×

\(L^ 2\)-boundedness of spherical maximal operators with multidimensional parameter sets. (English) Zbl 0671.42017

For \(s>0\), let \(M_ sf(x)\) be the spherical mean of f on the sphere \(\{y\in {\mathbb{R}}^ n\); \(| y-x| =s\}\). The author introduces a k- dimensional parameter set S as follows. Let T be a k-dimensional \(C^ 1\)-surface in \({\mathbb{R}}_+^{n+1}\) such that \(T\cap {\mathbb{R}}^ n\times \{0\}\neq \emptyset\) and no tangent plane of t is parallel to \({\mathbb{R}}^ n\times \{0\}\), and let \(\rho: {\mathbb{R}}_+\to {\mathbb{R}}_+\) be a differentiable function such that for some constants \(c_ 1\), \(c_ 2\) with \(c_ 1>c_ 2>0\), \(\rho '(r)\geq c_ 2\) and \(\rho (r)\leq c_ 1r\) for \(0<r\leq 1\). Set \(S=\{(\rho (r)y,r)\); (y,r)\(\in T\), \(0<r\leq 1\}\), which he calls a k-dimensional parameter set. For such a set S he considers the maximal operator defined by \({\mathcal M}f(x)=\sup_{(u,s)\in S}| M_ sf(x-u)|.\) He gives the \(L^ 2\)-boundedness of \({\mathcal M}\) under the assumption, (i) \(n\geq 3\) and dim \(S\leq n-2\), or (ii) \(n\geq 3\), \(\dim S=n-1\) and there is an (n-1)-dimensional subspace V of \({\mathbb{R}}^ n\) such that for each (u,s)\(\in S\), \(\{\) y-u; (y,s)\(\in S\}\subset V\). These extend (on \(L^ 2({\mathbb{R}}^ n))\) the theorem of Stein, where \(S=\{(0,s)\); \(s>0\}\), and its generalizations to dim S\(=1\) by Greenleaf and Sogge-Stein.
Reviewer: K.Yabuta

MSC:

42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite
Full Text: DOI