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We consider the Schrödinger operators $-\Delta +V\left(x\right)$ in ${ℝ}^n$ where the nonnegative potential $V\left(x\right)$ belongs to the reverse Hölder class $B_q$ for some $q\ge n/2$. We obtain the optimal $L^p$ estimates for the operators $(-\Delta +V{)}^{i\gamma },{\nabla }^2(-\Delta +V{)}^{-1},\nabla (-\Delta +V{)}^{-1/2}$ and $\nabla (-\Delta +V{)}^{-1}$ where $\gamma \in ℝ$. In particular we show that $(-\Delta +V{)}^{i\gamma }$ is a Calderón-Zygmund operator if $V\in B_{n/2}$ and $\nabla (-\Delta +V{)}^{-1/2},\nabla (-\Delta +V{)}^{-1}\nabla$ are Calderón-Zygmund operators if $V\in B_n$.

Let $ℒ=$ -div $\left(A\right(x\left)\nabla \right)$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ${ℝ}^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p\>2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla {\left(ℒ\right)}^{-1/2}$ on the $L^p$ space. As an application, for $1\<p\<3+ϵ$, we establish the $L^p$ boundedness of Riesz transforms on Lipschitz domains for operators with $VMO$ coefficients. The range of $p$ is sharp. The closely related boundedness of ...

We establish a Carleman type inequality for the subelliptic operator $ℒ={\Delta }_z+|x{|}^2{\partial }_t^2$ in ${ℝ}^{n+1}$, $n\ge 2$, where $z\in {ℝ}^n$, $t\in ℝ$. As a consequence, we show that $-ℒ+V$ has the strong unique continuation property at points of the degeneracy manifold $\left\{\right(0,t)\in {ℝ}^{n+1}|t\in ℝ\}$ if the potential $V$ is locally in certain $L^p$ spaces.

We consider initial-boundary value problems for a parabolic system in a Lipschitz cylinder. When the space dimension is three, we obtain estimates for the solutions when the lateral data taken from the best possible range of L-spaces.

For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an $O\left(ϵ\right)$ estimate in $H^1$ for solutions with Dirichlet condition.