Integral Operators in Non-Standard Function Spaces

A Sequel: Inequalities, Sharp Estimates, Bounded Variation, and Approximation

2025
Book

Authors: Alexander Meskhi; Humberto Rafeiro; Stefan Samko
Book Series: Operator Theory: Advances and Applications
Publisher: Springer Nature Switzerland
Part of: Springer Professional "Business + Economics & Engineering + Technology"; Springer Professional "Engineering + Technology"; Springer Professional "Business + Economics"

About this book

This volume, as a sequel to Volumes I-IV of “Integral Operators in Non-Standard Function Spaces”, is devoted to the authors’ most recent advances in harmonic analysis and their applications.

This volume focusses on Rellich inequalities in the variable exponent and multilinear settings, trace inequalities for linear and multilinear fractional integrals, sharp weighted estimates for norms of operators of harmonic analysis, criteria governing Sobolev-type inequalities for (generalized) fractional integrals associated with non-doubling measures, sharp Olsen-type inequalities, studies on Herz-type spaces, approximation in subspaces of Morrey spaces, introduction of variable exponent bounded variation spaces in the Riesz sense, and characterization of weighted Sobolev spaces via weighted Riesz bounded variation spaces.

The book is aimed at an audience ranging from researchers in operator theory and harmonic analysis to experts in applied mathematics and post graduate students. In particular, it is hoped that this book will serve as a source of inspiration for researchers in abstract harmonic analysis, function spaces, PDEs and boundary value problems.

Frontmatter

Chapter 1. Rellich Inequality

Abstract

Weighted Rellich inequality is obtained in variable exponent Lebesgue spaces. Two-sided estimates of the measure of non-compactness for the two-weighted Hardy operator are derived in $L^{p(\cdot )}$ spaces. Further, weighted multilinear Rellich and Hardy estimates are obtained in one-dimensional case.

Alexander Meskhi, Humberto Rafeiro, Stefan Samko

Chapter 2. Trace Inequalities for Fractional Integrals

Abstract

We establish necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality

$$\displaystyle {} \left \| I_{\alpha } f \,\right \|_{ L^{p}_v( \mathbb {R}^n) } \leqslant C \left \| f\,\right \|_{ L^{p,1}(\mathbb {R}^n) }, \;\; f\in L^{p,1}(\mathbb {R}^n), $$

for the Riesz potential $I_{\alpha }$, where $L^{p,1}(\mathbb {R}^n)$ is the Lorentz space. The condition on $\,v$ is of D. Adams type. The same problem is studied for potentials defined on spaces of homogeneous type. We also find necessary and sufficient conditions on a weight v for the boundedness of multilinear Riemann–Liouville operators from $\prod _{j=1}^m L^{p_j}$ to $L^q_v$.

Alexander Meskhi, Humberto Rafeiro, Stefan Samko

Chapter 3. Integral Transforms Associated with Measures

Abstract

We give a complete characterization of a measure $\mu $ governing the boundedness of fractional integral operator

$$\displaystyle J_{\gamma , \mu }f(x)= \int _{X}\, \frac {f(y)}{d(x,y)^{1-\gamma }} d\mu (y), \;\; 0<\gamma <1, $$

defined on a quasi-metric measure space $(X, d, \mu )$ from one grand Lebesgue spaces $L^{p), \theta _1}_{\mu }(X)$ into another $L^{q), \theta _2}_{\mu }(X)$. Necessary and sufficient conditions on a measure $\mu $ guaranteeing the boundedness of the multilinear fractional integral operator $T_{\gamma , \mu }^{(m)}$ (defined with respect to a measure $\mu $) from the product of Lorentz spaces $\prod _{k=1}^m L^{r_k, s_k}_{\mu }(X)$ to the Lorentz space $L^{p,q}_{\mu }(X)$ are established. From these results we have the similar results for linear fractional integrals $J_{\gamma , \mu }$ (i.e., for $m=1$). Furthermore, we study the generalized fractional integral transform $T_{\varphi }$ associated to a measure on a quasi-metric space. We give a characterization of those measures for which these operators are bounded between $L_p$-spaces defined on nonhomogeneous spaces. We also establish necessary and sufficient conditions for the compactness of fractional integral operators from $L^p_{\mu }(X)$ to $L^q_{\mu }(X)$ with $1<p<q<\infty $, where $\mu $ is a measure on a quasi-metric measure space X. Additionaly, we investigate the multilinear variants of the quantities which measure the noncompactness of multilinear operators taking values in Banach spaces with the uniform approximation property.

Alexander Meskhi, Humberto Rafeiro, Stefan Samko

Chapter 4. Sharp Bounds in Weighted Inequalities

Abstract

In this chapter, we establish sharp weighted norm estimates (Buckley-type estimates), generally speaking, for positive kernel operators on spaces of homogeneous type. We discuss the operators involving one-sided maximal and fractional integral operators, Hilbert transforms, multiple integral operators, potential operators, etc. Some of the results are derived via two-weight estimates for integral operators.

Alexander Meskhi, Humberto Rafeiro, Stefan Samko

Chapter 5. Olsen Inequality

Abstract

We establish the multilinear inequality

$$\displaystyle {} \big \| g \, {\mathcal {I}}_{\alpha } (\vec {f}) \big \|_{ L^{q}_r } \leqslant C \big \| g \big \|_{ L^q_{\ell }} \prod _{j=1}^m \big \| f_j\big \|_{L^{p_j}_{s_j}}, $$

where

$$\displaystyle {\mathcal {I}}_{\alpha }(\vec {f}\, ) (x) = \int \limits _{({\mathbb {R}}^n)^m}\frac {f_1(y_1)\cdots f_m(y_m)}{(|x-y_1|+ \cdots + |x-y_m|)^{mn-\alpha }} d\vec {y}, \;\; x\in {\mathbb {R}}^n, $$

with $0<\alpha < mn$, where $L^{q}_r$, $L^{q}_{\ell }$, $L^{p_j}_{s_j}$, $j=1,\ldots , m$, are Morrey space with indices satisfying certain homogeneity conditions. This inequality is sharp in the sense that the Morrey norm in $\| g \|_{ L^q_{\ell }}$ can not be replaced by the smaller one.

Alexander Meskhi, Humberto Rafeiro, Stefan Samko

Chapter 6. More on Herz-Type Spaces

Abstract

Next, utilizing the atomic decomposition of Herz-type Hardy spaces with variable smoothness and integrability, we derive boundedness results for central Calderón–Zygmund operators on Herz-type Hardy spaces with variable smoothness and integrability.

Alexander Meskhi, Humberto Rafeiro, Stefan Samko

Chapter 7. Approximation in Subspaces of Morrey Spaces

Abstract

We start by introducing a new subspace of Morrey spaces whose elements can be approximated by infinitely differentiable compactly supported functions. Consequently, we give an explicit description of the closure of the set of such functions in Morrey spaces. A generalization of known embeddings of Morrey spaces into weighted Lebesgue spaces is also obtained.

Alexander Meskhi, Humberto Rafeiro, Stefan Samko

Chapter 8. Bounded Variation Spaces and Related Topics

Abstract

Finally, we introduce weighted Riesz bounded variation spaces defined on an open subset of n-dimensional Euclidean space and use them to characterize weighted Sobolev spaces when the weight belongs to the Muckenhoupt class. As an application, using Rubio de Francia’s extrapolation theory, a similar characterization of the variable exponent Sobolev spaces via variable exponent Riesz bounded variation spaces is obtained.

Alexander Meskhi, Humberto Rafeiro, Stefan Samko

Title: Integral Operators in Non-Standard Function Spaces
Authors: Alexander Meskhi
Humberto Rafeiro
Stefan Samko
Publisher: Springer Nature Switzerland
Electronic ISBN: 978-3-032-03691-9
Print ISBN: 978-3-032-03690-2
DOI: https://doi.org/10.1007/978-3-032-03691-9

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Integral Operators in Non-Standard Function Spaces

A Sequel: Inequalities, Sharp Estimates, Bounded Variation, and Approximation

About this book

Table of Contents

Frontmatter

Chapter 1. Rellich Inequality

Chapter 2. Trace Inequalities for Fractional Integrals

Chapter 3. Integral Transforms Associated with Measures

Chapter 4. Sharp Bounds in Weighted Inequalities

Chapter 5. Olsen Inequality

Chapter 6. More on Herz-Type Spaces

Chapter 7. Approximation in Subspaces of Morrey Spaces

Chapter 8. Bounded Variation Spaces and Related Topics

Backmatter

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