Related papers: On Limits at Infinity of Weighted Sobolev Function…
Consider the following nonlinear elliptic equation of $p(x)$-Laplacian type with nonstandard growth \begin{equation*} \left\{ \begin{aligned} &{\rm div} a(Du, x)=\mu \quad &\text{in}& \quad \Omega, &u=0 \quad &\text{on}& \quad…
This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - \mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with homogeneous Dirichlet…
The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the…
We study the convergence and divergence of the wavelet expansion of a function in a Sobolev or a Besov space from a multifractal point of view. In particular, we give an upper bound for the Hausdorff and for the packing dimension of the set…
In this paper we analyse functions in Besov spaces $B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$, and functions in fractional Sobolev spaces $W^{r,q}(\mathbb{R}^N,\mathbb{R}^d),r\in (0,1),q\in [1,\infty)$. We prove for…
It is shown that most of the well-known basic results for Sobolev-Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in $\mathbb{R}^n$, continue to be valid on a wide class of Riemannian manifolds with…
We study sufficient conditions on weight functions under which norm approximations by analytic polynomials are possible. The weights we study include radial, non-radial, and angular weights.
In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A_p-condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for…
We provide a complete characterization of compactness of Sobolev embeddings of radially symmetric functions on the entire space $\mathbb{R}^n$ in the general framework of rearrangement-invariant function spaces. We avoid any unnecessary…
Critical Sobolev-type inequality for a class of weighted Sobolev spaces on the entire space is established. We also investigate the existence of extremal function for the associated variational problem. As an application, we prove the…
Given a bounded Lipschitz domain $\omega\subset\mathbb{R}^{d-1}$ and a lower semicontinuous function $W:\mathbb{R}^N\to\mathbb{R}_+\cup\{+\infty\}$ that vanishes on a finite set and that is bounded from below by a positive constant at…
We characterize a weighted norm inequality which corresponds to the embedding of a class of absolutely continuous functions into the fractional order Sobolev space. The auxiliary result of the paper is of independent interest. It comprises…
Embedding theorems for symmetric functions without zero boundary condition have been studied on flat Riemannian manifolds, such as the Euclidean space. However, these theorems have only been established on hyperbolic spaces for functions…
This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain…
In this article, we develop the theory of weighted $L^2$ Sobolev spaces on unbounded domains in $\mathbb R^n$. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the…
For 1<p<infty, and weight w in A_p, and function f in L^p(w), we show that the r-variation of the Walsh-Fourier sums are finite, for r sufficiently large as function of w. (That r is a function of w is necessary.) This strengthens a result…
In this paper, we investigated the boundedness of multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ associated with rectangles or related to more general basis with multiple weights…
On the space of weighted radial Sobolev space, the following generalization of Moser-Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If $\beta \in [0,1)$ and $w_0(x) = |\log |x||^\beta $ then $$ \sup_{\int_B…
We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal…
We show that if $p_-\geq 2$, then a sufficient condition for the density of smooth functions with compact support, in the variable exponent Sobolev space $W^{1,p(\cdot)}(\mathbb R^n)$, is that the Riesz potentials of compactly supported…