arXiv subjects

Natasha Samko

Publications and source records attributed to Natasha Samko.

4 recordsLinked to original sources

arXiv2025

Hardy-Hilbert type inequalities on homogeneous groups-An introduction and generalization to the kernel case

There is a lot of information available concerning Hardy-Hilbert type inequalities in one or more dimensions. In this paper we introduce the development of such inequalities on homogeneous groups. Moreover, we point out a unification of several of the Hardy-Hilbert type inequalities in the classical case to a general kernel case. Finally, we generalize these results to the homogeneous group case.

math.FA↗

arXiv2023

Sharpness of some Hardy-type inequalities

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.

math.CA↗

arXiv2011

On a Muckenhoupt-type condition for Morrey spaces

As is known, the class of weights for Morrey type spaces $\mathcal{L}^{p,\lb}(\rn) $ 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 $L^p(\Om)$. For instance, in the case of power weights $|x-a|^ν, \ a\in \mathbb{R}^1,$ the singular operator (Hilbert transform) is bounded in $L^p(\mathbb{R})$, if and only if $-1<ν 1$ we also provide some $\lb$-dependent \textit{à priori} assumptions on weights and give some estimates of weighted norms $\|χ_B\|_{p,\lb;w}$ of the characteristic functions of balls.

math.FA↗

arXiv2008

Weighted Hardy and singular operators in Morrey spaces

We study the weighted boundedness of the Cauchy singular integral operator $S_\Gm$ in Morrey spaces $L^{p,λ}(\Gm)$ 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 $L^{p,λ}(0,\ell), \ell>0$. 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. Key words and phrases: Morrey space, singular operator, Hardy operator, Hardy-Littlewood maximal operator, weighted estimate.

math.FA↗