Let Ω be some domain in the hyperbolic space Hn (with n ≥ 2) and S1 the geodesic ball that has the same first Dirichlet eigenvalue as Ω. We prove the Payne-Pólya-Weinberger conjecture for Hn, i.e., that the second Dirichlet eigenvalue on Ω is smaller or equal than the second Dirichlet eigenvalue on S1. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing f...

A new diffusion process taking values in the space of all probability measures over [0,1] is constructed through Dirichlet form theory in this paper. This process is reversible with respect to the Ferguson-Dirichlet process (also called Poisson Dirichlet process), which is the reversible measure of the Fleming-Viot process with parent independent mutation. The intrinsic distance of this process...

The paper is devoted to the spaces of generalized smoothness on so-called h–sets. First we find the quarkonial representations of isotropic spaces of generalized smoothness on R and on an h–set. Then we investigate the representations of such spaces via differences, which are very helpful when we want to find an explicit representation of the domain of a Dirichlet form on h–sets. We prove that ...

The theory of Dirichlet forms deserves to be better known. It is an area of Markov process theory that uses the energy of functionals to study a Markov process from a quantitative point of view. For instance, the recent notes of Saloff-Coste [S-C] use Dirichlet forms to analyze Markov chains with finite state space, by making energy comparisons. In this way, information about a simple chain is ...

We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients σ(x) in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions s(x) over the domain Ω. Although homogenization is a non-linear and non-injective ope...

We consider the Dirichlet series ∞ ∑ k=2 akk −1−x =: f(x), x > 0, with coefficients ak ≥ 0 for all k. Among others, we prove exact estimates of certain weighted Lp-norms of f on the unit interval (0, 1) for any 0 < p <∞, in terms of the coefficients ak . Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for int...

Let ψ $\psi$ be a continuous decreasing function defined on all large positive real numbers. We say that m × n $m\times n$ matrix A $A$ is -Dirichlet if for every sufficiently number t $t$ one can find p ∈ Z $\bm {p} \in {\mathbb {Z}}^m$ , q ∖ { 0 } {q} {Z}}^n\setminus \lbrace \bm {0}\rbrace$ satisfying ∥ − < ( ) $\Vert A\bm {q}-\bm {p}\Vert ^m< \psi ({t})$ and {q}\Vert ^n<{t}$ . This property ...

Let (X , d,m) be a metric measure space with a local regular Dirichlet form. We establish necessary and sufficient conditions for upper heat kernel bounds with subdiffusive space-time exponent to hold. This characterization is stable under rough isometries, that is it is preserved under bounded perturbations of the Dirichlet form. Further, we give a criterion for stochastic completeness in term...

In this paper we first study the generalized weighted Hardy spaces $H^{p}_{L,w}(X)$ for $0 < p \le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on space of homogeneous type $X$ in both cases finite and infinite measure. We show that defined via maximal functions atomic decompositions coincide. Then prove regularity estimates Green inhomogeneous Dirich...

Abstract We discuss random interpolating sequences in weighted Dirichlet spaces ${{\mathcal{D}}}_\alpha $, $0\leq \alpha \leq 1$, when the radii of sequence points are fixed a priori and arguments uniformly distributed. Although conditions for deterministic interpolation these depend on capacities, which very hard to estimate general, we show that is driven by surprisingly simple distribution c...