arXiv subjects

Stefan Samko

Publications and source records attributed to Stefan Samko.

5 recordsLinked to original sources

arXiv2019

On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators

We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $C_0^\infty(\mathbb{R}^n)$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy-Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.

math.FA↗

arXiv2017

Approximation in Morrey spaces

A new subspace of Morrey spaces whose elements can be approximated by infinitely differentiable compactly supported functions is introduced. 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 Lesbesgue spaces is also obtained.

math.FA↗

arXiv2009

Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup

Under the standard assumptions on the variable exponent $p(x)$ (log- and decay conditions), we give a characterization of the variable exponent Bessel potential space $\mathfrak B^α[L^{p(\cdot)}(\mathbb R^n)]$ in terms of the rate of convergence of the Poisson semigroup $P_t$. We show that the existence of the Riesz fractional derivative $\mathbb{D}^\al f$ in the space $L^{p(\cdot)}(\rn)$ is equivalent to the existence of the limit $\frac{1}{\ve^\al}(I-P_\ve)^\al f$. In the pre-limiting case $\sup_x p(x)<\frac{n}{\al}$ we show that the Bessel potential space is characterized by the condition $\|(I-P_\ve)^\al f\|_{p(\cdot)}\leqq C \ve^\al$

math.FA↗

arXiv2009

Hardy type inequality in variable Lebesgue spaces

We prove that in variable exponent spaces $L^{p(\cdot)}(Ω)$, where $p(\cdot)$ satisfies the log-condition and $Ω$ is a bounded domain in $\mathbf R^n$ with the property that $\mathbf R^n \backslash \barΩ$ has the cone property, the validity of the Hardy type inequality $$| 1/δ(x)^α\int_Ωϕ(y) dy/|x-y|^{n-α}|_{p(\cdot)} \leqq C |ϕ|_{p(\cdot)}, \quad 0<\al<\min(1,\frac{n}{p_+})$$, where $δ(x)=\mathrm{dist}(x,\partialΩ)$, is equivalent to a certain property of the domain $\Om$ expressed in terms of $\al$ and $χ_\Om$.

math.FA↗

arXiv2008

Fractional, Maximal and Singular Operators in Variable Exponent Lorentz Spaces

We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The main goal of the paper is to show that the boundedness of these operators in the spaces $\mathcal{L}^{p(\cdot), q(\cdot)}$ is possible without the local log-condition on the exponents, typical for the variable exponent Lebesgue spaces; instead the exponents $p(s)$ and $q(s)$ should only satisfy decay conditions of log-type as $s\to 0$ and $s\to\infty$. To prove this, we base ourselves on the recent progress in the problem of the validity of Hardy inequalities in variable exponent Lebesgue spaces.

math.FA↗