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Papers by Natasha Samko

Some Weighted Norm Inequalities

Weighted Inequalities of Hardy Type, 2017

Journal of mathematical sciences, Feb 2, 2024

We describe some chosen ideas and results for more than 100 years prehistory and history of the r... more We describe some chosen ideas and results for more than 100 years prehistory and history of the remarkable development concerning Hardy-type inequalities. In particular, we present a newer convexity approach, which we believe could partly have changed this development if Hardy had discovered it. In order to emphasize the current very active interest in this subject, we finalize by presenting some examples of the recent results, which we believe have potential not only to be of interest for a broad audience from a historical perspective, but also to be useful in various applications.

Real analysis exchange, 2007

An error in Section 4 of [NS] is corrected.

Criterion for the Fredholmness of Singular Operators with Piecewise Continuous Coefficients in Generalized Hölder Spaces with Weight

Birkhäuser Basel eBooks, 2003

In a previous paper we found conditions for a singular integral operator with piecewise continuou... more In a previous paper we found conditions for a singular integral operator with piecewise continuous coefficients to be Fredholm in a weighted generalized Holder space H 0 ω (Γ, ρ) together with a formula for the index. The conditions were given in terms of Boyd-type indices of the space H 0 ω (Γ, ρ). In this paper we prove that those conditions are also necessary for a singular integral operator to be Fredholm.

Weighted Inequalities of Hardy Type

WORLD SCIENTIFIC eBooks, Apr 12, 2017

Hardy's Inequality and Related Topics Some Weighted Norm Inequalities The Hardy-Steklov Opera... more Hardy's Inequality and Related Topics Some Weighted Norm Inequalities The Hardy-Steklov Operator Higher Order Hardy Inequalities Fractional Order Hardy Inequalities Integral Operators on the Cone of Monotone Functions.

Some New Scales of Weight Characterizations of Hardy-type Inequalities

In this paper we present, discuss and illustrate some new scales of conditions to characterize mo... more In this paper we present, discuss and illustrate some new scales of conditions to characterize modern forms of Hardy′s inequalities which can not be found in the newest books in this area. Moreover, some results of importance as motivation for these scales are presented and discussed in a historical perspective.

New and Complementary Results

arXiv (Cornell University), Sep 29, 2011

As is known, the class of weights for Morrey type spaces L p,λ (R n ) for which the maximal and/o... more As is known, the class of weights for Morrey type spaces L p,λ (R n ) for which the maximal and/or singular operators are bounded, is different from the known Muckenhoupt class A p of such weights for the Lebesgue spaces L p (Ω). For instance, in the case of power weights |x-a| ν , a ∈ R 1 , the singular operator (Hilbert transform) is bounded in L p (R), if and only if -1 < ν < p -1, while it is bounded in the Morrey space L p,λ (R), 0 ≤ λ < 1, if and only if the exponent α runs the shifted interval λ -1 < ν < λ + p -1. A description of all the admissible weights similar to the Muckenhoupt class A p is an open problem. In this paper, for the one-dimensional case, we introduce the class A p,λ of weights, which turns into the Muckenhoupt class A p when λ = 0 and show that the belongness of a weight to A p,λ is necessary for the boundedness of the Hilbert transform in the one-dimensional case. In the case n > 1 we also provide some λ-dependent à priori assumptions on weights and give some estimates of weighted norms χ B p,λ;w of the characteristic functions of balls.

Oscillating Zygmund-Bary-Stechkin characteristics and Fredholmness of singular integral operators in the Holder spaces with oscillating characteristics and oscillating weights

We consider quasi-monotonic functions of the Zygmund-Bary-Stechkin class Z with the main emphasis... more We consider quasi-monotonic functions of the Zygmund-Bary-Stechkin class Z with the main emphasis on properties of the index numbers of functions in this class (of Boyd type indices). A special attention is paid to functions whose lower and upper index numbers do not coincide with each other (non-equilibrated functions). It is proved that the bounds for functions in Z known in terms of these indices, are exact in a certain sense. We also single out some special family of non-equilibrated functions in Z which oscillate in a certain way between two power functions. Given two numbers 0 < a < b < 1, we explicitly construct examples of functions in Z for which a and b serve as the index numbers. Moreover, for a certain class of function oscillating between two power functions there is given an explicit formula for calculation of the indices. The developed properties of functions in this class are applied to an investigation of the normal solvability of some singular integral operators in weighted spaces with prescribed oscillating modulus of continuity and oscillating weights.

Some remarks and new developments concerning Hardy-type inequalities

arXiv (Cornell University), Aug 18, 2008

We study the weighted boundedness of the Cauchy singular integral operator S Γ in Morrey spaces L... more We study the weighted boundedness of the Cauchy singular integral operator S Γ in Morrey spaces L p,λ (Γ) on curves satisfying the arc-chord condition, for a class of "radial type" almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces L p,λ (0, ℓ), ℓ > 0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.

Necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces

Mathematische Nachrichten, Apr 3, 2018

Mathematical Methods in The Applied Sciences, Jul 1, 2016

arXiv (Cornell University), Feb 18, 2023

The current status concerning Hardy-type inequalities with sharp constants is presented and descr... more The current status concerning Hardy-type inequalities with sharp constants is presented and described in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure dx with the Haar measure dx/x. There are also derived some new two-sided Hardy-type inequalities for monotone functions, where not only the two constants are sharp but also where the involved function spaces are (more) optimal. As applications, a number of both well-known and new Hardy-type inequalities are pointed out. And, in turn, these results are used to derive some new sharp information concerning sharpness in the relation between different quasi-norms in Lorentz spaces.

Singular operators in variable spaces Lp (·)(,P) with oscillating weights

Mathematische Nachrichten, 2007

On compactness of integral operators with a generalized weak singularity in weighted spaces of continuous functions with a given continuity modulus

$Research paper thumbnail of Zygmund-type estimates for fractional integration and differentiation operators of variable order$

Zygmund-type estimates for fractional integration and differentiation operators of variable order

Russian Mathematics, 2011

ABSTRACT We consider non-standard generalized Hölder spaces of functions defined on a segment of ... more ABSTRACT We consider non-standard generalized Hölder spaces of functions defined on a segment of the real axis, whose local continuity modulus has a majorant varying from point to point. We establish some properties of fractional integration operators of variable order acting from variable generalized Hölder spaces to those with a “better” majorant, as well as properties of fractional differentiation operators of variable order acting from the same spaces to those with a “worse” majorant. Keywords and phrasesfractional integration operators–fractional differentiation operators–generalized continuity modulus–generalized Hölder spaces

Analysis and mathematical physics, May 30, 2024

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO ... more For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range. Non-standard function spaces • Generalized Morrey spaces • Weighted singular integral operators • Weighted commutators and their applications • Elliptic PDE with discontinuous coefficients Mathematics Subject Classification 46E30 • 42B35 • 42B25 • 47B38 B Natasha Samko

Mathematical Methods in the Applied Sciences

In this paper, we discuss the study of some signal processing problems within Bayesian frameworks... more In this paper, we discuss the study of some signal processing problems within Bayesian frameworks and semigroups theory, in the case where the Banach space under consideration may be nonseparable. For applications, the suggested approach may be of interest in situations where approximation in the norm of the space is not possible. We describe the idea for the case of the abstract Cauchy problem for the evolution equation and provide more detailed example of the diffusion equation with the initial data in the nonseparable Morrey space.

On some results in weighted Morrey spaces with applications to PDE