Singular solutions of elliptic boundary value problems in polyhedra
We study weighted function spaces of Lebesgue, Besov and Triebel-Lizorkin type where the weight function belongs to some Muckenhoupt class. The singularities of functions in these spaces are characterised by means of envelope functions.
In this paper we study the Cauchy problem for the nonlinear Dirac equation in the Sobolev space Hs. We prove the existence and uniqueness of global solutions for small data in Hs with s > 1...
We give necessary and sufficient conditions for the equality in weighted Sobolev spaces. We also establish a Rellich-Kondrachov compactness theorem as well as a Lusin type approximation by Lipschitz functions in weighted Sobolev spaces.
We prove local embeddings of Sobolev and Morrey type for Dirichlet forms on spaces of homogeneous type. Our results apply to some general classes of selfadjoint subelliptic operators as well as to Dirichlet operators on certain self-similar fractals, like the Sierpinski gasket. We also define intrinsic BV spaces and perimeters and prove related isoperimetric inequalities.
Our aim in this paper is to deal with the boundedness of the Hardy-Littlewood maximal operator on grand Morrey spaces of variable exponents over non-doubling measure spaces. As an application of the boundedness of the maximal operator, we establish Sobolev's inequality for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces. We are also concerned with Trudinger's inequality and the continuity for Riesz potentials.
We derive a new criterion for a real-valued function to be in the Sobolev space . This criterion consists of comparing the value of a functional with the values of the same functional applied to convolutions of with a Dirac sequence. The difference of these values converges to zero as the convolutions approach , and we prove that the rate of convergence to zero is connected to regularity: if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization...
Let be a Riemannian manifold, which possesses a transitive Lie group of isometries. We suppose that , and therefore , are compact and connected. We characterize the Sobolev spaces
The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from . The analysis interestingly combines use of Poincaré inequalities and of some Hardy type inequalities.