1

) .

In the case Φ(u) = u

, 1 ≤ p < ∞, the generalized Morrey–Orlicz space M

( R

) turns into the generalized Morrey space M

( R

). For the basic properties of M

( R

)-spaces we refer to [60].

2 Boundedness of some linear operators between distinct Morrey–Orlicz spaces

In this section we consider boundedness of Hardy type operators and the Riesz potential in Morrey–Orlicz spaces.

First we give a brief historical overview of the main results concerning the boundedness of Hardy type operators in Lebesgue spaces, Morrey spaces and Orlicz spaces. Then we present our main result (Theorem 1) on the bounded- ness of some weighted Hardy type operators between distinct Morrey–Orlicz spaces. This result is proved in Paper A.

Next we consider the Riesz potential. Many authors have investigated boundedness of this operator in classical spaces such as Lebesgue spaces, Orlicz spaces and Morrey spaces. We start by giving a brief review of the development which has taken place within the area and then we present the main results in this thesis concerning boundedness of the Riesz potential in Morrey–Orlicz spaces and central Morrey–Orlicz spaces.

First we present our result on non-weighted boundedness of this operator between distinct Morrey–Orlicz spaces, which is a generalization of Adams type result (Theorem 2). Then we consider weighted fractional integral op- erator and basing on our result for Hardy operators (see subsection 2.1) we provide conditions for this operator to be bounded between distinct Morrey–

Orlicz spaces (Theorem 3). These results are presented in Paper A.

Next we prove the strong-type and weak-type estimates for the Riesz potential in central Morrey–Orlicz and weak central Morrey–Orlicz spaces.

8

In Theorem 4 we give necessary conditions for boundedness of this opera- tor in central Morrey–Orlicz spaces, and in Theorem 5 we obtain sufficient conditions for such boundedness. These results are presented in Paper B.

2.1 Hardy type operators

Suppose that p > 1, f (x) ≥ 0 and f

is integrable over (0, ∞). Then the classical Hardy’s integral inequality from 1925 (see [37]) reads:

Z



 1 x

Z

f (y) dy





p

dx ≤

 p

p − 1



Z

0

f (x)

dx.

This inequality means that the classical Hardy operator Hf (x) =

R

f (t) dt is bounded in L

(0, ∞) and its norm is kHk

≤

. One can even show that its norm is kHk

=

, for proof we refer to [46, Example 1, pp.

20–21].

The history of development of the classical Hardy inequality during the period 1906–1928 is described in [45] (see also [46]). In particular, it was shown the importance of contributions of famous mathematicians other than G. H. Hardy, such as E. Landau, G. P´ olya, M. Riesz and I. Schur.

The boundedness of the one-dimensional Hardy operator between dif- ferent weighted Lebesgue spaces has been studied by many authors until 1980-ies and finally the classical Hardy inequality was extended to the gen- eral Hardy inequality



 Z



 Z

f (t) dt





q

u(x) dx





1 q

≤ C



 Z

a

f (x)

v(x) dx





1 p

, f ≥ 0,

with parameters a, b, p, q, such that −∞ ≤ a < b ≤ ∞, 0 < q ≤ ∞, 1 ≤ p ≤ ∞, and with u(x), v(x) given weight functions (i.e. functions which are measurable and positive almost everywhere on (a, b)). For formulations and proofs of those results and for the history of the problem we refer to the books [25], [46] and references therein.

In 1995 M. Christ and L. Grafakos proved in [20] that n-dimensional Hardy operator

Hf (x) = 1

|x|

Z

|y|≤|x|

f (y) dy

9

is bounded in L

( R

) for 1 < p ≤ ∞. Then, in 1997 P. Drabek, H. Heinig and A. Kufner extended this result to the weighted n-dimensional case in

[23] 

 Z

Rⁿ

|Hf(x)|

u(x) dx





1 q

≤ C



 Z

Rⁿ

|f(x)|

v(x) dx





1 p

for 1 < p ≤ q < ∞ and general n-dimensional weights u and v.

The investigation of Hardy integral inequality in Orlicz spaces started in 1960s. In 1964 N. Levinson [47] proved that if Φ is a twice differentiable convex increasing function on [0, ∞) with Φ(t)Φ

(t) ≥ (1 −

)Φ

(t)

for all t > 0, then

Z

0

Φ



 1 x

Z

0

|f(t)| dt



 dx ≤ (p

)

Z

0

Φ( |f(x)|) dx.

Later this result was modified by M. Carro and H. Heinig [18]. Results concerning the boundedness of Hardy operators in Orlicz spaces can be found in papers by E. K. Godunova [29], V. Kokilashvili [40] and G. Palmieri [63].

In 2000s some authors considered boundedness of Hardy type operators in Morrey type spaces. For example, conditions under which the Hardy type operator

H

f (x) = |x|

Z

|y|≤|x|

f (y) dy

is bounded between distinct Morrey type spaces were presented in [10].

V. I. Burenkov and R. Oinarov in [11] considered a more general Hardy type operator

H

f (x) = u( |x|) Z

|y|≤|x|

f (y) dy, x ∈ R

,

where u is a fixed non-negative measurable function on (0, ∞), which is not equivalent to 0. They proved that the operator H

is bounded from weighted Lebesgue space to Morrey type space.

In Paper A we consider boundedness of n-dimensional weighted Hardy operators between distinct Morrey–Orlicz spaces:

H

f (x) = |x|

w( |x|) Z

|y|≤|x|

f (y) w( |y|) dy, H

f (x) = |x|

w( |x|)

Z

|y|>|x|

f (y)

|y|

w( |y|) dy.

(6)

10

Here w : (0, ∞) → (0, ∞) is continuous measurable function such that w(2r) ≤ Cw(r) for some constant C > 0, and

is almost increasing on (0, ∞) for some a ∈ R.

We consider generalized Morrey–Orlicz spaces M

( R

) and M

( R

), which are defined in (5) and such that the functions ϕ and ψ satisfy the following conditions:

ϕ is increasing on (0, ∞) and ϕ(r)

r

is decreasing on (0, ∞), ψ is almost increasing on (0, ∞), ϕ(r) ≤ Aψ(r) for some constant A > 0 and any r > 0.

The main result in this thesis concerning boundedness of the Hardy type operators H

and H

from M

( R

) to M

( R

) reads:

Theorem 1. Let 0 < α < n, Φ be an Orlicz function,

be almost decreas- ing on (0, ∞) and

Z

ϕ(t) dt

t ≤ Cϕ(r) (7)

for some constant C > 0 and all r > 0. Let also V (r) = Φ

(r) [U

(r)]

be continuous, increasing, unbounded, concave on [0, ∞) with V (0) = 0, and Ψ = V

, where U(r) =

.

(i) The Hardy operator H

is bounded from M

( R

) to M

( R

), pro- vided

r

w(r) Φ

(U(r)) is almost increasing on (0, ∞) and Z

0

t

w(t) Φ

(U(t)) dt t ≤ C

r

w(r) Φ

(U(r)) , (8)

for some constant C

> 0 and all r > 0.

(ii) The Hardy operator H

is bounded from M

( R

) to M

( R

), pro- vided

Φ

(U(r))

w(r) is almost decreasing on (0, ∞) and Z

r

Φ

(U(t)) w(t)

dt t ≤ C

Φ

(U(r)) w(r) ,

(9)

for some constant C

> 0 and all r > 0.

11

In the case when w(t) ≡ 1 we don’t need to require the condition (8) for the M

( R

) → M

( R

) − boundedness of the operator H

= H

|

. Remark 1. Let 0 < α < n,

be almost decreasing on (0, ∞) and ϕ satisfy condition (7). Let also V (r) = Φ

(r) [U

(r)]

be continuous, increasing, unbounded, concave on [0, ∞) with V (0) = 0, and Ψ = V

, where U(r) =

ϕ(r)

rⁿ

. Then the Hardy operator H

is bounded from M

( R

) to M

( R

).

As we will see further on, we can apply the result in Theorem 1 on the boundedness of Hardy operators H

and H

between distinct Morrey–

Orlicz spaces to prove boundedness of the weighted Riesz potential operator in Morrey–Orlicz spaces. In Section 3 we continue to consider Hardy type operators and we investigate the boundedness of those operators in H¨older type spaces.

2.2 Riesz potential

The fractional integral operator or Riesz potential I

, 0 < α < n, defined for x ∈ R

by

I

f (x) = Z

Rⁿ

f (y)

|x − y|

dy, (10)

plays a role in various branches of analysis, including potential theory, har- monic analysis, Sobolev spaces and partial differential equations, see, for example [2], [32], [76] and [77].

The study of boundedness of I

between L

-spaces was initiated by S. Sobolev in 1938. He proved in [74] that the operator I

is bounded from L

( R

) to L

( R

) for 1 < p <

if and only if

=

−

(cf. [76, Theo- rem 1, pp. 119–121]). Boundedness of the Riesz potential between Orlicz spaces was done by I. B. Simonenko [73] in 1964 and, later on, by A. Cianchi [21] in 1999 in full generality. Another characterizations for the strong and weak boundedness of the Riesz potential on Orlicz spaces were also given by V. S. Guliyev, F. Deringoz, and S. Hasanov [34] in 2017. The results on boundedness of the Riesz fractional integral operator from M

( R

) to M

( R

) were first obtained by S. Spanne with the Sobolev exponent

1

q

=

−

, and this result was published in 1969 by J. Peetre [65].

Spanne–Peetre theorem. Let 0 < α < n, λ > 0 and 1 < p <

. If

1

q

=

−

and

=

, then there exists a constant C > 0 independent of f such that

kI

f k

≤ Ckfk

for every f ∈ M

( R

). Constant C depends on p, α, λ and n.

12

In 1975 a stronger result with a better exponent

=

−

was obtained by D. R. Adams [1] (see also [19]) in the following statement:

Adams theorem. Let 0 < α < n, λ > 0 and 1 < p <

. If

=

1

p

−

, then I

is bounded from M

( R

) to M

( R

).

It is easy to see from (3) that Adams theorem improves the Spanne–Peetre theorem.

In 1994 V. S. Guliyev [30] and E. Nakai [56] extended Spanne–Peetre’s re- sult and proved the boundedness of I

in generalized Morrey spaces.

A. Eridani and H. Gunawan [26] obtained Adams type result for generalized Morrey spaces in 2002. After that many authors investigated boundedness of the Riesz potential in Morrey type spaces, see for instance [7], [9], [31], [32], [53] and references therein. More information concerning the development of this area can be found in the survey paper [6].

Then it was natural to consider the boundedness of operator I

in Morrey–

Orlicz spaces. In 2008 E. Nakai [60] studied the M

( R

) → M

( R

)–

boundedness of the fractional integral operator and obtained Adams type result.

In Paper A we study boundedness of Riesz potential I

and weighted frac- tional integral operator w( |x|) I

f

(x) between distinct Morrey–Orlicz spaces.

First we consider M

( R

) → M

( R

) −boundedness of the operator I

, using Hedberg’s method [38] and obtain Adams type result. Note, that the assumptions on functions Φ, ϕ and Ψ, ψ, which define Morrey–Orlicz spaces M

and M

, respectively, are the same as in the Theorem 1. Our main result on non-weighted boundedness of the operator I

reads as follows:

Theorem 2. Let 0 < α < n, Φ be an Orlicz function such that its comple- mentary function Φ

∈ ∆

. If there exists a constant C > 0 such that

Z

t

Φ

 ϕ(t) t

 dt

t ≤ Cr

Φ

 ϕ(r) r



for all r > 0, (11)

then the Riesz fractional integral operator I

is bounded from M

( R

) to M

( R

).

The proof of this theorem is given in the Paper A (cf. [12, Theorem 1]).

Next we consider the weighted fractional integral operator w( |x|) I

f

(x). We deal with the following classes of weights:

V

: |w(x) − w(y)|

|x − y|

≤ C w (max {x, y})

(max {x, y})

,

13

V

: |w(x) − w(y)|

|x − y|

≤ C w (min {x, y}) (max {x, y})

,

where w : (0, ∞) → (0, ∞) is a continuous weight function, 0 < µ ≤ 1, x, y > 0 and x 6= y. Observe that if w(t) ∈ V

, then

∈ V

. Typical examples of such weights are

1. w(t) = t

∈ V

and w(t) = t

∈ V

, where β ≥ 0.

2. w(t) = t

(ln(e + t))

∈ V

and w(t) = t

(ln(e + t))

∈ V

, where β > 1 and γ ≥ 0.

The weighted Riesz fractional integral operator can be estimated by the non-weighted Riesz fractional integral operator and the weighted Hardy type operators. We will use the following pointwise estimates, obtained in [69]:

Pointwise estimate. Let 0 < α < n, w ∈ V

∪ V

with µ = min {1, n − α}

be a weight and f : R

→ [0, ∞) be a given measurable function. Then the following pointwise estimates hold

w( |x|)

 I

1

w f



(x) ≤ I

f (x) + c

( H

f (x) + H

f (x), if w ∈ V

, H

f (x) + H

f (x), if w ∈ V

, where H

= H

|

and w

(x) = |x|

w(x).

In the next theorem we will require on the above weight function w addi- tionally that w(2r) ≤ Cw(r) for some constant C > 0 and all r > 0, and

is almost increasing for some a ∈ R. Note, that if w ∈ V

, then w is almost increasing on (0, ∞). Therefore we don’t need to require assumption on the function

to be almost increasing since it is obviously satisfied with a = 0.

Our main result on boundedness of the weighted Riesz potential reads:

Theorem 3. Let 0 < α < n, Φ be an Orlicz function such that its com- plementary function Φ

∈ ∆

. Assume that µ = min {1, n − α} and weight w ∈ V

∪V

. Assume also that

is almost decreasing on (0, ∞), ϕ satisfies condition (7) and assumption (11) holds.

1. If w ∈ V

, then the weighted Riesz potential operator w( |x|) I

f
(x) is bounded from M

( R

) to M

( R

) provided conditions (8) hold.

2. If w ∈ V

, then the weighted Riesz potential operator w( |x|) I

f
(x) is bounded from M

( R

) to M

( R

) provided

r

w(r) Φ

 ϕ(r) r



is almost decreasing on (0, ∞)

14

and Z

r

t

w(t) Φ

 ϕ(t) t

 dt

t ≤ C r

w(r) Φ

 ϕ(r) r

 , for some constant C > 0 and all r > 0.

The proof of this theorem is given in the Paper A (cf. [12, Theorem 3]).

Result on the boundedness of the Riesz potential I

in central Morrey spaces was proved in [27, Proposition 1.1]: if 1 < p <

, 0 < λ < 1,

=

1

p

−

and

=

, then the Riesz potential I

is bounded from M

(0) to M

(0) (see also [7, Theorem 5.1 (j)], where the result is proved even for more general local Morrey type spaces). In [33] V. S. Gulyiev proved boundedness of I

in generalized central Morrey spaces. Moreover, Y. Komori-Furuya and E. Sato showed in [41, Proposition 1] that Adams type result does not hold in central Morrey spaces, that is, if

=

−

and

<

−

<

, then I

is not bounded from M

(0) to M

(0) because

<

.

In Paper B we generalize the last results to central Morrey–Orlicz spaces.

In Theorem 4 we give necessary conditions for boundedness of I

from M

(0) to M

(0) and in this case one of our main results reads:

Theorem 4. Let 0 < α < n, Φ, Ψ be Orlicz functions and 0 ≤ λ, µ < 1.

(i) If the Riesz potential I

is bounded from M

(0) to M

(0), then there are positive constants C

, C

such that

(a) u

Φ

(u

) ≤ C

Ψ

(u

) for any u > 0.

(b) s

(u

) ≤ C

u

s

(u

) for any u > 0.

(ii) If there exists a small constant c > 0 such that c ≤

with v

= 1 and

lim inf

t→∞

Φ

(ct

) Ψ

(t

) = ∞, then I

is not bounded from M

(0) to M

(0).

In Theorem 5 we obtain sufficient conditions for the strong and weak boundedness of the Riesz potential in central Morrey–Orlicz spaces:

Theorem 5. Let 0 < α < n, Φ, Ψ be Orlicz functions and either 0 < λ, µ < 1, λ 6= µ or λ = µ = 0. Assume that there exist constants C

, C

≥ 1 such that

Z

u

t

Φ

(t

) dt

t ≤ C

Ψ

(u

) for all u > 0 (12)

15

and Z

u

t

Φ

r

t

 dt

t ≤ C

Ψ

r

u

 for all u > 0 and for all r > 0. (13)

(i) If Φ

∈ ∆

, then I

is bounded from M

(0) to M

(0), that is, there exists a constant C

≥ 1 such that kI

f k

≤ C

kfk

for all f ∈ M

(0).

(ii) The operator I

is bounded from M

(0) to W M

(0), that is, there exists a constant c

≥ 1 such that kI

f k

≤ c

kfk

for all f ∈ M

(0).

3 Hardy operators in H¨ older type spaces

Let Ω be a domain (i.e., an open connected set) in R

, Ω ⊆ R

, n ≥ 1 and 0 < λ ≤ 1. Condition of the form

|f(x) − f(y)| ≤ c|x − y|

, for some c > 0 and all x, y ∈ Ω, occurred for the first time in the work of R. Lipschitz on trigonometric series in 1864 and on ordinary differential equations in 1876 and also in H¨older’s Ph.D. thesis “Beitr¨agen zur Potential Theorie” in 1882.

By C

(Ω) we denote all bounded continuous functions f defined on Ω such that the seminorm

[f ]

: = sup

x,y∈Ω,x6=y

|f(x) − f(y)|

|x − y|

is finite. These spaces are called H¨older spaces or sometimes Lipschitz spaces.

If we equip them with the norm kfk

= sup

x∈Ω

|f(x)| + [f]

,

then C

(Ω) are Banach spaces. The definition of this norm goes back to J. Schauder [72] (cf. [66, p. 440]). If λ = 1 then we say that f is a Lipschitz function on Ω. For more properties of H¨older spaces we refer for instance to [24], [44], [55], [66], [70], [75], [82] and references therein.

By ˜ C

(Ω) we denote all continuous functions on Ω such that [f ]

< ∞.

We also define the following closed subspaces:

C

(Ω) = {f ∈ C

(Ω) : f (0) = 0 },

16

C ˜

(Ω) = {f ∈ ˜ C

(Ω) : f (0) = 0 }.

Note that [f ]

is a norm in the space ˜ C

(Ω).

We will also use the H¨older spaces on compactification of R

. Let ˙ R

denote the compactification of R

by a single infinite point. By C

( ˙ R

) for 0 < λ ≤ 1 we denote all bounded continuous functions f on R

such that there exists lim

|x|→∞

f (x) = f ( ∞),

|f(x) − f(y)| ≤ C |x − y|

(1 + |x|)

(1 + |y|)

for all x, y ∈ R

and

|f(x) − f(∞)| ≤ C

(1 + |x|)

for all x ∈ R

. This is a Banach space with respect to the norm

kfk

= kfk

+ sup

x,y∈ ˙Rⁿ,x6=y

|f(x) − f(y)|

 (1 + |x|)(1 + |y|)

|x − y|



. We will also deal with the following closed subspaces:

C

( ˙ R

) = {f ∈ C

( ˙ R

) : f (0) = 0 }, C

( ˙ R

) = {f ∈ C

( ˙ R

) : f ( ∞) = 0}

and

C

( ˙ R

) = C

( ˙ R

) ∩ C

( ˙ R

).

For more properties of H¨older spaces on compactification of R

we refer to the book [75, Chapter 2].

In the Paper C or in its published version [13] of this thesis we study Hardy type operators (6) with w = 1, that is,

H

f (x) = |x|

Z

|y|≤|x|

f (y) dy,

H

f (x) = |x|

Z

|y|>|x|

f (y)

|y|

dy,

(14)

where α ≥ 0 and x ∈ B

(we will sometimes include also r = ∞, which in fact means that B

= R

). We write H = H

and H = H

in the case α = 0.

The operator H may be considered in both with and without compactification settings, but a consideration of H requires the compactification due to the needed convergence of integrals at infinity.

17

We prove that the operator H is bounded in C

(B

) and the operator H

is bounded from ˜ C

(B

) into ˜ C

(B

), 0 < r ≤ ∞, in the case α > 0, where 0 < λ + α ≤ 1 in both cases. This result is obtained in [13] as the Theorem 2.2:

Theorem 6. Let α ≥ 0, λ > 0, λ + α ≤ 1 and 0 < r ≤ ∞.

(i) In the case α = 0 the Hardy operator H

is bounded in C

(B

) and [H

f |

]

≤ C[f]

.

(ii) In the case α > 0 the operator H

is bounded from ˜ C

(B

) into ˜ C

(B

).

In the case of spaces with compactification we obtain the following result, proved in [13, Theorem 3.2, Theorem 3.3]:

Theorem 7. If 0 ≤ λ < 1, then the operator H is bounded in C

( ˙ R

). If 0 < λ < 1, then the operator H is bounded from C

( ˙ R

) to C

( ˙ R

).

In Paper D, presented in published paper [15], we further develop the approach from Paper C to obtain weighted results for Hardy operators in H¨older spaces. We consider the n-dimensional weighted Hardy operators H

and H

defined in (6) with power weight (w( |u|) = |u|

), that is,

H

f (x) = |x|

Z

|y|≤|x|

f (y)

|y|

dy, H

f (x) = |x|

Z

|y|>|x|

f (y)

|y|

dy,

(15)

where x ∈ R

, α ∈ [0, 1) and γ ∈ R. We write H

= H

and H

= H

in the case α = 0.

It should be noted that results concerning boundedness of weighted Hardy operators in H¨older spaces don’t in general include the corresponding results in the non-weighted case as a particular case.

We need to introduce more spaces of H¨older type. We say that f ∈ C ˜

( R

) if

[f ]

: = sup

x∈Rⁿ

|f(x) − f(0)|

|x|

< ∞.

We also need the space ˜ C

( R

, ̺) of differentiable in R

\ {0} functions f, where ̺ = ̺( |x|) is a radial weight. We equip the space ˜ C

( R

, ̺) with the seminorm

[f ]

: = sup

x∈Rⁿ

̺( |x|)|grad f(x)|.

18

Treated as factor spaces over the constants the spaces ˜ C

( R

), ˜ C

( R

) and C ˜

( R

, ̺) are normed spaces with the corresponding seminorms serving as norms. We also use the following H¨older spaces:

C

( R

),

C

( R

) and

ν

C

( R

, ̺) defined by the norms kfk

= sup

x∈Rⁿ

|f(x)|

|x|

+ [f ]

, kfk

= sup

x∈Rⁿ

|f(x)|

|x|

+ [f ]

and

kfk

= sup

x∈Rⁿ

|f(x)|

|x|

+ [f ]

,

respectively. When ν = 0 we have our earlier spaces C

( R

), C

( R

) and C

( R

, ̺), respectively.

By ˜ C

( R

), ˜ C

( R

), ˜ C

( R

, ̺),

C

( R

),

C

( R

) and

C

( R

, ̺) we denote the subspaces of the corresponding spaces, defined by the additional condition f (0) = 0.

First we consider boundedness of the operators (15) in H¨older spaces without compactification and obtain estimates via the spherical means

Φ(t) = 1

|S

| Z

Sⁿ⁻¹

f (tσ) dσ.

Such estimates give substantially stronger results than estimates via the cor- responding norm of the function f, which appears in spherical means Φ. We obtain the following results, proved in [15, Theorem 4.1]:

Theorem 8. Let α ≥ 0, λ > 0 and λ + α ≤ 1.

(i) The inequality

kH

k

≤ CkΦk

holds if and only if γ < n + λ. This implies that the operator H

is bounded from ˜ C

( R

) to

C

( R

, |x|

).

(ii) The inequality

kH

k

≤ CkΦk

holds if and only if γ < n. This implies that the operator H

is bounded from C

( R

) to

C

( R

, |x|

).

19

(iii) The inequalities

kH

k

≤ CkΦk

and

kH

k

≤ CkΦk

hold if and only if γ > λ. This implies that the operator H

is bounded from ˜ C

( R

) to

C

( R

, |x|

) and from C

( R

) to

C

( R

, |x|

), respectively.

Theorem 9. Let 0 < λ ≤ 1.

(i) The inequality

kH

k

≤ CkΦk

holds if and only if γ < n. This implies that the operator H

is bounded from C

( R

) to C

( R

, |x|

).

(ii) The inequality

kH

k

≤ CkΦk

holds if and only if γ > λ. This implies that the operator H

is bounded from C

( R

) to C

( R

, |x|

).

Then we consider the case of H¨older type spaces on compactification of R

. We present weighted estimates for Hardy operators (15) in the case α = 0 and our main result reads:

Theorem 10. Let 0 ≤ λ < 1, then the operator H

is bounded in C

( ˙ R

) if and only if γ < n − λ.

Let 0 < λ < 1, then the operator H

is bounded in C

( ˙ R

) if and only if γ > λ.

20

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