an:02011915
Zbl 1036.42019
Seeger, Andreas; Tao, Terence; Wright, James
Endpoint mapping properties of spherical maximal operators.
EN
J. Inst. Math. Jussieu 2, No. 1, 109-144 (2003).
1474-7480 1475-3030
2003
j
42B25 42B20
maximal function; spherical means; \(\delta\)-entropy; Minkowski dimension; \(L^p\) estimates; weak type estimates
Let \(n\) be an integer with \(n\geq 2\). Define the spherical means operator \(\mathcal A_t\) by \(\mathcal A_tf(x)=\int_{S^{n-1}} f(x-ty)\,d \sigma(y)\) \((t>0)\), and for a set \(E\subset(0,\infty)\) its maximal operator \(\mathcal M_E\) by \(\mathcal M_Ef(x)=\sup_{t\in E}| \mathcal A_tf(x)| \). E.M.~Stein showed that for \(E=(0,\infty)\) \(\mathcal M_E\) is bounded on \(L^p(\mathbb R^n)\) if and only if \(p>n/(n-1)\) \((n\geq3)\), and Bourgain showed it in the case \(n=2\). Seeger, Wainger and Wright showed that \(\mathcal M_E\) is bounded from the radial part \(L^p| _{\roman{radial} }\) to the Lorentz space \(L^{p,q}\), \(1<p<n/(n-1)\), \(p\leq q\leq\infty\), if and only if \(E\) satisfies
Condition \(\mathcal C_{p,q}\):
\[
\sup_j\bigl( \sum_{n\geq0} [N(E^{j+\ell},2^j)]^{q/p}2^{-\ell(n-1)q/p'} \bigr)^{1/q}< \infty\;(p\leq q<\infty),
\]
\[
\sup_{k\in \mathbb Z, \delta>0}N(E^k, 2^k\delta)^{1/p} \delta^{(n-1)/p'}<\infty\;(q=\infty).
\]
Here, \(N(E, \delta)\) is the \(\delta\)-entropy number of \(E\), i.e., the minimal number of intervals of length \(\delta\) needed to cover \(E\), and \(E^k=[2^k, 2^{k+1})\cap E\).
Under some additional regularity assumption (denoted by \(\mathcal R_p\)), the authors give analogues of the radial case. That is, for \(1<p<n/(n-1)\) \(\mathcal M_E\) is bounded on \(L^p(\mathbb R^n)\) iff \(E\) satisfies Condition \(\mathcal C_{p,q}\), and is of weak type \((p,p)\) iff \(E\) satisfies Condition \(\mathcal C_{p,\infty}\).
The endpoint case \(p=n/(n-1)\) is also discussed.
Kôzô Yabuta (Nishinomiya)