On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider a strongly elliptic self-adjoint operator $L$, where $L$ is negative definite. Now consider the heat semigroup $e^{tL}$, which is $L^2$ contractive. I am curious whether $e^{tL}$ is bounded on $L^1(\Omega)$, that is, could we say something like $$ \Vert e^{tL}f\Vert_{L^1(\Omega)} \leq C\Vert f\Vert_{L^1(\Omega)}, f \in L^1(\Omega) ?$$ If yes, will the constant $C$ be independent of $t$? Most books list these types of results when $p \in (1, \infty)$. I would be really grateful if someone can provide a reference. If results like the above hold only under special conditions on the operator $L$, I would like to know that too. Thanks.

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    $\begingroup$ For positive $f$ and neumann BC for the laplacian, you can prove the conservation of the $L^1$ norm by using the divergence theorem. $\endgroup$ – guacho Apr 29 '15 at 5:54
  • $\begingroup$ This can be done on $C(\bar\Omega)$ by using the maximum principle and then on $L^1$ by duality. See for instance Theorem 3.10 in Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. $\endgroup$ – Michael Renardy Apr 29 '15 at 13:43
  • $\begingroup$ @MichaelRenardy Thanks, I just checked the theorem. Is anything known for Neumann boundary conditions? $\endgroup$ – mathgirl Apr 29 '15 at 15:54
  • $\begingroup$ That is left as an exercise to the reader. $\endgroup$ – Michael Renardy Apr 29 '15 at 16:03

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