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In $mathbb R^d$, $d geq 3$, consider the divergence and the non-divergence form operators begin{equation} tag{$i$} - abla cdot a cdot abla + b cdot abla, end{equation} begin{equation} tag{$ii$} - a cdot abla^2 + b cdot abla, end{equation} where $a=I+c mathsf{f} otimes mathsf{f}$, the vector fields $ abla_i mathsf{f}$ ($i=1,2,dots,d$) and $b$ are form-bounded (this includes the sub-critical class $[L^d + L^infty]^d$ as well as vector fields having critical-order singularities). We characterize quantitative dependence on $c$ and the values of the form-bounds of the $L^q rightarrow W^{1,qd/(d-2)}$ regularity of the resolvents of the operator realizations of ($i$), ($ii$) in $L^q$, $q geq 2 vee ( d-2)$ as (minus) generators of positivity preserving $L^infty$ contraction $C_0$ semigroups. The latter allows to run an iteration procedure $L^p rightarrow L^infty$ that yields associated with ($i$), ($ii$) $L^q$-strong Feller semigroups.
This paper discusses some regularity of almost periodic solutions of the Poissons equation $-Delta u = f$ in $mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poissons equation. Proc.
We prove that a probability solution of the stationary Kolmogorov equation generated by a first order perturbation $v$ of the Ornstein--Uhlenbeck operator $L$ possesses a highly integrable density with respect to the Gaussian measure satisfying the n
In this paper, we develop the Littman-Stampacchia-Weinberger duality approach to obtain global W^1,p estimates for a class of elliptic problems involving Leray-Hardy operators and measure sources in a distributional framework associated with a dual formulation with a specific weight function.
This paper establishes global weighted Calderon-Zygmund type regularity estimates for weak solutions of a class of generalized Stokes systems in divergence form. The focus of the paper is on the case that the coefficients in the divergence-form Stoke
This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In cite{IsazaLinaresPonce} it was established that if the initial datun $u_0in H^l((b,infty))$ for some $linmathbb Z^+$ and $bin