A Regularity Theory for Parabolic Equations with Anisotropic Nonlocal Operators in \(\boldsymbol{L_{{q}}(L_{{p}})}\) Spaces
Jae-Hwan Choi, Jae-Hoon Kang, Daehan Park
Abstract
.In this paper, we present an \(L_q(L_p)\)-regularity theory for parabolic equations of the form \( \partial_t u(t,x)=\mathcal {L}^{\vec {a},\vec {b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. \) Here, \(\mathcal{L}^{\vec{a},\vec{b}}(t)\) represents anisotropic nonlocal operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: \( \mathcal {L}^{\vec {a},\vec {0}}(t)u(x)=\sum_{i=1}^{d} \int_{\mathbb {R}} ( u(x^{1},\dots,x^{i-1},x^{i}+y^{i},x^{i+1},\dots,x^{d}) - u(x) ) \frac {a_{i}(t,y^{i})}{|y^{i}|^{1+\alpha_{i}}} \mathrm {d}y^{i}. \) To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calderón–Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic nonlocal operators and parabolic equations with isotropic nonlocal operators.Keywordsanisotropic nonlocal operatorSobolev regularityLevy processMSC codes45K0535B6547G2060H30