We prove a fractional version of the Hardy-Sobolev-Maz’ya inequality for arbitrary domains and norms with p ≥ 2. This inequality combines the fractional Sobolev and the fractional Hardy inequality into a single inequality, while keeping the sharp constant in the Hardy inequality.
In this paper some embedding theorems related to fractional integration and differentiation in harmonic mixed norm spaces on the half-space are established. We prove that mixed norm is equivalent to a “fractional derivative norm” and that harmonic conjugation is bounded in for the range , . As an application of the above, we give a characterization of by means of an integral representation with the use of Besov spaces.
This paper examines developments in the study of boundary-value problems between about 1860 and 1890, in the context of the general evolution of this theory from the physical models in which the subject has its roots to a free-standing part of pure mathematics. The physically-motivated work of Carl Neumann and his method of the arithmetic mean appear as an initial phase in this development, one which employs physical models as an integral part of its reasoning and which concentrates on geometrical...
We show that -dimensional complete and noncompact metric measure spaces with nonnegative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are isometric to the model metric measure -space (i.e. the Euclidean metric -space). We also show that the Euclidean metric spaces are the only complete and noncompact metric measure spaces of nonnegative weighted Ricci curvature satisfying some prescribed Sobolev type inequality.
We prove that any positive function on ℂℙ¹ which is constant outside a countable -set is the order function of a fundamental solution of the complex Monge-Ampère equation on the unit ball in ℂ² with a singularity at the origin.
We prove for a large class of symmetric pseudo differential operators that they generate a Feller semigroup and therefore a Dirichlet form. Our construction uses the Yoshida-Hille-Ray Theorem and a priori estimates in anisotropic Sobolev spaces. Using these a priori estimates it is possible to obtain further information about the stochastic process associated with the Dirichlet form under consideration.