Inequalities for some maximal functions. II

Abstract:

Let $S$ be a smooth compact hypersurface in ${{\mathbf {R}}^n}$, and let $\mu$ be a measure on $S$, absolutely continuous with respect to surface measure. For $t$ in ${{\mathbf {R}}^ + },{\mu _t}$ denotes the dilate of $\mu$ by $t$, normalised to have the same total variation as $\mu$: for $f$ in $\mathcal {S}({{\mathbf {R}}^n}),{\mu ^\# }f$ denotes the maximal function ${\sup _{t > 0}}|{\mu _t}\ast f|$. We seek conditions on $\mu$ which guarantee that the a priori estimate \[ \left \| \mu ^\# f\right \|_p \leq C\left \| f \right \|_p, \quad f \in S(\mathbf {R}^n),\] holds; this estimate entails that the sublinear operator ${\mu ^\# }$ extends to a bounded operator on the Lebesgue space ${L^p}({{\mathbf {R}}^n})$. Our methods generalise E. M. Stein’s treatment of the "spherical maximal function" [5]: a study of "Riesz operators", $g$-functions, and analytic families of measures reduces the problem to that of obtaining decay estimates for the Fourier transform of $\mu$. These depend on the geometry of $S$ and the relation between $\mu$ and surface measure on $S$. In particular, we find that there are natural geometric maximal operators limited on ${L^p}({{\mathbf {R}}^n})$ if and only if $p \in (q,\infty ];q$ is some number in $(1,\infty )$, and may be greater than $2$. This answers a question of S. Wainger posed by Stein [6]>.