Operators and Inequalities in various Function Spaces and their Applications

LICENTIATE T H E S I S

Department of Engineering Sciences and Mathematics Division of Mathematical Science

Operators and inequalities in various function spaces and their applications

ISSN 1402-1757 ISBN 978-91-7583-723-9 (print)

ISBN 978-91-7583-724-6 (pdf) Luleå University of Technology 2016

Evgeniya Burtseva Operators and inequalities in various function spaces and their applications

Evgeniya Burtseva

Mathematics

Operators and inequalities in various function spaces and their applications

Evgeniya Burtseva

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

SE-971 87 Lule˚ a, Sweden

Printed by Luleå University of Technology, Graphic Production 2016 ISSN 1402-1757

ISBN 978-91-7583-723-9 (print) ISBN 978-91-7583-724-6 (pdf) Luleå 2016

2010 Mathematics Subject Classification: 46E30, 47G10, 47G40 Keywords: generalized weighted Morrey space, classical Morrey spaces, potential operator, generalized potential operator, singular integral operator, operator theory, integral inequalities.

Abstract

In this thesis we study the theory of operators and inequalities in some non- standard function spaces, with emphasis on mapping properties of various operators between various function spaces. We consider the main operators of harmonic analysis: Singular and Potential type operators, Maximal operators and Hardy type operators. All these operators are studied in Morrey type spaces, including also weighted versions of such spaces. The Hardy type operators, both in weighted and non-weighted settings, are also studied in H¨older type spaces.

This thesis consists of five papers (papers A - E) and an introduction.

In paper A we prove the boundedness of the Riesz fractional integration operator from a generalized Morrey space to a certain Orlicz-Morrey space, which covers the Adams result for Morrey spaces. We also give a generaliza- tion to the case of weighted Riesz fractional integration operators for some class of weights.

In paper B we study the boundedness of the Cauchy Singular Integral operator on curves in complex plain in generalized Morrey spaces. We also consider the weighted case with radial weights. We apply these results to the study of Fredholm properties of singular integral operators in weighted generalized Morrey spaces.

In paper C we prove the boundedness of the Potential operator in weighted generalized Morrey space in terms of Matuszewska-Orlicz indices of weights and apply this result to the Helmholtz equation with a free term in such a space. We also give a short overview of some typical situations when Poten- tial type operators arise when solving PDEs.

In paper D some new inequalities of Hardy type are proved. More exactly, the boundedness of multidimensional weighted Hardy operators in H¨older spaces are proved in cases with and without compactification.

In paper E the mapping properties are studied for Hardy type and gen- eralized potential type operators in weighted Morrey type spaces.

Preface

This thesis is composed of five papers (A-E). These publications are put into a more general frame in an introduction that also serves as a basic overview of the field.

A E. Burtseva and N. Samko. Weighted Adams type theorem for the Riesz fractional integral in generalized Morrey space. Fract. Calc.

Appl. Anal. 19 (2016), no. 4, 954-972.

B E. Burtseva. Singular integral operators in generalized Morrey spaces on curves in the complex plane. Research report 2016-08, ISSN: 1400- 4003, Department of Engineering Sciences and Mathematics, Lule˚a Uni- versity of Technology. (Submitted)

C E. Burtseva, S. Lundberg, L.-E. Persson and N. Samko. Potential Type Operators in PDEs and their applications. AIP Conf. Proc., to appear 2016.

D E. Burtseva, S. Lundberg, L.-E. Persson and N. Samko. Multidimen- sional Hardy type inequalities in H¨older spaces. Research report 2016- 09, ISSN: 1400-4003, Department of Engineering Sciences and Mathe- matics, Lule˚a University of Technology. (Submitted)

E E. Burtseva and N. Samko. Generalized Potential Operators in Morrey type spaces. (Submitted)

Acknowledgements

I would like to express my deepest appreciation to my supervisors Professor Natasha Samko, Professor Lars-Erik Persson and Professor Peter Wall for their generous professional and personal support. I am sincerely grateful for their understanding, kindness, patience and wise advices.

First of all I want to thank my main supervisor Professor Natasha Samko for involving me into a new mathematical field and opening a new exciting and fascinating world for me. I am eternally grateful to Natasha for her total support and encourage which inspires and gives me a lot of energy and enthusiasm to work. Her careful and persistent help contributed enormously to the production of this thesis.

I am sincerely thankful to Professor Lars-Erik Persson for the successful collaboration, exclusive support and his truly fatherly care.

I express my deep gratitude to Professor Peter Wall for his permanent support and, above all, for giving me the opportunity to have all the condi- tions for fruitful scientific work.

I am also grateful to the Department of Engineering Sciences and Math- ematics at Lule˚a University of Technology for such a good opportunity to realise my ambitions.

Introduction

In this Licentiate thesis we deal with the following function spaces: Mor- rey type spaces, H¨older type spaces and their weighted versions. Both Morrey and H¨older spaces are non-separable spaces.

We consider Singular operators and their commutators; Potential type operators, Maximal operators and Hardy type operators.

1 Spaces

1.1 Morrey type spaces

Morrey type spaces are known as function spaces well suited for applications to partial differential equations and there exist many studies of ”Morrey regularity” of their solutions. Morrey spaces were introduced in 1938 by C. Morrey [33] in relation to regularity properties of solutions to partial differential equations. Classical Morrey spaces are presented in various books, see for instance [13], [25], [53], [54]. Morrey spaces, classical or generalized, have a long history which is well presented in various sources, see for instance the papers [37], [45], [55] and references therein. We also refer to [2] for the latest research on the theory of Morrey spaces associated with Harmonic Analysis.

Let Ω⊆ Rⁿbe an open set. We denoteB(x, r) = B(x, r)^∼ ∩Ω, x ∈ Ω, r > 0.

1.1.1 Classical Morrey spaces

Definition 1. Let 1 ≤ p < ∞ and λ ≥ 0. The Morrey space L^p,λ(Ω) is defined as

L^p,λ(Ω) =

⎧⎪

⎪⎨

⎪⎪

⎩

f ∈ L^p(Ω) : sup

x∈Ω;r>0

1 r^λ



∼B(x,r)

|f(y)|^pdy <∞

⎫⎪

⎪⎬

⎪⎪

⎭

. (1)

This is a Banach space with respect to the norm

f_L^p,λ_(Ω) := sup

x∈Ω;r>0

⎛

⎜⎜

⎝1 r^λ



B(x,r)∼

|f(y)|^pdy

⎞

⎟⎟

⎠

1p

. (2)

The space L^p,λ(Ω) is trivial when λ > n, 

L^p,λ(Ω) ={0}

and L^p,0(Ω) = L^p(Ω) and L^p,n(Ω) = L^∞(Ω). In the case λ ∈ (0, n], the space L^p,λ(Ω) is non-separable.

1.1.2 Generalized Morrey spaces

The generalized Morrey spaces L^p,ϕare obtained by replacing r^λby a function ϕ(r) in the definition of the Morrey space.

Definition 2. Let 1 ≤ p < ∞. The generalized Morrey space L^p,ϕ(Rⁿ) is defined by the norm

fp,ϕ= sup

x∈Rⁿ,r>0

⎛

⎜⎝ 1 ϕ(r)



B(x,r)

|f(y)|^p dy

⎞

⎟⎠

1p

. (3)

Everywhere in the sequel it is assumed that ϕ : R+ → R+ is a measurable function satisfying the following assumptions:

1. ϕ(r) is continuous in a neighborhood of the origin;

2. ϕ(0) = 0;

3. inf

r>δϕ(r) > 0 for every δ > 0 and

ϕ(r)≥ crⁿ (4)

for 0 < r ≤ l, if l < ∞, and 0 < r ≤ N with an arbitrary N > 0, if l = ∞, the constant c depends on N in the latter case. The condition (4) makes the space L^p,ϕ(Rⁿ) non-trivial (see [40, Corollary 3.4]).

1.1.3 Orlicz-Morrey spaces

We also introduce Orlicz-Morrey type spaces to investigate the bounded- ness of fractional integral operators and generalized potential type opera- tors. Orlicz-Morrey type spaces unify Orlicz and Morrey type spaces. The generalized Orlicz-Morrey type space L^Φ,ϕ(Rⁿ) is defined by the following condition:

sup

x,r ϕ(r)⁻¹



B(x,r)

Φ (kf (y)) dy <∞

for some k > 0, where Φ is Young function. We work with the following Orlicz-Morrey space norm (5), used for instance in [35]:

fΦ,ϕ:= inf



λ : sup 1 ϕ(r)



B(x,r)

Φ

f (y) λ



dy≤ 1



. (5)

Remark 1. In the case Φ(u) = u^p and ϕ(r) replaced by ϕ^1p, the Orlicz- Morrey spaces L^Φ,ϕ(Rⁿ) turns into the generalized Morrey space L^p,ϕ(Rⁿ).

The following theorem provides conditions for a function f to belong to the Orlicz-Morrey space L^p,ϕ(Rⁿ).

Theorem 2. [7, Theorem 2.1, paper A] Let 1≤ p < ∞, the function ϕ(t) be almost increasing, ϕ(0) = 0, and function ϕ(r)/rⁿ be almost decreasing. Let the function f (x) on Rⁿ have a radial almost decreasing dominant g(|x|) :

|f(x)| ≤ Cg(|x|), x ∈ Rⁿ

and a function Φ be a Young function satisfying the Δ₂ condition.

Then the conditions

1. Φ◦ g is almost decreasing;

2. ^r

0

tⁿ⁻¹Φ◦ gdt ≤ Cϕ(r);

3. rⁿΦ◦ g(r) ≤ Cϕ(r) are sufficient for f ∈ L^Φ,ϕ(Rⁿ).

1.1.4 H¨older type spaces

By C^λ(Ω), 0 < λ ≤ 1, where Ω is an open set in Rⁿ, Ω ⊆ Rⁿ, n ≥ 1, we denote the class of H¨older continuous functions, defined by the seminorm

[f ]_λ:= sup

x,x+h∈Ω

|h|<1

|f(x + h) − f(x)|

|h|^λ <∞.

Equipped with the norm

f_C^λ = sup

x∈Ω|f(x)| + [f]λ

C^λ(Ω) is a Banach space. We shall deal with the case Ω = B_R, where B_R = B(0, R) :={x ∈ Rⁿ :|x| < R}, 0 < R ≤ ∞.

We define the generalized H¨older space C^ω(·)(Ω) as the set of continuous in Ω functions having the following finite norm

f_C^ω(·) = sup

x∈Ω|f(x)| + [f]ω(·)

with the seminorm

[f ]_ω(·) = sup

x,x+h∈Ω

|h|<1

|f(x + h) − f(x)|

ω(|h|) ,

where ω : [0, 1]→ R+ is a non-negative increasing function in C([0, 1]) such that ω(0) = 0 and ω(t) > 0 for 0 < t≤ 1.

We deal also with the space ˜C₀^λconsisting of functions f for which [f ]_λ<∞ and f (0) = 0. This space contains functions which are unbounded at infinity.

Note that [f ]_λ is a norm in this space.

We also denote the H¨older spaces with compactification. Let ˙Rⁿ denote the compactification ofRⁿ by a single infinite point.

Definition 3. Let 0 ≤ λ < 1. We say that f belongs to C^λ( ˙Rⁿ), for all x, y∈ Rⁿ, if

|f(x) − f(y)| ≤ C |x − y|^λ (1 +|x|)^λ(1 +|y|)^λ. C^λ( ˙Rⁿ) is a Banach space with respect to the norm

f_C^λ_{( ˙R}ⁿ₎=f_{C( ˙R}ⁿ₎+ sup

x,y∈Rⁿ|f(x) − f(y)|

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

|x − y|

_λ . We will also deal with the following subspaces:

C₀^λ( ˙Rⁿ) ={f ∈ C^λ( ˙Rⁿ) : f (0) = 0},

C_∞^λ( ˙Rⁿ) ={f ∈ C^λ( ˙Rⁿ) : f (∞) = 0}

and

C_∞,0^λ = C_∞^λ ∩ C₀^λ. 1.1.5 Weighted spaces

For a non-negative weight function (t) the weighted space X() is defined in the standard way:

X() ={f : f ∈ X} (6)

with

fX;:=fX. (7)

We will deal with almost monotone weights of radial type. For the properties of such functions and their numerical characteristics, i.e. Matuszewska-Orlicz indices we refer to [21], [30], [31], [42], [47], [49] and [51], see also the references therein.

2 Operators

2.1 Hardy type operators

There are a lot of results which are devoted to study the boundedness of Hardy type operators between weighted Lebesgue spaces and most of the results are for the one-dimensional case, see e.g. the books [24], [26], [27]

and the references given therein. But for applications it is also often re- quired to consider the boundedness between other function spaces. Un- fortunately, there exist not so many results concerning the boundedness of Hardy type operators in other function spaces. However, some results of this type can be found in Chapter 11 of the book [27], where it is reported on Hardy type inequalities in Orlicz, Lorentz and rearrangement invariant spaces and also on some really first not complete results in general Banach function spaces. Moreover, in [48] some corresponding Hardy type inequali- ties in weighted Morrey spaces were proved; in [40] the weighted estimates for multidimensional Hardy type operators were proved in generalized Morrey spaces. We refer to the paper [29], where the following two-weighted es- timates, L^p,ϕ_loc(X, w₁) → L^q,ψ_loc(X, w₂) and VL^p,ϕ_loc(X, w₁) → V L^q,ψ_loc(X, w₂), for the Hardy type operators are obtained for the case of an arbitrary underlying quasi-metric measure space (X, μ, ) with the growth condition.

In paper D (see [10]) of this thesis we study the following Hardy type operators

H^αf (x) =|x|^α−n



|y|<|x|

f (y)dy (8)

and

H^αf (x) =|x|^α



|y|>|x|

f (y)

|y|ⁿdy, (9)

where α≥ 0 and x ∈ BR, 0 < R≤ ∞ for the operator H^α, and R = ∞ for the operator H^α. We write H = H^α and H = H^α in the case α = 0. The

operator H^α, α = 0, may be considered in both with and without compacti- fication settings, but a consideration ofH requires the compactification due to the needed convergence of integrals at infinity.

We prove that the operator H^α is bounded in C^λ(B_R) if α = 0 and the operator H^α is bounded from ˜C₀^λ(B_R) into ˜C₀^λ+α(B_R), 0 < R ≤ ∞, in the case α > 0, where 0 < λ + α ≤ 1 in both cases. In the case of spaces with compactification we also prove the boundedness of operator H in C^λ( ˙Rⁿ), and the C_∞,0^λ ( ˙Rⁿ)→ C_∞^λ( ˙Rⁿ)− boundedness of H.

We also consider the following weighted Hardy type operators

H_γf (x) =|x|^γ−n



|z|≤|x|

f (z)

|z|^γ dz and Hγf (x) =|x|^γ



|z|≥|x|

f (z)

|z|^γ+ndz.

We prove the boundedness of H_γ in the space C^λ(B_R) for γ < n, and in the space C₀^λ(B_R) under the condition γ < n + λ. We also consider such weighted Hardy type operators in the spaces with compactification. In this case we prove that the operator H_γ is bounded in C^λ( ˙Rⁿ) for 0≤ λ < 1, if γ < n−λ;

and the operatorHγ is bounded in C^λ( ˙Rⁿ) for 0 < λ < 1, if γ > λ.

The boundedness of some multidimensional Hardy type operators from generalized Morrey to Orlicz-Morrey spaces was proved in papers A and E (see also [7] and [11] respectively). To prove the weighted boundedness of Riesz fractional integral and generalized potential operator we use point- wise estimates. In view of these point-wise estimates such weighted potential operators can be estimated by some weighted Hardy type operators. Below we describe Hardy type operators we use in our study.

In Paper A we consider the following Hardy type operators:

H_w^α =|x|^α−nw(|x|)



|y|<|x|

f (y)

w(|y|)dy, (10)

and

H^α_w=|x|^αw(|x|)



|y|>|x|

f (y)

|y|ⁿw(|y|)dy. (11) Our main result on the boundedness of such Hardy type operators in terms of Matuszewska-Orlicz indices reads:

Theorem 3. Let 1 < p < ⁿ_α, Φ be a Young function satisfying the Δ₂ condition, ϕ be almost increasing function, satisfying the conditions

min{m(ϕ), m∞(ϕ)} > 0 and ϕ ∈ Zn−αp.

1. The operators H^α = H_w^α|w≡1 and H^α_−α = H^α_w|w=|x|^−α are bounded from L^p,ϕ(Rⁿ) to L^Φ,ϕ(Rⁿ).

2. Suppose that w ∈ W , w(2t) ≤ Cw(t) and ^ϕ_w^1p ∈ W ([0, l]). Then the operator H_w^α is bounded from L^p,ϕ(Rⁿ) to L^Φ,ϕ(Rⁿ), if

min

 m

ϕ^1p w

 , m_∞

ϕ^1p w



>−n

p^, (12)

where ¹_p+ _p¹_ = 1.

3. Suppose that _w¹ ∈ W , or w ∈ W and w(2t) ≤ Cw(t). Then the operator H^α_wα is bounded from L^p,ϕ(Rⁿ) to L^Φ,ϕ(Rⁿ) if

max

 M

ϕ^p1 w

 , M_∞

ϕ^p1 w



< n

p − α. (13)

Here w_α=|x|^−αw(x).

In paper E we consider the following generalized Hardy type operators:

H_μ^af (x) =|x|^μ−na(|x|)



|y|<|x|

f (y)

|y|^μ dy, (14)

H^a_aμf (x) =|x|^μ



|y|>|x|

a(|y|)f(y)

|y|^n+μ dy. (15)

Our main Theorem on the boundedness of such operators reads:

Theorem 4. Let 1 < p < ∞, Φ be a Young function satisfying the Δ2

condition, ϕ be almost increasing function, satisfying the condition min{m(ϕ), m∞(ϕ)} > 0.

Let the function a(r) ∈ W (R+) satisfying the Δ₂ condition, be almost in- creasing, the function ^a(r)_rn be almost decreasing and aϕ^1p ∈ Zn/p. Then:

1. The operator H_μ^a is bounded from L^p,ϕ(Rⁿ) to L^Φ,ϕ(Rⁿ), if μ < n

p^ + min m

ϕ^1p

, m_∞ ϕ^p1

, (16)

where ¹_p+ _p¹_ = 1.

2. The operatorH^a_aμ is bounded from L^p,ϕ(Rⁿ) to L^Φ,ϕ(Rⁿ) if μ >−n

p + max

 M

 aϕ^1p

 , M_∞

 aϕ^1p

. (17)

For more information concerning Hardy type inequalities in Morrey type spaces, and their applications we refer to [7], [29], [41], [49], see also references therein.

2.2 Potential type operators

Let

I^αf (x) =



Rⁿ

f (y)

|x − y|^n−αdy, 0 < α < n, (18) be the Riesz fractional integration operator.

The mapping properties for the Riesz fractional integration operator within the frameworks of L^p,λ− spaces were first obtained by S. Spanne with the Sobolev exponent ¹_q = ¹_p − ^α_n, this result was published in [39]. A stronger result with a better exponent ¹_q = ¹_p− _n−λ^α is due to Adams (see [1]).

Results on the boundedness of Riesz fractional integration operators are also known in Morrey spaces with variable exponents, see [3], [22] and [23].

We refer also to the recent survey [46], where mapping properties of the Riesz fractional integration operators in various function spaces are discussed.

Weighted estimates for the Riesz potential operator in Morrey spaces were studied for instance in [40], [47]. In particular, the L^p,λ → L^q,λ weighted boundedness of the potential operator I^α, defined in (18) was considered by N. Samko in [47]. In [40] this result was extended to the case of generalized Morrey type spaces.

For generalized Morrey spaces in various papers there were found con- ditions in terms of integral inequalities imposed on exponents p, q and the function ϕ when Spanne or Adams type result holds within the frameworks of generalized Morrey space, see for instance [16], [18], [19], [38], [40] and ref- erences therein. However, when the function ϕ(r) defining the Morrey space is different from a power function, it is more natural to estimate the Riesz fractional integration operator not in the Lebesgue-Morrey norm I^αfL^p,ϕ, but in the Orlicz-Morrey normI^αf_L^Φ,ϕ. In such a general setting, i.e. from L^Φ,ϕ to L^Ψ,ψ the Riesz fractional integration operators were considered in [17], where the obtained estimates correspond to Spanne’s result.

In paper A we study the L^p,ϕ → L^Φ,ϕ mapping of Riesz fractional inte- gration operator I^α. We obtain a stronger statement which implies Adams’

result, in the case of classical Morrey space. Moreover, we generalize this result to the case of weighted Riesz fractional integration operators for a certain class of weights.

To prove the non-weighted boundedness of Riesz fractional integration operator we use the Hedberg approach. Our main result on non-weighted boundedness of the operator I^α reads as follows:

Denote: U (r) := ^ϕ(r)_rn and V (r) := r^p1[U⁻¹(r)]^α, r > 0.

Theorem 5. Let 1 < p < ⁿ_α and ϕ(t) satisfy the conditions:

U is decreasing, t^αpU (t) is almost decreasing and ϕ∈ Zn−αp, the function V is increasing, concave and V (0) = 0, V (+∞) = +∞.

Then the operator I^α is bounded from the space L^p,ϕ(Rⁿ) into the Orlicz- Morrey space L^Φ,ϕ(Rⁿ), where Φ = V⁻¹ is the inverse to V.

Due to the classes of the weights we use in our study, we can prove the following point-wise estimate for the weighted operator wI^{α 1}_w. The classes V_±^μ are defined in paper A.

Lemma 6. [49, Lemma 5.2] Let w∈ V₋^μ∪ V₊^μ with μ = min{1, n − α} be a weight and f a non-negative function. Then the following point-wise estimate holds

wI^α1

wf (x)≤ I^αf (x) + c

H_w^αf (x) +H^α_−αf (x), if w∈ V₊^μ,

H^αf (x) +H^α_wαf (x), if w∈ V₋^μ. (19) Thus, the problem of the boundedness of the weighted Riesz fractional in- tegration operator is reduced to study the boundedness of the non-weighted one and weighted Hardy type operators in such spaces. Our main result in this case reads:

Theorem 7. Let 0 < α < n, 1 < p < ⁿ_α, ϕ(r) be a non-negative almost increasing measurable function satisfying the condition min{m(ϕ), m∞(ϕ)} >

0. Let the Young function Φ satisfy Δ₂ condition and let the weight w ∈ W (R¹₊)∩ W (R¹₊) satisfy the condition

w∈ V₋^μ∪ V₊^μ, μ = min{1, n − α}.

Then under the conditions of the Theorem 5, the following conditions:

min

 m

ϕ^1p w

 , m_∞

ϕ^1p w



>−n p^ for w∈ V₊^μ and

max

 M

 ϕ^1p

w

 , M_∞

 ϕ^1p

w



< n p − α

for w ∈ V₋^μ, are sufficient for the weighted Riesz fractional integration oper- ator wI^{α 1}_w be bounded from L^p,ϕ(Rⁿ) to L^Φ,ϕ(Rⁿ).

The generalized Riesz potential operator I^a is defined by I^af (x) =



Rⁿ

a(|x − t|)f(t) dt

|x − t|ⁿ (20)

for any suitable function f onRⁿ. In the case a(t) = t^αwe get classical Riesz potential operator. Generalized Riesz potential operators attracted attention last years. Mostly, such potential operators were studied in Orlicz spaces.

We refer in particular to [20], [36]. Similar generalized potential operators were also studied in rearrangement invariant spaces in [44].

In paper E we consider the generalized Riesz potential operator I^a and prove the weighted and non-weighted boundedness of such operator from generalized Morrey space L^p,ϕ(Rⁿ) to Orlicz-Morrey space L^Φ,ϕ(Rⁿ). This is a generalization of our results for operator of fractional integration I^α, presented in paper A.

For the study of non-weighted boundedness of I^a we use ”Hedberg’s trick”. Then we consider the weighted boundedness of generalized potential operator. First we study the case of power weights and prove the bound- edness of generalized potential operator with such weights. Then using the properties of quasi-monotone weights we estimate the generalized potential operator with such weights by the operator with power weights.

We denote U (r) := ^ϕ(r)_rn , and V (r) := r^1pa[U⁻¹(r)], r > 0, and obtain the following result in terms of indices of weight w and characteristic of the space ϕ:

Theorem 8. Let 1 < p < n, a(t) ∈ W (R+) be almost increasing, ^a(t)_tn be almost decreasing and min{m(a), m∞(a)} > 0. Assume that ϕ(r) is a non- negative almost increasing measurable function satisfying conditions (4) and min{m(ϕ), m∞(ϕ)} > 0, U is decreasing, a^p(t)U (t) is almost decreasing, aϕ^1p ∈ Zn/p, the function V is increasing, concave and V (0) = 0, V (+∞) = +∞. Let the Young function Φ(r) and the function a(r) ∈ W (R+) satisfy Δ₂ condition and the weight w belong to W (R¹₊)∩ W (R¹₊).

Then the generalized weighted Riesz potential operator wI^{a 1}_w is bounded from L^p,ϕ(Rⁿ) to L^Φ,ϕ(Rⁿ) under either the condition

0 < max{M(w), M∞(w)} < n p^ + min

 m

 ϕ^1p

 , m_∞

 ϕ^1p

, (21) or

−n

p + max M

aϕ^p1

, M_∞ aϕ^1p

< min{m(w), m∞(w)} < 0. (22)

Here U⁻¹ is the function inverse to U.

It is well known that the Potential type operators appear in solutions of various partial differential equations, for instance Poisson’s and Helmholtz equations. In paper C (see [9]) we prove the boundedness of Potential op- erator in weighted generalized Morrey space in terms of Matuszewska-Orlicz indices of weights and apply this result to the Hemholtz equation inR³ with a free term in such a space. Our main result reads as follows:

Theorem 9. Let 1 < p < ³₂, q > p, and

w∈ [W (R+)∩ W (R+)]∩ [V¹₋(R+)∪ V¹₊(R+)].

Let also the functions ϕ and ψ satisfy the assumptions:

M (ϕ) < 3− 2p, and ϕ(r) ≤ cr^{3−1p −21q} and ϕ^1/p

r^3p−2 ∈ L^q,ψ. (23) Under the conditions

2−3− M(ϕ)

p < m(w)≤ M(w) < 3

p^ +m(ϕ)

p , (24)

and

2− 3− M∞(ϕ)

p < m_∞(w)≤ M∞(w) < 3

p^ +m_∞(ϕ)

p , (25)

for every f ∈ L^p,ϕ(R³, w), there exists a twice Sobolev differentiable particular solution u∈ L^q,ψ(R³, w) of the Helmholtz equation:

(Δ + k²I)u(x) = f (x).

2.3 Singular integral operators

The theory of the Riemann boundary value problem and singular integral equations on curves in the complex plane, including Fredholm properties, is well known, see the books [12], [32] and [34]. In particular this theory was extensively developed in such spaces as Lebesgue, Orlicz and recently in variable exponent Lebesgue spaces and their weighted versions, see [4], [5], [14], [15] and [23]. For the case of composite curves we refer to [6], [14] and [15].

In paper B (see [8]) we deal with singular integral operators in generalized Morrey spaces. We study the boundedness of a singular integral operator S_Γ in the space L^p,ϕ(Γ, ), where Γ is a composite curve which is a union of a

finite number of non-intersecting curves Γ_k without self-intersection, satis- fying arc-chord condition. The boundedness of singular integral operators in classical Morrey spaces on a single curve was studied in [48]. For the weighted boundedness of a general class of multidimensional singular inte- gral operators in generalized Morrey spaces, we refer to [43]. To prove the boundedness of the singular integral operator S_Γ in the weighted generalized Morrey spaceL^p,ϕ(Γ, ρ), we first prove the non-weighted boundedness of the maximal operator along such a curve in L^p,ϕ(Γ). Then we derive the non- weighted boundedness of S_Γ via the Alvarez-P´erez-type point-wise estimate.

We also refer to [52], where two-weight estimates for the maximal operator in local Morrey spaces are proved. After that we study the weighted case.

For simplicity we fix the weight at a single point t^k₀ on each curve Γ_k. Denote: ϕ = (ϕ₁, ϕ₂, ..., ϕ_m) and  = (₁, ₂, ..., _m) , where _k = w_k(|t − t^k₀|), wk ∈ W, t^k₀ ∈ Γk, k = 1, 2, ..., m, W is some class of quasi-monotone functions. Our main result on the boundedness of S_Γ reads:

Theorem 10. Let Γ = ^m

k=1

Γ_k, where Γ_k is a curve satisfying the arc-chord condition. Let 1 < p <∞, ϕk(r)≥ cr,

0 < m(ϕ_k)≤ M(ϕk) < 1, k = 1, 2, ..., m, (26)

and ϕ_k(r)

r be an almost decreasing function. (27) Then the operator S_Γ is bounded in the generalized Morrey spaceL^p,ϕ(Γ, ), if

m

ϕ_k w_k^p



> 1− p, M

ϕ_k w^p_k



< 1. (28)

We apply the obtained results to the study of Fredholm properties of the singular integral operators in weighted generalized Morrey spaces. We study the Fredholmness of the following singular integral operator:

Au : = a(t)u(t) + b(t) (SΓu) (t), (S_Γu) (t) = 1 πi



Γ

u(τ )

τ− tdτ, t∈ Γ (29) in weighted generalized Morrey space L^p,ϕ(Γ, ), where Γ is a set of non- intersecting oriented closed curves Γ_k without self-intersection, satisfying arc- chord condition. Γ may be such a single curve or union of such curves.

Fredholmness of such operators in classical weighted Morrey spaces was studied in [50]. The case of generalized Morrey spaces on an interval was

studied in [28]. We apply the methods from these papers to extend the results obtained there to the case of generalized weighted Morrey spaces on composite curves.

Theorem 11. Let 1 < p < ∞, ϕk(r) ≥ cr. Let Γ = ^m

k=1

Γ_k be a set of finite number of closed curves satisfying the arc-chord condition, without self-intersection. Let a(t), b(t) ∈ C(Γ) and (t) be a weight, i.e.  = (₁, ..., _m) ,where _k = w_k

|t − t^k₀|

, t^k₀ ∈ Γk, k = 1, 2, ..., m. Then the operatorA is Fredholm in the space L^p,ϕ(Γ, ) if inf_t∈Γ|a(t) ± b(t)| = 0, and

m

ϕ_k w^p_k



> 1− p, M

ϕ_k w^p_k



< 1, k = 1, 2, ..., m. (30) The index of the operator A in the space L^p,ϕ(Γ, ) is equal to the L^p,ϕ- index of the function g:

Ind_Lp,ϕ(Γ,)A = −indL^p,ϕ(Γ,)g(t) :=κ

(and the d-characteristic is equal to (κ, 0), if κ ≥ 0 and (0, |κ|), if κ ≤ 0), where g(t) = ^a(t)+b(t)_a(t)−b(t).

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Paper A

E. Burtseva and N. Samko. Weighted Adams type theorem for the Riesz fractional integral in generalized Morrey space. Fract. Calc. Appl. Anal. 19 (2016), no. 4, 954-972.

Adams type theorem for the Riesz fractional integral in generalized Morrey space

Evgeniya Burtseva and Natasha Samko

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

SE-971 87 Lule˚ a, Sweden

Abstract

We prove the boundedness of the Riesz fractional integration oper- ator from a generalized Morrey space L^p,ϕ to a certain Orlicz-Morrey space L^Φ,ϕ, which covers the Adams result for Morrey spaces. We also give a generalization to the case of weighted Riesz fractional integra- tion operators for some class of weights.

MSC 2010: Primary 46E30; Secondary 42B35, 42B25, 47B38

Key Words and Phrases: generalized Morrey space, Orlicz-Morrey space, weighted Riesz fractional integration operator, weighted Hardy inequalities, weighted Hardy operators, Bary-Stechkin classes

1 Introduction

We consider weighted Riesz fractional integration operator I^α in generalized Morrey spaces L^p,ϕ.

The classical Morrey spaces L^p,λ introduced in [15] in relation to the study of partial differential equations, are presented in various books, see for instance [4], [13], [27], [28]. We also refer to [2] for the latest research on the theory of Morrey spaces associated with Harmonic Analysis. The generalized Morrey spaces L^p,ϕ are obtained by replacing r^λ by a function

ϕ(r) in the definition of the Morrey space, see the definitions in the section

“Preliminaries”. Morrey spaces, classical or generalized, have a long history which is well presented in various sources, see for instance the survey paper [21] and references therein.

In the case of classical Morrey spaces with ϕ(r) = r^λ there is known the L^p,λ → L^q,λ boundedness with Sobolev exponent ¹_q = ¹_p−^α_n (Spanne’s result, [18]), and with better exponent ¹_q = _p¹ − _n−λ^α (Adams result [1]). Results on the boundedness of Riesz fractional integration operators are also known in Morrey spaces with variable exponents, see [3], [11] and [12]. We refer also to the recent survey [22] and the paper [26], where mapping properties of the Riesz fractional integration operators in various function spaces are discussed.

For generalized Morrey spaces in various papers there were found con- ditions in terms of integral inequalities imposed on exponents p, q and the function ϕ, where Spanne or Adams type result holds within the frameworks of generalized Morrey space, see for instance [8, 9, 5, 16, 19] and references therein. However, when the function ϕ(r) defining the Morrey space is differ- ent from a power function, it is more natural to estimate the Riesz fractional integration operator not in the Lebesgue-Morrey normI^αfL^p,ϕ, but in the Orlicz-Morrey norm I^αf_L^Φ,ϕ, even if f itself is in the Lebesgue-Morrey space, not in the Orlicz-Morrey space.

In such a general setting, i.e. from L^Φ,ϕ to L^Ψ,ψ the Riesz fractional integration operators were considered in [7], where the obtained estimates correspond to Spanne’s result.

In the setting I^α : L^p,ϕ → L^Φ,ϕ we obtain a stronger statement corre- sponding to Adams’ result.

Moreover, we generalize this statement to the case of weighted Riesz fractional integration operators for a certain class of weights. Weighted Riesz potentials in Morrey spaces were studied for instance in [19, 23].

The paper is organized as follows:

In Section 2 we give definitions related to generalized Morrey and Orlicz- Morrey spaces and prove some auxiliary statements which play a key role in the proofs of the main results. In Section 3 we prove the main results. In Section 3.1 we study the non-weighted case and prove Adams type theorem on L^p,ϕ(Rⁿ) → L^Φ,ϕ(Rⁿ)-boundedness for the Riesz fractional integration operator I^α. In Section 3.2 we prove the boundedness of Hardy type operators from generalized Morrey space to Orlicz-Morrey space. And finally, in Section

3.3 we obtain conditions for the L^p,ϕ(Rⁿ) → L^Φ,ϕ(Rⁿ)-boundedness of the weighted Riesz fractional integration operator and formulate the main results for the weighted case. The main results are given in Theorems 3.8, 3.10 and 3.11. For readers convenience, in Section 4 we give necessary definitions and properties of some classes of weight functions and also some known estimates we need for our study.

2 Preliminaries

2.1 Definitions of generalized Morrey and Orlicz-Morrey spaces

Let 1 ≤ p < ∞. The generalized Morrey space L^p,ϕ(Rⁿ) is defined by the norm

fp,ϕ = sup

x∈Rⁿ,r>0

⎛

⎜⎝ 1 ϕ(r)



B(x,r)

|f(y)|^p dy

⎞

⎟⎠

1p

. (2.1)

Everywhere in the sequel it is assumed that ϕ : R+ → R+ is a measurable function satisfying the following assumptions:

1. ϕ(r) is continuous in a neighborhood of the origin;

2. ϕ(0) = 0;

3. inf

r>δϕ(r) > 0 for every δ > 0 and

ϕ(r)≥ crⁿ (2.2)

for 0 < r ≤ l, if l < ∞, and 0 < r ≤ N with an arbitrary N > 0, if l = ∞, the constant c depending on N in the latter case. The condition (2.2) makes the space L^p,ϕ(Rⁿ) non-trivial (see [19, cor 3.4]).

The classical Morrey space corresponds to the case ϕ(r) = r^λ, 0 < λ < n.

It is denoted by L^p,λ(Rⁿ).

The generalized Orlicz-Morrey space L^Φ,ϕ(Rⁿ) is defined by the following condition:

sup

x,r ϕ(r)⁻¹



B(x,r)

Φ(kf (y))dy <∞

for some k > 0, where Φ is Young function, we recall its definition below, see Definition 2.1. A natural way to introduce a norm in this space is either the form:

fΦ,ϕ:= sup

x∈Rⁿ,r>0

1

ϕ(r)fχB(x,r)_L^Φ_(Rⁿ₎, (2.3) where L^Φ(Rⁿ) is the standard Orlicz space defined by the norm:

fΦ := inf

 λ :



Rⁿ

Φ

f (y) λ



dy ≤ 1

 ,

or in the form:

fΦ,ϕ := inf



λ : sup 1 ϕ(r)



B(x,r)

Φ

f (y) λ



dy ≤ 1



. (2.4)

We work with the norm (2.4), used for instance in [17].

Definition 2.1. A function Φ : [0; +∞] → [0; +∞] is called a Young function if Φ is convex, left-continuous, lim

r→+0Φ(r) = Φ(0) = 0 and lim

r→+∞Φ(r)

= Φ(∞) = ∞.

From the convexity and Φ(0) = 0 it follows that any Young function is increasing.

Remark 2.2. In the case Φ(u) = u^p and ϕ(r) replaced by ϕ^1p, the Orlicz-Morrey spaces L^Φ,ϕ(Rⁿ) turns into the Morrey space L^p,ϕ(Rⁿ).

2.2 Inclusion of some functions into generalized Mor- rey spaces and to Orlicz-Morrey spaces

Sufficient conditions for inclusion of functions to generalized Morrey spaces are given in the following theorem.

Theorem 2.3. Let 1 ≤ p < ∞, the function ϕ(t) be almost increasing, ϕ(0) = 0, and function ϕ(r)/rⁿ be almost decreasing. Let a function f (x) on Rⁿ have a radial almost decreasing dominant g(|x|) :

|f(x)| ≤ Cg(|x|), x ∈ Rⁿ.