## HW due October 11

Exercise: Let $L$ be an elliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure $\mu$ . Let $\alpha$ be a multi-index. If $K$ is a compact set of $\mathbb{R}^n$, show that there exists a positive constant $C$ such that for $f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R})$, $\sup_{x \in K} |\partial^{\alpha} \mathbf{P}_t f(x)| \le C \left( 1 +\frac{1}{t^{|\alpha|+\kappa}} \right) \| f \|_{\mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R})},$ where $\kappa$ is the smallest integer larger than $\frac{n}{4}$

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