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In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for

We show that the range of the Cauchy transform on the classical Bergman space is the reproducing kernel Hilbert space of functions in the bi-analytic Dirichlet space subject to a first-order differential equation. The closed formula of its reproducing kernel is given explicitly. Our investigation includes an explicit characterization of the range of the weighted Cauchy transform on the weighted Bergman–Dirichlet spaces, extending Dyn’kin’s result showing that the single-valued functions in the Dirichlet space vanishing at the origin are the Cauchy transform of some functions in the Bergman space of anti-holomorphic functions.

In this paper, we introduce a family of Dirichlet spaces Dnn∈N . This family satisfies the continuous inclusions Dn⊂⋯⊂D2⊂D1⊂D0=D , where D is the classical Dirichlet space. Next, we define and study the operator Xf(z):=f′(z)-f′(0) and its adjoint operator Yf(z)=z2f′(z) on the Dirichlet space D , and we establish an uncertainty inequality of Heisenberg type for this space. A more general uncertainty inequality for the space Dn is also given when we considered the operators Xn=Xn and Yn=Yn .

In this paper, we consider and study the n -dimensional extension of the Bergman–Dirichlet and Bargmann–Dirichlet spaces introduced recently in El Hamyani et al. (Ann Glob Anal Geom 49(1):59–72, 2016 ). We give a complete description of the considered spaces, including the explicit closed formulas for their reproducing kernel functions. Moreover, we investigate their asymptotic behavior when the curvature goes to 0.

We introduce new functional spaces generalizing the weighted Bergman and Dirichlet spaces on the complex disk D R = D ( 0 , R ) as well as the Bargmann–Fock spaces on the whole complex plane C . We give a complete description of the considered spaces. Mainly, we are interested in giving explicit formulas for their reproducing kernel functions and their asymptotic behavior as R goes to infinity.

The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,\\mathsf d,\\mathfrak m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong \\mathrm {CD}^{*}(K,N) condition of Bacher-Sturm.

We prove a conjecture of Brevig, Ortega-Cerdk, Seip and Zhao about contractive inequalities between Dirichlet and Hardy spaces and discuss its consequent connection with the Riesz projection. Keywords: contractive inequalities, Dirichlet spaces, Besov spaces, Hardy spaces, Bergman spaces, Riesz's projection.

Using Newtonian potentials and balayages of positive Borel measures, we describe the family of superharmonically weighted Dirichlet spaces Dω for which every Blaschke sequence is a zero set. We also investigate random zero sets and random Blaschke products in Dω spaces.

This work is devoted to the comparison of de Branges–Rovnyak spaces H(b) and harmonically weighted Dirichlet spaces Dμ. We completely characterize which H(b) spaces are also harmonically weighted Dirichlet spaces Dμ, when μ is a finite sum of atoms. This is a generalization of a previous result by Costara–Ransford (Costara and Ransford J. Funct. Anal. 265(12), 3204–3218 2013): we make no assumptions on the Pythagorean pair (b, a), and we produce new examples.

We characterize the membership in the Schatten ideals $\\mathcal {S}_p$ , $0

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