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Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding heat semigroup.

My question is, what are updated upper bounds on $\| e^{t\Delta}\|_{L^\infty \to L^\infty}$? The literature in this general area is enormous, and it is rather difficult to find the state of the art. This is mainly a reference request.

Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding heat semigroup.

My question is, what are updated upper bounds on $\| e^{t\Delta}\|_{L^\infty \to L^\infty}$? I guess it is realistic to expect very good bounds in the regime of small $t$. The literature in this general area is enormous, and it is rather difficult to find the state of the art. This is mainly a reference request.

Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding heat semigroup.

My question is, what are updated bounds on $\| e^{t\Delta}\|_{L^\infty \to L^\infty}$? The literature in this general area is enormous, and it is rather difficult to find the state of the art. This is mainly a reference request.

Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding heat semigroup.

My question is, what are updated upper bounds on $\| e^{t\Delta}\|_{L^\infty \to L^\infty}$? The literature in this general area is enormous, and it is rather difficult to find the state of the art. This is mainly a reference request.