Continuity modulus of stochastic homeomorphism flows for SDEs with non-Lipschitz coefficients

Let Xt(x) solve the following Itô-type SDE (denoted by EQ.(σ,b,x)) in Rd

     

Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure

Let D⊂Rd be a bounded domain and let

     

On estimates of the density of Feynman-Kac semigroups of α-stable-like processes

Suppose that α∈(0,2) and that X is an α-stable-like process on Rd. Let F be a function on Rd belonging to the class Jd,α (see Introduction) and be _∑s?tF(Xs−,Xs), t>0, a discontinuous additive functional of X. With neither F nor X being symmetric, under certain conditions, we show that the Feynman-Kac semigroup defined by

     

Boundedness of singular integral operators with variable kernels

Let K be a generalized Calderón-Zygmund kernel defined on Rn×(Rn?{0}). The singular integral operator with variable kernel given by

     

Iterates of a Berezin-type transform

Let B be the open unit ball of Rn and dV denote the Lebesgue measure on Rn normalized so that the measure of B equals 1. Suppose f∈L¹(B,dV). The Berezin-type transform of f is defined by

     

Characterization of graphs having extremal Randi? indices

The higher Randi? index R_t(G) of a simple graph G is defined as

     

Random martingales and localization of maximal inequalities

Let (X,d,μ) be a metric measure space. For ∅≠R⊆(0,∞) consider the Hardy-Littlewood maximal operator

     

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Rayleigh functions σl(ν) are defined as series in inverse powers of the Bessel function zeros λν,n≠0,

     

The products on the unit sphere and even-dimension spaces

The distribution δ(k)(r−1) focused on the unit sphere Ω of Rm is defined by

     

Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let ?n∈C^∞(Rd?{0}) be a non-radial homogeneous distance function of degree n∈N satisfying ?n(tξ)=tn?n(ξ). For f∈S(Rd+1) and δ>0, we consider convolution operator associated with the smooth cone type multipliers defined by

     

On generalized Lie derivations of Lie ideals of prime algebras

Let A be a prime algebra of characteristic not 2 with extended centroid C, let R be a noncentral Lie ideal of A and let B be the subalgebra of A generated by R. If f,d:R→A are linear maps satisfying that

     

Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions

Let Ω⊂RN be a bounded domain with Lipschitz boundary, with a>0 on . Let σ be the restriction to ∂Ω of the (N−1)-dimensional Hausdorff measure and let be σ-measurable in the first variable and assume that for σ-a.e. x∈∂Ω, B(x,⋅) is a proper, convex, lower semicontinuous functional. We prove in the first part that for every p∈(1,∞), the operator Ap:=div(a|∇u|^p−2∇u) with nonlinear Wentzell-Robin type boundary conditions

     

An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control

Given a bounded domain Ω⊂Rd and two integro-differential operators L¹, L² of the form we study the fully nonlinear Bellman equation

(0.1)     

The local variational principle of topological pressure for sub-additive potentials

Let (X,T) be a topological dynamical system and be a sub-additive potential on C(X,R). Let U be an open cover of X. Then for any T-invariant measure μ, let . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle:

     

Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities

In this paper we classify the centers, the cyclicity of its Hopf bifurcation and their isochronicity for the polynomial differential systems in R² of arbitrary degree d?3 odd that in complex notation z=x+iy can be written as

     

Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls

For 0?σ<1/2 we characterize Carleson measures μ for the analytic Besov-Sobolev spaces on the unit ball Bn in Cn by the discrete tree condition

     

On the Small Ball Inequality in all dimensions

Let hR denote an L^∞ normalized Haar function adapted to a dyadic rectangle R⊂d[0,1]. We show that for choices of coefficients α(R), we have the following lower bound on the L^∞ norms of the sums of such functions, where the sum is over rectangles of a fixed volume:

     

Approximate convexity of Takagi type functions

Given a bounded function Φ:R→R, we define the Takagi type function TΦ:R→R by