Let L=??+V be a Schrödinger operator, where the potential V belongs to reverse Hölder class. By subordinative formula, we introduce fractional heat semigroup {e?tL?}t>0, 0<?<1, associated with L. aid of fundamental solution equation: ?tu+Lu=?tu??u+Vu=0, estimate gradient and time-fractional derivatives kernel K?,tL(·,·), respectively. This method is independent Fourier transform, can a...

We prove a multi-parameter dyadic embedding theorem for the Hardy operator on multi-tree. also show that large class of Dirichlet spaces in bi-disc and tri-disc, this proves those holomorphic functions bi- tri-disc. completely describe Carleson measures such embeddings. The result below generalizes Arcozzi et al. from bi-tree to tri-tree Carleson–Chang condition box condition. One our descripti...

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly divergence form operator have been focus recent studies. In this setting we show that an with coefficients satisfying vanishing Carleson condition in upper half space is asymptotically optimal $$A_\infty $$ weight. particular, for such operators logarithm kernel (locally) mean oscill...

We prove that the generalized Carleson operator with polynomial phase function of degree two is of weak type (2,2). For this, we introduce a new approach to the time-frequency analysis of the quadratic phase.

In this work we characterize boundedness and compactness of weighted composition operators acting between Dirichlet type spaces by using Carleson measures. We also find essential norm estimates for these operators.

We investigate the function theory on function spaces on a dyadic tree which model Dirichlet spaces of holomorphic functions. Most of the specific questions addressed deal with Carleson measures on those spaces.

We describe some recent results on the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplacian. More precisely we focus on Carleson-type estimates and boundary Harnack principles.