In this article, we discuss some recent developments on pairs of dual wavelet frames in Sobolev spaces constructed by the Oblique Extension Principle (OEP) from refinable function vectors. Our investigation sheds a new light on understanding the OEP method for constructing MRA dual wavelet frames.

In this paper, we establish two embedding inequalities for the weighted Sobolev space and homogeneous endpoint Besov by using Hausdorff capacity. To do this, shall determine dual spaces of Choquet spaces.

This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal B-splines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast ...

‎We present a Picone's identity for the‎ ‎$mathcal{A}_{p(x)}$-Laplacian‎, ‎which is an extension of the classic‎ ‎identity for the ordinary Laplace‎. ‎Also‎, ‎some applications of our‎ ‎results in Sobolev spaces with variable exponent are suggested.

We study superharmonic functions related to elliptic equations with structural conditions involving a variable growth exponent. We establish pointwise estimates for such functions in terms of a Wolff type potential. We apply these estimates to prove a variable exponent version of the Hedberg–Wolff theorem on the dual of Sobolev spaces with zero boundary values.

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is Weyl quantization homogeneous quadratic form with non-negative definite real part. The solution semigroup shown to be strongly continuous on several spaces: Shubin-Sobolev spaces, Schwartz space, tempered distributions, equal index Beurling type Gelfand-Shilov spaces and their dual ultradistribu...

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 multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the unif...

Motivated by [B. Han and Q. Mo, Adv. Comp. Math. 18 (2003) 211-245] and [B. Han and Z. Shen, Constr. Approx. 29 (2009) 369-406], we propose dual two-direction frames in dual Sobolev spaces (H(R), H−s(R)), with s > 0. Based on the dual two-direction frames from a pair of two-direction refinable functions, dual multiwavelet frames with symmetry {Ψ(x) := (ψ 1(x), ψ 2(x)) T }`=1 and {Ψ̃(x) : = (ψ̃ 1(...

We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and FaberKrahn estimates for Hörmander vector fields.