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YCor
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Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary, and let $L$ be a negative definite second order elliptic differential operator defined with $\mathcal{D}(L) \subset H^2(\Omega)$, given by either Dirichlet or Neumann boundary conditions. My question is, for $f \in L^1(\Omega)$, do we have the following for small time $t$: $$ \Vert e^{tL}f \Vert_{L^1(\Omega)} \leq C\Vert f\Vert_{L^1(\Omega)}? $$

The reason why I expect this to hold is, I expect something like $e^{tL}f \to f$ pointwise almost everywhere. Is this true? In particular, does $\Omega$ need to be bounded for this? Thanks in advance.

Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary, and let $L$ be a negative definite second order elliptic differential operator defined with $\mathcal{D}(L) \subset H^2(\Omega)$, given by either Dirichlet or Neumann boundary conditions. My question is, for $f \in L^1(\Omega)$, do we have the following for small time $t$: $$ \Vert e^{tL}f \Vert_{L^1(\Omega)} \leq C\Vert f\Vert_{L^1(\Omega)}? $$

The reason why I expect this to hold is, I expect something like $e^{tL}f \to f$ pointwise almost everywhere. Is this true? In particular, does $\Omega$ need to be bounded for this? Thanks in advance.

Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary, and let $L$ be a negative definite second order elliptic differential operator defined with $\mathcal{D}(L) \subset H^2(\Omega)$, given by either Dirichlet or Neumann boundary conditions. My question is, for $f \in L^1(\Omega)$, do we have the following for small time $t$: $$ \Vert e^{tL}f \Vert_{L^1(\Omega)} \leq C\Vert f\Vert_{L^1(\Omega)}? $$

The reason why I expect this to hold is, I expect something like $e^{tL}f \to f$ pointwise almost everywhere. Is this true? In particular, does $\Omega$ need to be bounded for this?

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user95820
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Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary, and let $L$ be a negative definite second order elliptic differential operator defined with $\mathcal{D}(L) \subset H^2(\Omega)$, given by either Dirichlet or Neumann boundary conditions. My question is, for $f \in L^2(\Omega)$$f \in L^1(\Omega)$, do we have the following for small time $t$: $$ \Vert e^{tL}f \Vert_{L^1(\Omega)} \leq C\Vert f\Vert_{L^1(\Omega)}? $$

The reason why I expect this to hold is, I expect something like $e^{tL}f \to f$ pointwise almost everywhere. Is this true? In particular, does $\Omega$ need to be bounded for this? Thanks in advance.

Let $\Omega$ be a domain in $\mathbb{R}^n$, and let $L$ be a negative definite second order elliptic differential operator defined with $\mathcal{D}(L) \subset H^2(\Omega)$, given by either Dirichlet or Neumann boundary conditions. My question is, for $f \in L^2(\Omega)$, do we have the following for small time $t$: $$ \Vert e^{tL}f \Vert_{L^1(\Omega)} \leq C\Vert f\Vert_{L^1(\Omega)}? $$

The reason why I expect this to hold is, I expect something like $e^{tL}f \to f$ pointwise almost everywhere. Is this true? In particular, does $\Omega$ need to be bounded for this? Thanks in advance.

Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary, and let $L$ be a negative definite second order elliptic differential operator defined with $\mathcal{D}(L) \subset H^2(\Omega)$, given by either Dirichlet or Neumann boundary conditions. My question is, for $f \in L^1(\Omega)$, do we have the following for small time $t$: $$ \Vert e^{tL}f \Vert_{L^1(\Omega)} \leq C\Vert f\Vert_{L^1(\Omega)}? $$

The reason why I expect this to hold is, I expect something like $e^{tL}f \to f$ pointwise almost everywhere. Is this true? In particular, does $\Omega$ need to be bounded for this? Thanks in advance.

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user95820
user95820
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Short time $L^1$ bounds for semigroups obtained from elliptic operators

Let $\Omega$ be a domain in $\mathbb{R}^n$, and let $L$ be a negative definite second order elliptic differential operator defined with $\mathcal{D}(L) \subset H^2(\Omega)$, given by either Dirichlet or Neumann boundary conditions. My question is, for $f \in L^2(\Omega)$, do we have the following for small time $t$: $$ \Vert e^{tL}f \Vert_{L^1(\Omega)} \leq C\Vert f\Vert_{L^1(\Omega)}? $$

The reason why I expect this to hold is, I expect something like $e^{tL}f \to f$ pointwise almost everywhere. Is this true? In particular, does $\Omega$ need to be bounded for this? Thanks in advance.