Related papers: Martingale approach to Sobolev embedding theorems
A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed…
We prove some infinitesimal analogs of classical results of Menger, Schoenberg and Blumenthal giving the existence conditions for isometric embeddings of metric spaces in the finite-dimensional Euclidean spaces.
We establish equivalence between the boundedness of specific supremum operators and the optimality of function spaces in Sobolev embeddings acting on domains in ambient Euclidean space with a prescribed isoperimetric behavior. Our approach…
We prove asymptotic formulas for the behavior of approximation, Gelfand, Kolmogorov and Weyl numbers of Sobolev embeddings between Besov and Triebel-Lizorkin spaces defined on quasi-bounded domains.
We extend in this article the classical imbedding theorems for fractional Lebesgue-Sobolev's spaces into the so-called Grand Lebesgue spaces, with sharp constant evaluation.
A finitely-additive measure $\lambda $ on an infinite-dimensional real Hilbert space $E$ which is invariant with respect to shifts and orthogonal mappings has been defined. This measure can be considered as the analog of the Lebesgue…
We apply a symmetrization procedure to the setting of Jacobi expansions and study potential spaces in the resulting situation. We prove that the potential spaces of integer orders are isomorphic to suitably defined Sobolev spaces. Among…
Necessary and sufficient conditions are offered for Sobolev type spaces built on rearrangement-invariant spaces to be continuously embedded into (generalized) Campanato and Morrey spaces on open subsets of the $n$-dimensional Euclidean…
We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on $\RR^n$ and different classes of measures: Gaussian measures on $\RR^n$, symmetric Bernoulli and symmetric uniform probability measures on…
Let X be a compact Hausdorff space. We study finite-to-one mappings r:X->X, onto X, and measures on the corresponding projective limit space X_\infinity(r). We show that the invariant measures on X_\infinity(r) correspond in a one-to-one…
We define the local trace function for subspaces of $\ltworn$ which are invariant under integer translation. Our trace function contains the dimension function and the spectral function defined by Bownik and Rzeszotnik and completely…
We prove boundary inequalities in arbitrary bounded Lipschitz domains on the trace space of Sobolev spaces. For that, we make use of the trace operator, its Moore-Penrose inverse, and of a special inner product. We show that our trace…
As a generalization of the Fourier transform, the fractional Fourier transform was introduced and has been further investigated both in theory and in applications of signal processing. We obtain a sampling theorem on shift-invariant spaces…
The notion of the ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the…
An apparently new concept of maximal mean difference quotient is defined for functions in the Lebesgue space $L_{loc}(R^n)$. Our definitions are meaningful for vector valued functions on general measure metric spaces as well and seem to…
In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded…
In the paper, we introduce the notion of a local regular supermartingale relative to a convex set of equivalent measures and prove for it an optional Doob decomposition in the discrete case. This Theorem is a generalization of the famous…
In this paper we characterize the trace spaces of a class of weighted function spaces of intersection type with mixed regularities. To a large extent we can overcome the difficulty of mixed scales by employing a microscopic improvement in…
A complete description of traces on $\mathbb{R}^{n}$ of functions from the weighted Sobolev space $W^{l}_{1}(\mathbb{R}^{n+1},\gamma)$, $l \in \mathbb{N}$, with weight $\gamma \in A^{\rm loc}_{1}(\mathbb{R}^{n+1})$ is obtained. In the case…
We study the distortion of intermediate dimension under supercritical Sobolev mappings and also under quasiconformal or quasisymmetric homeomorphisms. In particular, we extend to the setting of intermediate dimensions both the…