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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