# Harnack inequality for linear parabolic equations

I am trying to understand the proof of the Harnack inequality using the ideas of J.Nash as proved in the paper "A new proof of Moser's Parabolic Harnack Inequality using the old Ideas of Nash" by E.B.Fabes and D.W.Stroock.

On page 331, they obtain the following bounds: $$\|P^{\Psi}_t f\|_{\infty} \leq \frac{C}{t^{\frac{n}{4}}} e^{\frac{2\alpha^2t}{\lambda}}\|f\|_2$$ and the adjoint bound $$\|(P^{\Psi}_t)^* f\|_{\infty} \leq \frac{C}{t^{\frac{n}{4}}} e^{\frac{2\alpha^2t}{\lambda}}\|f\|_2$$

They then claim that by duality, this implies $$\|P^{\Psi}_t f\|_{2} \leq \frac{C}{t^{\frac{n}{4}}} e^{\frac{2\alpha^2t}{\lambda}}\|f\|_1$$

Can anyone let me know how they go from $$L^{\infty}$$ bounds to obtaining the $$L^2$$ bound?

This is a standard duality argument. Let $$f\in L^1$$, $$g\in L^2$$ (I did not look at the paper, maybe you have to restrict to suitable dense subspaces). Then $$\left|\int(P_t f)g\right|=\left|\int fP_t^\ast g\right|\leq \|f\|_{1}\|P_t^\ast g\|_\infty\leq \|P_t^\ast\|_{L^2\to L^\infty}\|f\|_1\|g\|_2.$$ Thus $$\|P_t\|_{L^1\to L^2}\leq \|P_t^\ast\|_{L^2\to L^\infty}$$.