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Besicovitch type maximal operators and applications to Fourier analysis. (English) Zbl 0756.42014

Averaging over tubes of unit length and width \(\delta<1\) leads to two maximal functions \(f^*_ \delta\), \(f^{**}_ \delta\). The first is defined on the sphere \(S_{d-1}\) in \(\mathbb{R}^ d\) and for a fixed direction one takes the maximum over all translates of the tube (the Kakeya maximal function). The second is defined on \(\mathbb{R}^ d\) and at a given point \(x\) one considers all \(\delta\) tubes centered at \(x\) and varying direction (the Nikodým maximal function). It is a natural conjecture that the \(L^ p\) bound on \(f^*_ \delta\), \(f^{**}_ \delta\) is given essentially by the formula \((1/\delta)^{d/p- 1+\varepsilon}\) for \(1\leq p\leq d\). This fact is known for \(d=2\) but open in higher dimensions. In particular, it seems unknown whether a set in \(\mathbb{R}^ 3\) containing a line in every direction needs to have full Hausdorff dimension. The estimate is verified here if \(p<p(d)\), where \((d+1)/2<p(d)<d/2+1\). As an application the author proves that there are no so-called Besicovitch \((d,k)\) sets provided that \(d\leq 2^{k-1+k}\). The author also applies these results to the restriction problem for the Fourier transform and spherical multipliers.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B15 Multipliers for harmonic analysis in several variables
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B08 Summability in several variables
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