Heat kernel of fractional Laplacian with Hardy drift via desingularizing weights

Abstract

We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular weight. Contents

On admissible singular drifts of symmetric α‐stable process

“…Moreover, even if $$\alpha =2$$, the form method can handle only the smaller class of vector fields $$|b|^2 \in \mathbf {F}_{\delta ^2} \big (\equiv \mathbf {F}_{\delta ^2}^1\big )$$ while giving a weaker result, on the regularity of the elements of the domain of $$\Lambda _2(b)$$, see detailed discussion in [15], see also [17]. The Hille Perturbation Theorem, while applicable to $$(-\Delta )^{\frac{\alpha }{2}} + b \cdot \nabla$$ in L 2 for all $$1<\alpha \le 2$$, can handle only the smaller class $$|b|^2 \in \mathbf {F}^{\alpha -1}_{\delta ^2}$$, see [19, Proposition 7] for details. See also Remark 3 below.…”

“…The calculational techniques used in the proof of the analogue of Proposition 1.8 in [18] are unavailable when $$\alpha <2$$. In this regard, we develop a new approach to the proof of these estimates taking advantage of the fact that the $$L^p$$ inequalities of [2, 26] are valid for abstract symmetric Markov generators, in particular, for a “weighted” fractional Laplace operator; we show that the latter is indeed a symmetric Markov generator using the method of proof of L 1 accretivity of non‐local operators in weighted spaces introduced in [19] (but for different weights and for different purpose). Armed with the $$L^p$$ inequalities for the weighed fractional Laplace operator, we repeat the principal steps of construction of the Feller semigroup but now in the weighted space, using the fact that the crucial pointwise estimate (C.0) does not depend on the choice of the weight on $$\mathbb {R}^d$$.…”

“…By duality, it suffices to prove $$\begin{equation*} {\left\Vert \eta e^{-t(\omega +A)}\eta ^{-1}f\right\Vert} _{1} \le \Vert f\Vert _1, \quad f\in L^1. \end{equation*}$$We employ the method introduced in [19]. Define truncated weights $$\begin{equation*} \eta _n:=\theta _n(\eta ), \quad n=1,2,\dots , \end{equation*}$$where $$\begin{equation*} \theta (s):={\left\lbrace \def\eqcellsep{&}\begin{array}{ll}s, & 0<s<1, \\[3pt] 2, & s>2, \end{array} \right.}$$…”

Methods of the Perturbation Theory for Fundamental Solutions to the Generalization of the Fractional Laplaciane

“…and for all r, ρ such that 0 < r < ρ. There exist extensive literature dedicated to the partial differential fractional Laplacian operator, general questions can be found in [4,[7][8][9], a wide review is presented in [4], an interesting approach to nonlinear heat equations in modulation spaces and Navier-Stokes equations can be found in [10]; some aspects of weights inequalities are described in [2,11]; fractional Laplacian is considered in [2,12], the list of selected works consists of 29 works . In the recent works [1,2], authors proved sharp two-sided estimations on the heat kernel of the fractional Laplacian with the perturbation of drift having critical-order singularity, also authors show that the operator with the heat kernel of the fractional Laplacian can be expressed as a Feller generator so that the probability measures uniquely determined by the Feller semigroup admits description as weak solutions to the corresponding SDE.…”

“…where the vector bx ðÞ¼cjxj À2s x is singular; and prove the HarnackÀÁis nonnegative in a ball with the center in point x 0 with a radius ρ for all r, ρ such that 0 < r < ρ. Let 1 ≤ p < ∞ and s ∈ 0, 1 ðÞ , and let u ∈ W p s be a weak solution to(2), where its fundamental solution satisfies condition(15).Then, the function u ∈ W p s is locally Holder continuous and oscillation of the function satisfies the estimation…”