arXiv subjects

V. D. Stepanov

Publications and source records attributed to V. D. Stepanov.

2 recordsLinked to original sources

arXiv2022

On weak associated reflexivity of weighted Sobolev spaces of the first order on real line

We study associate and double associate spaces of two-weighted Sobolev spaces of the first order on real half-line and we show that unlike the notion of duality the associativity is divided into two cases which we call "strong" and "weak" ones with the division of the second associativity into four cases. On the way we prove that the Sobolev space of compactly supported functions possess weak associated reflexivity and the double weak-strong associate space is vacuous. The case of power weights was recently characterized by reduction to Cesàro or Copson type spaces [18].

math.FA↗

arXiv2020

On weighted Hardy inequality with two-dimensional rectangular operator -- extension of the E. Sawyer theorem

A characterization is obtained for those pairs of weights $v$ and $w$ on $\mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(\mathbb{R}^2_+)$ to $L^q_w(\mathbb{R}^2_+)$ for $1<p\not= q<\infty$, which is an essential complement to E. Sawyer's result \cite{Saw1} given for $1<p\leq q<\infty$. Besides, we declare that the E. Sawyer theorem is actual if $p=q$ only, for $p<q$ the criterion is less complicated. The case $q<p$ is new.

math.FA↗