Im generally curious about $L^p$ estimates of integral averages. In particular, let $M$ be a (possibly noncompact) space and $f$ a (probably) smooth function with (probably, but in general maybe not) compact support. I want to consider the integral averages of f at fixed scale r>0: $$ f_r(x):=\frac{1}{vol(B_r(x))}\int_{B_r(x)}f(y)\mathrm{d}\mu(y).$$ Here M may be a Riemannian manifold, or some other space where $vol(B_r(x))$ may be non-constant, but where we assume noncollapsing so that $\inf_{x\in M}vol(B_r(x))>0$.
I would like to know what can generally be said about the $p$ norms of $f_r$, in terms of $f$ itself. For instance, it is trivial to prove a $(1,\infty)$ estimate: $$\|f_r\|_\infty\leqslant \frac{1}{\inf_{x\in M}vol(B_r(x))}\|f\|_1.$$
I am led to suspect that an estimate like $$\|f_r\|_2\leqslant \frac{1}{\inf_{x\in M}vol(B_r(x))^{1/2}}\|f\|_1$$ should hold, perhaps via an application of the $(1,\infty)$ estimate, Holders inequality, Jensen, or something of this nature. In any case, I havent been able to pin such a thing down. Writing out the $L^2$ norm of $f_r$ leads me an estimate like $$\|f_r\|^2_2\leqslant \frac{1}{\inf_{x\in M}vol(B_r(x))}\|f\|_1\|f_r\|_1$$ which would be perfect if there were a $(1,1)$ estimate with constant $1$. Any ideas or directions would be very appreciated!