$L^p$ generalization of maximal functions

Question

Is there any sort of generalization of the Hardy-Littlewood maximal function $$ Mf = \sup_{B} \frac{1}{|B|}\int_B |f(x)|dx $$ of the type $$ M_pf = \sup_{B} \left(\frac{1}{|B|}\int_B |f(x)|^pdx\right)^{1/p} $$ with $1\leq p\leq \infty$? Is there any reference where I can find information about them?

Answer 1

Yes. You can find it on page $92$ of Stein’s Singular Integrals and Differentiability Properties of Functions. He defines, for $\mu \ge 1$, the function $$M_\mu(f)(x) = \left(\sup_{r > 0} \frac{1}{m(B(x,r))} \int_{B(x,r)} |f(y)|^\mu\, dy\right)^{1/\mu}$$ and notes that $\|M_\mu(f)\|_p \le A_{p,\mu} \|f\|_p$ for $p > \mu$, but the inequality fails for $p \le \mu$.