Let $$(Lv)(t)=sum^{n} _{i,j=1} (-1)^{j} d_{j} left( s^{2alpha}(t) b_{ij}(t) mu(t) d_{i}v(t)right),$$ be a non-selfadjoint differential operator on the Hilbert space $L_{2}(Omega)$ with Dirichlet-type boundary conditions. In continuing of papers [10-12], let the conditions made on the operator $ L$ be sufficiently more general than [11] and [12] as defined in Section $1$. In this paper, we estim...

We consider the first Dirichlet eigenvalue of diffusion operators on the half line. A criterion for the equivalence of the first Dirichlet eigenvalue with respect to the maximum domain and that to the minimum domain is presented. We also describle the relationships between the first Dirichlet eigenvalue of transient diffusion operators and the standard Muckenhoupt’s conditions for the dual weig...

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 focus on developing nonparametric Bayes methods for collections of dependent random functions, allowing individual curves to vary flexibly while adaptively borrowing information. A prior is proposed, which is expressed as a hierarchical mixture of weighted kernels placed at unknown locations. The induced prior for any individual function is shown to fall within a reproducing kernel Hilbert s...

Let Tk be a sequence of triangulations of a polyhedron Ω ⊂ Rn and let Sk be the associated finite element space of continuous, piecewise polynomials of degree m. Let uk ∈ Sk be the finite element approximation of the solution u of a second order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence Sk ensures optimal...

The boundedness and compactness of weighted composition operators from Dirichlet type spaces to weighted type spaces on the unit ball of Cn are characterized in this paper. Mathematics subject classification (2010): Primary 47B38; Secondary 46E15.