Integral Operators in Non Standard Function Spaces Volume 1 Variable Exponent Lebesgue and Amalgam Spaces 1st Edition by Vakhtang Kokilashvili, Alexander Meskhi, Humberto Rafeiro, Stefan Samko 3319210149 9783319210148

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ISBN 10: 3319210149

ISBN 13: 9783319210148

Author: Vakhtang Kokilashvili, Alexander Meskhi, Humberto Rafeiro, Stefan Samko

This book, the result of the authors’ long and fruitful collaboration, focuses on integral operators in new, non-standard function spaces and presents a systematic study of the boundedness and compactness properties of basic, harmonic analysis integral operators in the following function spaces, among others: variable exponent Lebesgue and amalgam spaces, variable Hölder spaces, variable exponent Campanato, Morrey and Herz spaces, Iwaniec-Sbordone (grand Lebesgue) spaces, grand variable exponent Lebesgue spaces unifying the two spaces mentioned above, grand Morrey spaces, generalized grand Morrey spaces, and weighted analogues of some of them. The results obtained are widely applied to non-linear PDEs, singular integrals and PDO theory. One of the book’s most distinctive features is that the majority of the statements proved here are in the form of criteria. The book is intended for a broad audience, ranging from researchers in the area to experts in applied mathematicsand prospective students.

Integral Operators in Non Standard Function Spaces Volume 1 Variable Exponent Lebesgue and Amalgam Spaces 1st Table of contents:

Chapter 1 Hardy-type Operators in Variable Exponent Lebesgue Spaces

1.1 Preliminaries

1.1.1 Definitions and Basic Properties

1.1.2 Equivalent Norms

1.1.3 Minkowski Integral Inequality

1.1.4 Basic Notation

1.1.5 Estimates for Norms of Characteristic Functions of Balls

1.2 Convolution Operators

1.2.1 Convolution Operators Bounded in Lp(·)(Rn)

1.2.2 Estimation of Norms of Some Embeddings for Variable Exponent Lebesgue Spaces

1.2.3 Estimation of the Norm of Convolution Operators

1.3 Reduction of Hardy Inequalitiesto Convolution Inequalities

1.3.1 Equivalence Between Mellin Convolution on R+ and Convolutions on R. The Case of Constant p

1.3.2 The Case of Variable p

1.4 Variable Exponent Hardy Inequalities

1.5 Estimation of Constants in the Hardy Inequalities

1.6 Mellin Convolutions in Variable Exponents Spaces Lp(·)(R+)

1.7 Knopp–Carleman Inequalities in the Variable Exponent Setting

1.8 Comments to Chapter 1

Chapter 2 Maximal, Singular, and Potential Operators in Variable Exponent Lebesgue Spaces with Oscil

2.1 Preliminaries

2.2 Oscillating Weights of Bari–Stechkin Class

2.2.1 Some Classes of Almost Monotone Functions

2.2.2 ZBS Classes and MO Indices of Weights at the Origin

2.2.3 Examples of Weights

2.2.4 ZBS Classes and MO Indices of Weights at Infinity

2.3 Maximal Operator with Oscillating Weights

2.3.1 Weighted Pointwise Estimates

2.3.2 Weighted Boundedness; the Euclidean Case

2.3.3 A Non-Euclidean Case

2.4 Weighted Singular Operators

2.4.1 Calder´on–Zygmund-type Operators: the Euclidean Case

2.4.2 Singular Intervals with General Weights on Lyapunov Curves

2.4.3 Preliminaries Related to Singular Integrals on Curves

2.4.4 Singular Integrals with Cauchy Kernel on Carleson Curves

2.4.5 The Property of Γ to be a Carleson Curve is Necessary

2.5 Weighted Potential Operators

2.5.1 Non-weighted Estimates in the Prelimiting Case sup α(x)p(x) < n: the Euclidean Version

2.5.2 Non-weighted Estimates in the Limiting Case α(x)p(x)≡n: the Euclidean Version

2.5.3 Non-weighted Estimates in the Prelimiting Caseα(x)p(x) < n on Quasimetric Measure Spaces

Preliminaries on Quasimetric Measure Spaces

Different Versions of Fractional Operators

Sobolev-type Theorem for the Fractional Operators Iα(·) and Iα

Fractional Maximal Operators

2.5.4 Weighted Norm Estimates of Truncated Potential Kernels in the Euclidean Case

2.5.5 Fractional Integrals on Bounded Sets Ω ⊂ Rn with Oscillating Weights and Variable Order α(

2.5.6 Fractional Integrals on Rn with Power Weights Fixed at the Origin and Infinity and Constant α

2.5.7 Spherical Fractional Integrals on Sn with Power Weights

2.6 Generalized Potentials

2.6.1 Preliminaries

2.6.2 Estimation of the Variable Exponent Normof Truncated Generalized Potentials

2.6.3 An Appropriate Φ-Function

2.6.4 Proof of Theorem 2.72

2.6.5 Weighted Version

2.7 Weighted Extrapolation in the Setting of Quasimetric Measure Spaces

2.7.1 Preliminaries Related to Quasimetric Measure Spaces

2.7.2 Classes of the Weight Functions

2.7.3 Extrapolation Theorem

2.8 Application to Boundedness Problems in Lp(·)(Ω, ) for Classical Operators of Harmonic Analysis

2.8.1 Potential Operators and Fractional Maximal Function

2.8.2 Fourier Multipliers

2.8.3 Multipliers of Trigonometric Fourier Series

2.8.4 Majorants of Partial Sums of Fourier Series

2.8.5 Cauchy Singular Integral

2.8.6 Multidimensional Singular-type Operators

2.8.7 Fefferman–Stein Function and Vector-valued Operators

2.9 Comments to Chapter 2

Some general references on weighted results

Comments to Section 2.1

Comments to Section 2.2

Comments to Section 2.3

Comments to Section 2.4

Comments to Section 2.5

Comments to Section 2.6

Comments to Sections 2.7–2.8

Chapter 3 Kernel Integral Operators

3.1 Preliminaries

3.1.1 Variable Exponent Lebesgue Spaces

3.1.2 Variable Exponent Amalgam Spaces (VEAS)

3.1.3 Two-weighted Hardy Operator on the Real Line

3.1.4 Some Discrete Inequalities

3.2 Kernel Operators in Lp(·) Spaces

3.2.1 Boundedness Criteria

3.2.2 Compactness

3.2.3 Measure of Non-compactness

3.2.4 The Riemann–Liouville Operator with Variable Parameter

3.3 Boundedness in Variable ExponentAmalgam Spaces

3.3.1 General Operators on Amalgams

3.3.2 Two-weighted Hardy Operator

3.3.3 Kernel Operators in (Lp(·)(R+), lq)d and (Lp(·)(R), lq)

3.4 Maximal Functions and Potentials on VEAS

3.4.1 Maximal Operators in (Lp(·)(R), lq)

3.4.2 Fractional Integrals. Trace Inequality

3.5 Compactness of Kernel Operators on VEAS

3.6 Product Kernel Integral Operators with Measures

3.6.1 Hardy Operator with Respect to a Measure

3.6.2 Main Results

3.6.3 Proofs of the Main Results

3.6.4 A Fefferman–Stein-type Inequality

3.7 Comments to Chapter 3

Comments to Section 3.1

Comments to Sections 3.3, 3.4 and 3.5

Comments to Section 3.6

Chapter 4 Two-weight Estimates

4.1 Preliminaries

4.1.1 Some Properties of Variable Exponent Lebesgue Spaces

4.1.2 Variable Exponent Lebesgue Space on Quasimetric Measure Spaces

4.1.3 Carleson–H¨ormander Inequality

4.2 A Sawyer-type Condition on a Bounded Interval

4.3 A Sawyer-type Condition on an Unbounded Interval

4.4 Hardy-type Operators on Quasimetric Measure Spaces

4.5 Modular Conditions for Fractional Integrals

4.6 Modular Conditions for Maximal and Singular Operators

4.7 Norm-type Conditions for Maximal and Calderón-Zygmund Operators

4.7.1 Maximal Functions and Singular Integrals on SHT

4.7.2 Maximal Functions and Singular Integrals on R+

4.8 Potentials with Variable Parameters

4.8.1 Weighted Criteria for Potentials

4.8.2 Applications to Gradient Estimates

4.8.3 Potentials on Fractal Sets

4.9 Comments to Chapter 4

Chapter 5 One-sided Operators

5.1 Preliminaries

5.2 One-sided Extrapolation

5.3 One-sided Maximal Functions

5.4 One-sided Potentials

5.5 One-sided Calderón-Zygmund Operators

5.6 Weighted Criteria for One-sided Operators

5.6.1 Hardy–Littlewood One-sided Maximal Functions.One-weight Inequality

5.6.2 One-sided Fractional Maximal Operators. One-weight Inequality

5.7 Generalized One-sided Fractional Maximal Operators

5.7.1 The Two-weight Problem

5.7.2 Fefferman–Stein-type Inequalities

5.8 Two-weight Inequalities for Monotonic Weights

5.9 The Riemann–Liouville Operator on the Cone of Decreasing Functions

5.10 Comments to Chapter 5

Chapter 6 Two-weight Inequalities for Fractional Maximal Functions

6.1 Preliminaries

6.2 Generalized Maximal Function and Potentials

6.2.1 Fractional Maximal Function

6.2.2 Fractional Integrals

6.2.3 Diagonal Case

6.2.4 Further Remarks

6.3 Fractional Integral Operators on the Upper Half-space

6.3.1 Non-diagonal Case

6.3.2 Diagonal Case

6.4 Double Hardy Operator

6.5 Strong Fractional Maximal Functions in Lp(·) Spaces. Unweighted Case

6.6 Two-weight Estimates for Strong Fractional Maximal Functions

6.6.1 Formulation of Results

6.7 Comments to Chapter 6

Chapter 7 Description of the Range of Potentials, and Hypersingular Integrals

7.1 Preliminaries on Higher-order Hypersingular Integrals

7.2 Denseness of the Lizorkin Test Functions Space in Lp(·)(Rn)

7.3 Inversion of the Riesz Potential Operator on the Space Lp(·)(Rn)

7.4 Characterization of the Space of Riesz and Bessel Potentials of Functions in Lp(·)(Rn)

7.4.1 Preliminaries

7.4.2 Characterization of the Riesz Potentials on Lp(·)-Spaces

7.5 Function Spaces Defined by Fractional Derivatives in Lp(·)(Rn)

7.5.1 Definitions

7.5.2 Denseness of C∞0 in Lp(·),α(Rn)

7.6 Bessel Potentials Space of Functions in Lp(·)(Rn) and its Characterization

7.6.1 Basic Properties

7.6.2 Characterization of the Space Bα[Lp(·)(Rn)] via Hypersingular Integrals

7.6.3 Proof of Lemmas 7.14 and 7.15

7.7 Connection of the Riesz and Bessel Potentials with the Sobolev Variable Exponent Spaces

7.7.1 Coincidence with Variable Exponent Sobolev Spaces for α ∈ N

7.7.2 Denseness of C∞0 -Functions in W1,p(·)(Rn)

7.8 Characterization of the Variable Exponent Bessel Potential Space via the Rate of Convergence of

7.8.1 More on Fourier p(x)-Multipliers

7.8.2 On Finite Differences

7.8.3 More on the Function Kl,α(x)

7.8.4 Crucial Lemmas

7.8.5 A(x) and B(x) are Fourier p(·)-Multipliers

7.8.6 Main Theorems

7.9 Comments to Chapter 7

Comments to Section 7.1

Comments to Section 7.2

Comments to Section 7.3

Comments to Section 7.4

Comments to Sections 7.5, 7.6 and 7.7

Comments to Section 7.8

Chapter 8 More on Hypersingular Integrals and Embeddings into Hölder Spaces

8.1 Preliminaries on Hypersingular Integrals

8.2 Embeddings of Variable Sobolev Spaces into Hölder Spaces: the Euclidean Case

8.2.1 Hölder Spaces of Variable Order

8.2.2 Pointwise Inequalities for Sobolev Functions

8.2.3 Embedding Theorems for Hajłasz–Sobolev spaces

8.2.4 Extension to Higher Smoothness

8.3 Embeddings into Hölder Function Spaces on Quasimetric Measure Spaces

8.3.1 Variable Exponent Hölder Spaces on Quasimetric Spaces

8.3.2 Variable Exponent Hajłasz–Sobolev Spaces

8.3.3 Embeddings of Variable Exponent Hajłasz–Sobolev Spaces

8.3.4 Hypersingular Integrals in Variable Exponent Hajłasz–Sobolev Spaces

The Case of the Operator Dα(·)

The Case of the Operator Dα(·)

8.4 Comments to Chapter 8

Comments to Section 8.1

Comments to Section 8.2

Comments to Section 8.3

Chapter 9 More on Compactness

9.1 Two General Results on Compactness of Operators

9.1.1 Dominated Compactness Theorem

Preliminaries on Banach Function Spaces

Regular Integral Operators and Dominated Compactness Theorem in BFS

9.1.2 Compactness under Interpolation Theorem

9.1.3 Compactness of an Integral Operator with Integrable Almost Decreasing Radial Dominant Majorant

Non-weighted Case

Weighted Case

9.2 The Case Ω = Rn: Compactness of Convolution-type Operators with Coefficients Vanishing at Infin

9.3 Comments to Chapter 9

Comments to Section 9.1

Comments to Section 9.2

Chapter 10 Applications to Singular Integral Equations

10.1 Singular Integral Equations with Piecewise Continuous Coefficients

10.1.1 Introduction

10.1.2 Statement of the Main Result for the Spaces Lp(·)(Γ)

10.1.3 Singular Integral Operators in Banach Function Spaces X(Γ)

Banach Function Spaces, Suitable for Singular Integral Operators

X-Non-Singular Functions and X-Index of a PC-Function

10.1.4 Proof of Theorem 10.4

10.2 Boundedness and Fredholmness of PseudodifferentialOperators in Variable Exponent Spaces

10.2.1 Boundedness in Lp(·)(Rn, w) of Singular Integral-type Operators

Formulation of the Main Result

The Crucial Step: the Pointwise Estimate

Regularity of the Kernel

On the Kolmogorov Inequality

10.2.2 On Calculus of PDO on Rn

10.2.3 Operators with Slowly Oscillating Symbols

10.2.4 Boundedness of PDO in Hs,p(·)(Rn)

10.2.5 Fredholmness of PDO in Lp(·)(Rn) and Hs,p(·)(Rn)

Sufficient Conditions of Fredholmness in Lp(·)(Rn)

Necessary Conditions of the Fredholmness in Lp(·)(Rn)

Uniform Ellipticity

Ellipticity at Infinity

Fredholmness of PDO in Hs,p(·)(Rn)

10.2.6 Pseudodifferential Operators with Analytic Symbols in the Space Hs,p(·)(Rn)

10.3 Singular Integral Equations on Composite Carleson Curves via Mellin PDO

10.3.1 Introduction

10.3.2 Pseudodifferential Operators on R

Some Properties

Pseudodifferential Operators on Lebesgue Spaces with Variable Exponent

Local Invertibility at +∞

Local Invertibility at the Point x0 ∈ R

10.3.3 Mellin Pseudodifferential Operators

Main Property

Mellin PDOs in the Spaces Lp(·)n (R+, dμ)

Local Invertibility of Mellin PDOs

10.3.4 Singular Integral Operators on Some Classes of Carleson Curves

Curves, Weights, Coefficients

Representation of a Singular Integral Operator at the Node as a Mellin PDO

Boundedness of the Singular Integral Operator in Lp(·)(Γ, w).

The Fredholm Property of Singular Integral Operators in Lp(·)(Γ, w).

Local Invertibility

Simonenko’s Local Principle in Lp(·)(X)

Fredholmness of SIOs

Index Formula

10.3.5 Comparison of the Used Class of Oscillating Weights with the Bari–Stechkin-type Weights

Simonenko-type Class S2

10.4 Comments to Chapter 10

Comments to Section 10.1

Comments to Section 10.2

Comments to Section 10.3

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Tags: Vakhtang Kokilashvili, Alexander Meskhi, Humberto Rafeiro, Integral