Let $f$ satisfy the estimate $$ sup\,f\,_{B_r}\leq c\Big(\frac{1}{|B_r|}\int_{B_r}f^p\Big)^\frac{1}{p}, $$ for every $p\geq 2$. Here $B_r=B_r(0)$, a ball of radius $r$ centered at $0$ and $|B_r|$ is the Lebsgue measure of $B_r$. Let $q\in(0,2)$, then how does the following estimate follows: $$ {sup\,f}\,_{B_r}\leq c\frac{1}{|B_r|}\|f\|_{L^\infty(B_r)}^{1-q}\|f\|_{L^p(B_r)}^{q}. $$ More precisely, I could not get the second estimate from the last on page 7 in https://arxiv.org/pdf/2103.09646.pdf
Here $c$ is some given constant and $f\geq 0$. I was trying to use Young's and H'older both, but could not suceed yet. Can someone please help. Thanks.
I had a quick look at the paper and I saw that we also have $f$ is non-negative. Hence, if we raise $$\sup_{B_r} f \lesssim \bigg ( \frac 1 {\vert B_r \vert} \int_{B_r} \vert f(x)\vert^p \, dx \bigg )^{1/p} $$ to the power $p$ we get $$ (\sup_{B_r} f)^p \lesssim \frac 1 {\vert B_r \vert} \int_{B_r} \vert f(x)\vert^p \, dx .$$ Dividing both sides by $\| f\|_{L^\infty(B_r)}^{p-1}$ (noting $\| f\|_{L^\infty(B_r)}=\sup_{B_r} f$ by non-negativity) gives $$\sup_{B_r} f \lesssim \frac 1 {\vert B_r \vert}\| f\|_{L^\infty(B_r)}^{1-p} \int_{B_r} \vert f(x)\vert^p \, dx $$ as required.