arXiv subjects
Publications and source records attributed to Viktor Kolyada.
We prove embeddings of Sobolev and Hardy-Sobolev spaces into Besov spaces built upon certain mixed norms. This gives an improvment of the known embeddings into usual Besov spaces. Applying these results, we obtain Oberlin type estimates of Fourier transforms for functions in Sobolev spaces $W_1^1(\mathbb R^n).$
This paper gives a short survey of some basic results related to estimates of fractional integrals and Fourier transforms. It is closely adjoint to our previous survey papers \cite{K1998} and \cite{K2007}. The main methods used in the paper are based on nonincreasing rearrangements. We give alternative proofs of some results. We observe also that the paper represents the mini-course given by the author at Barcelona University in October, 2014.
We obtain sharp two-sided inequalities between $L^p-$norms $(1<p<\infty)$ of functions $Hf$ and $H^*f$, where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty).$ In an equivalent form, it gives sharp constants in the two-sided relations between $L^p$-norms of functions $H\f-\f$ and $\f$, where $\f$ is a nonnegative nonincreasing function on $(0,+\infty)$ with $\f(+\infty)=0.$ In particular, it provides an alternative proof of a result obtained by N. Kruglyak and E. Setterqvist (2008) for $p=2k (k\in \N)$ and by S. Boza and J. Soria (2011) for all $p\ge 2$, and gives a sharp version of this result for $1<p<2$.
We study the Lorentz spaces $L^{p,s}(R,μ)$ in the range $1 1$.