arXiv subjects

V. I. Kolyada

Publications and source records attributed to V. I. Kolyada.

4 recordsLinked to original sources

arXiv2023

Rearrangement estimates and limiting embeddings for anisotropic Besov spaces

The paper is dedicated to the study of embeddings of the anisotropic Besov spaces $B^{β_1,...,beta_n}_{p;θ_1,...,θ_n}(\Bbb R^n)$ into Lorentz spaces. We find the sharp asymptotic behaviour of embedding constants when some of the exponents $β_k$ tend to 1 ($β_k<1)$. In particular, these results give an extension of the estimate proved btý Bourgain, Brezis, and Mironescu for isotropic Besov spaces. Also, in the limit, we obtain a link with some known embeddings of anisotropic Lipschitz spaces. One of the key results of the paper is an anisotropic type estimate of rearrangements in terms of partial moduli of continuity.

math.FA↗

arXiv2014

Sections of functions and Sobolev type inequalities

We study functions of two variables whose sections by the lines parallel to the coordinate axis satisfy Lipschitz condition of the order $0<\a\le 1.$ We prove that if for a function $f$ the $\operatorname{Lip} \a-$ norms of these sections belong to the Lorentz space $L^{p,1}(\R) \,(p=1/\a),$ then $f$ can be modified on a set of measure zero so as to become bounded and uniformly continuous on $\R^2.$ For $\a=1$ this gives an extension of Sobolev's theorem on continuity of functions of the space $W_1^{2,2}(\R^2)$. We show that the exterior $L^{p,1}-$ norm cannot be replaced by a weaker Lorentz norm $L^{p,q}$ with $q>1$.

math.FA↗

arXiv2012

On Gagliardo-Nirenberg type inequalities

We present a Gagliardo-Nirenberg inequality which bounds Lorentz norms of the function by Sobolev norms and homogeneous Besov quasinorms with negative smoothness. We prove also other versions involving Besov or Triebel-Lizorkin quasinorms. These inequalities can be considered as refinements of Sobolev type embeddings. They can also be applied to obtain Gagliardo-Nirenberg inequalities in some limiting cases. Our methods are based on estimates of rearrangements in terms of heat kernels. These methods enable us to cover also the case of Sobolev norms with p = 1.

math.FA↗

arXiv2012

On limiting relations for capacities

The paper is devoted to the study of limiting behaviour of Besov capacities $\capa (E;B_{p,q}^\a) (0<\a<1)$ of sets in $\R^n$ as $\a\to 1$ or $\a\to 0.$ Namely, let $E\subset \R^n$ and $$J_{p,q}(\a, E)=[\a(1-\a)q]^{p/q}\capa(E;B_{p,q}^\a).$$ It is proved that if $1\le p<n, 1\le q<\infty,$ and the set $E$ is open, then $J_{p,q}(\a, E)$ tends to the Sobolev capacity $\capa(E;W_p^1)$ as $\a\to 1$. This statement fails to hold for compact sets. Further, it is proved that if the set $E$ is compact and $1\le p,q<\infty$, then $J_{p,q}(\a, E)$ tends to $2n^p|E|$ as $\a\to 0$ ($|E|$ is the measure of $E$). For open sets it is not true.

math.CA↗