This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted L classes. We establish: (1) Mapping properties for the double and single layer potentials, as well as the Newton potential; (2) Extrapolation-type solvability results: the fact that solv...

We show that a maximal inequality holds for the non-tangential maximal operator on Dirichlet spaces with harmonic weights on the open unit disc. We then investigate two notions of Carleson measures on these spaces and use the maximal inequality to give characterizations of the Carleson measures in terms of an associated capacity.

The problems of wave diffraction by a plane angular screen occupying an infinite 45 degree wedge sector with Dirichlet and/or Neumann boundary conditions are studied in Bessel potential spaces. Existence and uniqueness results are proved in such a framework. The solutions are provided for the spaces in consideration, and higher regularity of solutions are also obtained in a scale of Bessel pote...

This study examines how differences in corpus size influence the accuracy of Latent Semantic Analysis (LSA) spaces and Latent Dirichlet Allocation (LDA) spaces in two tasks: a word association task and a vocabulary definition test. Specific optimizations were considered in building each semantic model. Initial results indicate that larger corpora lead to greater accuracy and that LDA probabilis...

We introduce a new approach to Nehari’s problem. This approach is based on some kind of fixed point theorem and allows us to obtain some useful generalizations of Nehari’s and Adamyan – Arov – Krein (AAK) theorems. Among those generalizations: descriptions of Hankel operators in weighted 2 spaces; descriptions of Hankel operators from Dirichlet type spaces to weighted Bergman spaces; commutant ...

We study the backward shift operator on Hilbert spaces Hα (for α ≥ 0) which are norm equivalent to the Dirichlet-type spaces Dα. Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an...

We consider heat kernels on different spaces such as Riemannian manifolds, graphs, and abstract metric measure spaces including fractals. The talk is an overview of the relationships between the heat kernel upper and lower bounds and the geometric properties of the underlying space. As an application some estimate of higher eigenvalues of the Dirichlet problem is considered.

We present a discrete Hodge-Morrey-Friedrichs decomposition for piecewise constant vector fields on simplicial surfaces with boundary which is structurally consistent with the smooth theory. In particular, it preserves a deep linkage between metric properties of the spaces of harmonic Dirichlet and Neumann fields and the topology of the underlying geometry, which reveals itself as a discrete de...

——————————————————————————————————– Abstract In this Note, we study the characterization of the kernel of the Laplace operator with Dirichlet boundary conditions in exterior domains. We consider data in weighted Sobolev spaces. Résumé Nouvelle caractérisation du noyau du laplacien en domaine extérieur. Nous étudions dans cet article la caractérisation du noyau de l’opérateur laplacien avec des ...