Regularity of solutions to anisotropic nonlocal equations

We study harmonic functions associated to systems of stochastic differential equations of the form $$dX_t^i=A_{i1}(X_{t-})dZ_t^1+\cdots +A_{id}(X_{t-})dZ_t^d$$ d X t i = A i 1 ( X t - ) d Z t 1 + ⋯ + A id ( X t - ) d Z t d , $$i\in \{1,\dots ,d\}$$ i ∈ { 1 , ⋯ , d } , where $$Z_t^j$$ Z t j are independent one-dimensional symmetric stable processes of order $$\alpha _j\in (0,2)$$ α j ∈ ( 0 , 2 ) , $$j\in \{1,\dots ,d\}$$ j ∈ { 1 , ⋯ , d } . In this article we prove Hölder regularity of bounded harmonic functions with respect to solutions to such systems.

A boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions

In this paper, we present a boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions, which extends the classical Schwarz lemma for bounded harmonic functions to higher dimensions.

Plemelj–Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators

Let R be a compact Riemann surface and $$\Gamma $$ Γ be a Jordan curve separating R into connected components $$\Sigma _1$$ Σ 1 and $$\Sigma _2$$ Σ 2 . We consider Calderón–Zygmund type operators $$T(\Sigma _1,\Sigma _k)$$ T ( Σ 1 , Σ k ) taking the space of $$L^2$$ L 2 anti-holomorphic one-forms on $$\Sigma _1$$ Σ 1 to the space of $$L^2$$ L 2 holomorphic one-forms on $$\Sigma _k$$ Σ k for $$k=1,2$$ k = 1 , 2 , which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves $$\Gamma $$ Γ , to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of anti-holomorphic one-forms which are orthogonal to $$L^2$$ L 2 anti-holomorphic one-forms on R with respect to the inner product on $$\Sigma _1$$ Σ 1 . We show that the restriction of the Schiffer operator $$T(\Sigma _1,\Sigma _2)$$ T ( Σ 1 , Σ 2 ) to V is an isomorphism onto the set of exact holomorphic one-forms on $$\Sigma _2$$ Σ 2 . Using the relation between this Schiffer operator and a Cauchy-type integral involving Green’s function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.

Path Connected Components in the Spaces of Weighted Composition Operators with the Strong Operator Topology II

The path connected components are determined in the space of weighted composition operators on the space of bounded harmonic functions with the strong operator topology.

Brownian motion on stationary random manifolds

We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volume growth of the manifold. Other almost sure properties of these random manifolds are also studied.