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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.

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