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Eugene Shargorodsky

Publications and source records attributed to Eugene Shargorodsky.

At least 19 recordsLinked to original sources
arXiv2026

Lower estimates for the norm and the Kuratowski measure of noncompactness of Wiener-Hopf type operators

Let $X(\mathbb{R}^n)$ be a Banach function space and $Ω\subseteq\mathbb{R}^n$ be a measurable set of positive measure. For a Fourier multiplier $a$ on $X(\mathbb{R}^n)$, consider the Wiener-Hopf type operator $W_Ω(a):=r_ΩF^{-1}aF e_Ω$, where $F^{\pm 1}$ are the Fourier transforms, $r_Ω$ is the operator of restriction from $\mathbb{R}^n$ to $Ω$ and $e_Ω$ is the operator of extension by zero from $Ω$ to $\mathbb{R}^n$. Let $X_2(Ω)$ be the closure of $L^2(Ω)\cap X(Ω)$ in $X(Ω)$. We show that if $X(Ω)$ satisfies the so-called weak doubling property, then \[ \|a\|_{L^\infty(\mathbb{R}^n)} \le \|W_Ω(a)\|_{\mathcal{B}(X_2(Ω),X(Ω))}. \] Further, we prove that if $X(Ω)$ satisfies the so-called separated doubling property, then the Kuratowski measure of noncompactness of $W_Ω(a)$ admits the following lower estimate: \[ \frac{1}{2}\|a\|_{L^\infty(\mathbb{R}^n)} \le \|W_Ω(a)\|_{\mathcal{B}(X_2(Ω),X(Ω)),κ}. \] These results are specified to the case of variable Lebesgue spaces $L^{p(\cdot)}(C,w)$ with Muckenhoupt type weights $w$ over open cones $C\subseteq\mathbb{R}^n$ with the vertex at the origin.

math.FA↗
arXiv2025

Maximal Noncompactness of Wiener-Hopf Operators

Let $X(\mathbb{R})$ be a separable translation-invariant Banach function space and $a$ be a Fourier multiplier on $X(\mathbb{R})$. We prove that the Wiener-Hopf operator $W(a)$ with symbol $a$ is maximally noncompact on the space $X(\mathbb{R}_+)$, that is, its Hausdorff measure of noncompactness, its essential norm and its norm are all equal. This equality for the Hausdorff measure of noncompactness of $W(a)$ is new even in the case of $X(\mathbb{R})=L^p(\mathbb{R})$ with $1\le p<\infty$.

math.FA↗
arXiv2025

Morrey spaces over the unit circle cannot be renormed to become rearrangement-invariant

Let $1\le p<\infty$ and $0<λ<1$. We consider the classical Morrey space $L^{p,λ}(\mathbb{T})$ over the unit circle $\mathbb{T}$. We show that there are equimeasurable functions $f,g:\mathbb{T}\to\mathbb{R}$ such that $g\in L^{p,λ}(\mathbb{T})$ but $f\notin L^{p,λ}(\mathbb{T})$. This implies that the the space $L^{p,λ}(\mathbb{T})$ cannot be renormed to become rearrangement-invariant.

math.FA↗
arXiv2024

The essential norms of Toeplitz operators with symbols in $C+H^\infty$ on weighted Hardy spaces are independent of the weights

Let $1<p<\infty$, let $H^p$ be the Hardy space on the unit circle, and let $H^p(w)$ be the Hardy space with a Muckenhoupt weight $w\in A_p$ on the unit circle. In 1988, Böttcher, Krupnik and Silbermann proved that the essential norm of the Toeplitz operator $T(a)$ with $a\in C$ on the weighted Hardy space $H^2(\varrho)$ with a power weight $\varrho\in A_2$ is equal to $\|a\|_{L^\infty}$. This implies that the essential norm of $T(a)$ on $H^2(\varrho)$ does not depend on $\varrho$. We extend this result and show that if $a\in C+H^\infty$, then, for $1<p<\infty$, the essential norms of the Toeplitz operator $T(a)$ on $H^p$ and on $H^p(w)$ are the same for all $w\in A_p$. In particular, if $w\in A_2$, then the essential norm of the Toeplitz operator $T(a)$ with $a\in C+H^\infty$ on the weighted Hardy space $H^2(w)$ is equal to $\|a\|_{L^\infty}$.

math.FA↗
arXiv2024

On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces

Let $X$ be a Banach function space over the unit circle such that the Riesz projection $P$ is bounded on $X$ and let $H[X]$ be the abstract Hardy space built upon $X$. We show that the essential norm of the Toeplitz operator $T(a):H[X]\to H[X]$ coincides with $\|a\|_{L^\infty}$ for every $a\in C+H^\infty$ if and only if the essential norm of the backward shift operator $T(\mathbf{e}_{-1}):H[X]\to H[X]$ is equal to one, where $\mathbf{e}_{-1}(z)=z^{-1}$. This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.

math.FA↗
arXiv2024

The Liouville theorem for a class of Fourier multipliers and its connection to coupling

The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $Δf = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator $m(D)$, such that the solutions $f$ to $m(D)f=0$ are Lebesgue a.e.\ constant (if $f$ is bounded) or coincide Lebesgue a.e.\ with a polynomial (if $f$ grows like a polynomial). The class of Fourier multipliers includes the (in general non-local) generators of Lévy processes. For generators of Lévy processes we obtain necessary and sufficient conditions for a strong Liouville theorem where $f$ is positive and grows at most exponentially fast. As an application of our results above we prove a coupling result for space-time Lévy processes.

math.PR↗
arXiv2023

Bounded compact and dual compact approximation properties of Hardy spaces: new results and open problems

The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such spaces, to provide background for the open problems. Namely, we consider abstract Hardy spaces $H[X(w)]$ built upon translation-invariant Banach function spaces $X$ with weights $w$ such that $w\in X$ and $w^{-1}\in X'$, where $X'$ is the associate space of $X$. We prove that if $X$ is separable, then $H[X(w)]$ has the BCAP with the approximation constant $M(H[X(w)])\le 2$. Moreover, if $X$ is reflexive, then $H[X(w)]$ has the BCAP and the DCAP with the approximation constants $M(H[X(w)])\le 2$ and $M^*(H[X(w)])\le 2$, respectively. In the case of classical weighted Hardy space $H^p(w) = H[L^p(w)]$ with $1<p<\infty$, one has a sharper result: $M(H^p(w))\le 2^{|1-2/p|}$ and $M^*(H^p(w))\le 2^{|1-2/p|}$.

math.FA↗
arXiv2021

Algebras of convolution type operators with continuous data do not always contain all rank one operators

Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. The algebra $C_X(\dot{\mathbb{R}})$ of continuous Fourier multipliers on $X(\mathbb{R})$ is defined as the closure of the set of continuous functions of bounded variation on $\dot{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$ with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author \cite{FKK19} that if the space $X(\mathbb{R})$ is reflexive, then the ideal of compact operators is contained in the Banach algebra $\mathcal{A}_{X(\mathbb{R})}$ generated by all multiplication operators $aI$ by continuous functions $a\in C(\dot{\mathbb{R}})$ and by all Fourier convolution operators $W^0(b)$ with symbols $b\in C_X(\dot{\mathbb{R}})$. We show that there are separable and non-reflexive Banach function spaces $X(\mathbb{R})$ such that the algebra $\mathcal{A}_{X(\mathbb{R})}$ does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces $L^{p,1}(\mathbb{R})$ with $1<p<\infty$.

math.FA↗
arXiv2021

Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case

In a domain $Ω\subset \mathbb{R}^{\mathbf{N}}$ we consider a selfadjoint operator $\mathbf{T}=\mathfrak{A}^*P\mathfrak{A} ,$ where $\mathfrak{A}$ is a pseudodifferential operator of order $-l=-\mathbf{N}/2$ and $P=Vμ_Σ$ is a singular signed measure in $Ω$ concentrated on a Lipschitz surface $Σ$ of dimension $d<\mathbf{N}$, absolutely continuous with respect to the surface measure $μ_Σ$ on $Σ$. We establish eigenvalue estimates and asymptotics for this operator. It turns out that the order of these estimates and asymptotics is independent of the dimension $d$ of the surface. If there are several surfaces, possibly, of different dimensions, as well as an absolute continuous measure on $Ω$ the corresponding asymptotic coefficients add up.

math.AP↗
arXiv2020

Sharp estimates for conditionally centred moments and for compact operators on $L^p$ spaces

Let $(Ω, \mathcal{F}, \mathbf{P})$ be a probability space, $ξ$ be a random variable on $(Ω, \mathcal{F}, \mathbf{P})$, $\mathcal{G}$ be a sub-$σ$-algebra of $\mathcal{F}$, and let $\mathbf{E}^\mathcal{G} = \mathbf{ E}(\cdot | \mathcal{G})$ be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of $ξ- \mathbf{E}^\mathcal{G}ξ$ in terms of the moments of $ξ$. This allows us to find the optimal constant in the bounded compact approximation property of $L^p([0, 1])$, $1 < p < \infty$.

math.PR↗
arXiv2020

An estimate for narrow operators on $L^p([0, 1])$

We prove a theorem, which generalises C. Franchetti's estimate for the norm of a projection onto a rich subspace of $L^p([0, 1])$ and the authors' related estimate for compact operators on $L^p([0, 1])$, $1 \le p < \infty$.

math.FA↗
arXiv2020

On the essential norms of Toeplitz operators with continuous symbols

It is well known that the essential norm of a Toeplitz operator on the Hardy space $H^p(\mathbb{T})$, $1 < p < \infty$ is greater than or equal to the $L^\infty(\mathbb{T})$ norm of its symbol. In 1988, A. Böttcher, N. Krupnik, and B. Silbermann posed a question on whether or not the equality holds in the case of continuous symbols. We answer this question in the negative. On the other hand, we show that the essential norm of a Toeplitz operator with a continuous symbol is less than or equal to twice the $L^\infty(\mathbb{T})$ norm of the symbol and prove more precise $p$-dependent estimates.

math.FA↗
arXiv2020

On negative eigenvalues of two-dimensional Schroedinger operators with singular potentials

We present upper estimates for the number of negative eigenvalues of two-dimensional Schroedinger operators with potentials generated by Ahlfors regular measures of arbitrary dimension $α\in (0, 2]$.The estimates are given in terms of the integrals of the potential with a logarithmic weight and of its L $\log$ L type Orlicz norms. In the case $α= 1$, our estimates are stronger than the known ones about Schroedinger operators with potentials supported by Lipschitz curves.

math.SP↗
arXiv2018

The Brown-Halmos theorem for a pair of abstract Hardy spaces

Let $H[X]$ and $H[Y]$ be abstract Hardy spaces built upon Banach function spaces $X$ and $Y$ over the unit circle $\mathbb{T}$. We prove an analogue of the Brown-Halmos theorem for Toeplitz operators $T_a$ acting from $H[X]$ to $H[Y]$ under the only assumption that the space $X$ is separable and the Riesz projection $P$ is bounded on the space $Y$. We specify our results to the case of variable Lebesgue spaces $X=L^{p(\cdot)}$ and $Y=L^{q(\cdot)}$ and to the case of Lorentz spaces $X=Y=L^{p,q}(w)$, $1<p<\infty$, $1\le q<\infty$ with Muckenhoupt weights $w\in A_p(\mathbb{T})$.

math.FA↗
arXiv2018

Quantitative results on continuity of the spectral factorization mapping

The spectral factorization mapping $F\to F^+$ puts a positive definite integrable matrix function $F$ having an integrable logarithm of the determinant in correspondence with an outer analytic matrix function $F^+$ such that $F = F^+(F^+)^*$ almost everywhere. The main question addressed here is to what extent $\|F^+ - G^+\|_{H_2}$ is controlled by $\|F-G\|_{L_1}$ and $\|\log \det F - \log\det G\|_{L_1}$.

math.CV↗
arXiv2017

When does the norm of a Fourier multiplier dominate its $L^\infty$ norm?

One can define Fourier multipliers on a Banach function space by using the direct and inverse Fourier transforms on $L^2(\mathbb{R}^n)$ or by using the direct Fourier transform on $S(\mathbb{R}^n)$ and the inverse one on $S'(\mathbb{R}^n)$. In the former case, one assumes that the Fourier multipliers belong to $L^\infty(\mathbb{R}^n)$, while in the latter one this requirement may or may not be included in the definition. We provide sufficient conditions for those definitions to coincide as well as examples when they differ. In particular, we prove that if a Banach function space $X(\mathbb{R}^n)$ satisfies a certain weak doubling property, then the space of all Fourier multipliers $\mathcal{M}_{X(\mathbb{R}^n)}$ is continuously embedded into $L^\infty(\mathbb{R}^n)$ with the best possible embedding constant one. For weighted Lebesgue spaces $L^p(\mathbb{R}^n,w)$, the weak doubling property is much weaker than the requirement that $w$ is a Muckenhoupt weight, and our result implies that $\|a\|_{L^\infty(\mathbb{R}^n)}\le\|a\|_{\mathcal{M}_{L^p(\mathbb{R}^n,w)}}$ for such weights. This inequality extends the inequality for $n=1$ from \cite[Theorem~2.3]{BG98}, where it is attributed to J.~Bourgain. We show that although the weak doubling property is not necessary, it is quite sharp. It allows the weight $w$ in $L^p(\mathbb{R}^n,w)$ to grow at any subexponential rate. On the other hand, the space $L^p(\mathbb{R},e^x)$ has plenty of unbounded Fourier multipliers.

math.CA↗