In this paper, we are interested in some aspects of the biharmonic equation in the half-space R+ , with N ≥ 2. We study the regularity of generalized solutions in weighted Sobolev spaces, then we consider the question of singular boundary conditions. To finish, we envisage other sorts of boundary conditions.

We give a new constructive method for finding compactly supported prewavelets in L2 spaces in the multivariate setting. This method works for any dimensional space. When this method is generalized to the Sobolev space setting, it produces a pre-Riesz basis for Hs(IR) which can be useful for applications. AMS(MOS) Subject Classifications: Primary 42C15, Secondary 42C30

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.

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman...

In this paper we explore two constructions of the same family of metric measure spaces. The first construction was introduced by Laakso in 2000 where he used it as an example that Poincaré inequalities can hold on spaces of arbitrary Hausdorff dimension. This was proved using minimal generalized upper gradients. Following Cheeger’s work these upper gradients can be used to define a Sobolev spac...

Interest in Sobolev type equations has recently increased signi cantly, moreover, there arose a necessity for their consideration in quasi-Banach spaces. The need is dictated not so much by the desire to ll up the theory but by the aspiration to comprehend nonclassical models of mathematical physics in quasi-Banach spaces. Notice that the Sobolev type equations are called evolutionary if soluti...

Significance The Sobolev spaces, introduced in the 1930s, have become ubiquitous analysis and applied mathematics. They involve L p norms of gradient a function u . We present an alternative point view where deri...

We consider the fractional Laplacian with Hardy potential and study scale of homogeneous $L^p$ Sobolev spaces generated by this operator. Besides generalized reversed inequalities, analysis relies on a H\"ormander multiplier theorem which is crucial to construct basic Littlewood--Paley theory. The results extend those obtained recently in $L^2$ but do not cover negative coupling constants gener...