How to use Hardy-Littlewood-Sobolev inequality to estimate an integral involving two fuctions and Riesz Potential.

Question

Recently I've been studying some PDEs involving Riesz potential and I saw the following assertion:

If $u,v \in H^{1}(\mathbb{R}^{2})$, then $$\int_{\mathbb{R}^{2}}(I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}}|v|dx < +\infty, $$

where $I_{\beta}$ is the Riesz potential and $\beta \in (0,2)$.

I know about Hardy-Littlewood-Sobolev inequality, but I didn't get how to use it to show that the integral above is finite. Moreover, I'd like to know if is possible estimate the integral in terms of $L^{p}-$norm.

Sorry if the answer for this question is something standard, however, some papers in PDE assume that the most part of readers have the expertise required in the subject (which maybe be true) and don't fulfill some details.

Answer 1

Consider $u,v\in H^1(\mathbb{R}^2)$, and $\beta\in (0,2)$, then we have $$ I=\int_{\mathbb{R}^2} \left( I_\beta* |u|^{\beta/2+1}\right) |u|^{\beta/2} |v|\, dx\leq \| u\|_2^{\beta+1} \| v\|_2. $$ This follows basically from Hardy-Littlewood-Sobolev and Hölder's inequality: Fix $\beta$ as above and set $$ p=\dfrac{4}{\beta +2}, \qquad q=\dfrac{2p}{2-\beta p} , \qquad r=\dfrac{4}{\beta}. $$ Notice the following $$ I_\beta:L^p\to L^q, \qquad \frac{1}{q'}= \frac{1}{2}+\frac{1}{r}, $$ and so using first Hölder's inequality, then HLS, then Hölder again, \begin{equation} \begin{split} I & \leq \| I_\beta* |u|^{\beta/2+1} \|_q \| |u|^{\beta/2}|v|\|_{q'}\\ &\leq \| |u|^{\beta/2+1}\|_{p} \| |u|^{\beta/2}|v|\|_{q'}\\ &\leq \| u\|_2^{(\beta+2)/2} \| |u|^{\beta/2}\|_r \| v\|_{2}\\ &= \| u\|_2^{(\beta+2)/2} \| u\|_2^{\beta/2} \| v\|_2 \end{split} \end{equation}