In page 79 of Davies's book on Heat Kernels and spectral theory, the author proves that
$$\lVert e^{-Ht}f \rVert\_2 \leq c\_1t^{-\mu/ 4}\lVert f \rVert\_1$$
where the norms are $L^p$ norms. He states



> 
> by duality, it follows that
>  $$\lVert e^{-Ht}f \rVert\_\infty \leq c\_1t^{-\mu/ 4}\lVert f \rVert\_2$$
> 
> 
> 


Can someone explain what exactly this "duality" argument is? Above, $e^{-Ht}f$ we can take to be the solution of the heat equation with initial data $f$ (where $H$ is the Laplacian).



Duality argument to get $L^\infty-L^2$ inequality
You have shown that $e^{-Ht}\colon L^1\to L^2$ is bounded with the given operator norm. It follows that the dual operator
$$\left(e^{-Ht}\right)^\*\colon (L^2)^\* \to (L^1)^\*$$
is bounded with the same operator norm, because of the characterisation
$$\|T\|\_{L^1\rightarrow L^2}=\|T^\*\|\_{L^\infty\rightarrow L^2}=\sup\_{f,g}\frac{\langle Tf,g\rangle}{\|f\|\_1\|g\|\_2}\,.$$  

Since $(L^1)^\* = L^\infty$ and $(L^2)^\* = L^2$ (these are isomorphisms, actually), it remains to check that $\left(e^{-Ht}\right)^\* = e^{-Ht}$.


For this note that $$\int f(x)(e^{-Ht}g)(x)dx = \iint f(x)e(x,y;t)g(y)dxdy$$ where $e(x,y;t)$ is the heat kernel, which is real and symmetric in the variables $x,y$. It follows that $\left(e^{-Ht}\right)^\*$ is again integration against the heat kernel as claimed.


