I just wonder if someone could just give me the proof for the following estimation: $$ \| \nabla (e^{t\Delta} f) \|^{2}_{L^{2}} \leq \|{f}\|_{L^{1}} \ \|e^{t\Delta} f\|_{L^{\infty}} $$ where $e^{t\Delta}$ is the Gaussian Heat Kernel operator.
Thanks for any help. This estimate appears in a paper of Koch and Tataru: Well posedness of the Navier-Stokes equation.