I hope someone here can explain the following statement I read from the book stochastic differential equation and diffusion process written by Ikeda and Watanabe.
On page 186 of chapter IV of the book, it states the following:
I would like to ask how one can apply Hausdorff Young inequality to conclude statement 1?
The result that's usually called the Hausdorff–Young inequality (about the Fourier transform on $L^p$ spaces) is not what we want here. The authors refer to Young's inequality for convolutions, specifically: $$\|\nu_\lambda * f\|_p \le \|\nu_\lambda\|_1 \|f\|_p$$ Here, $$\|\nu_\lambda\|_1 = \int_0^\infty e^{-\lambda t} \|g_t\|_1 \,dt= 1/\lambda$$ because $g_t$ is nonnegative and $\|g_t\|_1=1$.