# Measurability of the heat semigroup in $L^\infty$

Let $$S(t)$$ be the $$C_0$$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $$L^2(\Omega)$$, where $$\Omega$$ is a bounded open subset of $$R^n$$.

It is known that $$S(t)$$ doesn't induce a $$C_0$$-semigroup in $$L^\infty(\Omega)$$ because of the lack of continuity of $$t \mapsto S(t)x,\; x\in L^\infty(\Omega)$$ w.r.t the $$L^\infty$$-norm.

So my question is the following: is the integral $$\int_0^T \|S(t)x\|_{L^\infty(\Omega)} dt$$ well defined for some class of $$x\in L^2(\Omega)$$ and for every $$T>0$$?

Thanks!

• Many thanks to Prof. Alex for the editing. – Rabat Mar 16 at 20:52
• The integral is indeed well-defined. Firste, $|S(t) x|$ is bounded (pointwise) by the convolution of the Gauss–Weierstrass kernel $k_t$ and $|x|$, and by Cauchy–Schwarz, this convolution is bounded by $\|k_t\|_2 \|x\|_2 = c t^{-1/2} \|x\|_2$. Second, $S(t) x$ is jointly continuous as a function on $(0, \infty) \times \Omega$, and thus $\|S(t) x\|_\infty$ is continuous on $(0, \infty)$. – Mateusz Kwa?nicki Mar 16 at 21:27
• I leave the above as a comment, as I believe this question is more suitable for Math.SE. – Mateusz Kwa?nicki Mar 16 at 21:28
• Thank you for these useful information. Could you please indicate me any reference for that, especially for the continuity. – Rabat Mar 16 at 22:23
• Joint continuity of $S(t) x$ is a consequence of the dominated convergence theorem and joint continuity of heat kernel. The latter property is classical, and I suppose it can be found in most textbooks on heat equation or parabolic PDEs (or Brownian motion); but I do not have a reference off the top of my head. – Mateusz Kwa?nicki Mar 16 at 23:28