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Endpoint mapping properties of spherical maximal operators. (English) Zbl 1036.42019
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.

42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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