Boundedness and compactness of commutators related with uniformly parabolic operators
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키워드
초록
Let $$P_{A}=\partial_{t}-\sum_{i,j=1}^{n}\partial_{i}(a_{i j}(x)\partial_{j})+V(x)$$ be the uniformly parabolic operator, where nonnegative potential function $V(x)$ belongs to the reverse H\"older class $RH_q$ with $q\geq n/2,\; n\geq 3$. We deduce some regularity estimates for the fundamental solution of $P_{A}$. As applications, we obtain the $L^p$-boundedness and $L^p$-compactness of commutators $[b,V^{\alpha}{P_{A}}^{-\alpha}]$, where $b \in VMO (\mathbb{ R }^{n+1}_{+})$ and $0 \leq \alpha \leq 1$.
Abstract
Let $$P_{A}=\partial_{t}-\sum_{i,j=1}^{n}\partial_{i}(a_{i j}(x)\partial_{j})+V(x)$$ be the uniformly parabolic operator, where nonnegative potential function $V(x)$ belongs to the reverse H\"older class $RH_q$ with $q\geq n/2,\; n\geq 3$. We deduce some regularity estimates for the fundamental solution of $P_{A}$. As applications, we obtain the $L^p$-boundedness and $L^p$-compactness of commutators $[b,V^{\alpha}{P_{A}}^{-\alpha}]$, where $b \in VMO (\mathbb{ R }^{n+1}_{+})$ and $0 \leq \alpha \leq 1$.