Bourgain, J. Besicovitch type maximal operators and applications to Fourier analysis. (English) Zbl 0756.42014 Geom. Funct. Anal. 1, No. 2, 147-187 (1991). 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. Reviewer: M.Milman (Boca Raton) Cited in 6 ReviewsCited in 113 Documents 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 Keywords:Besicovitch type maximal operator; Bochner-Riesz summability; Kakeya maximal function; Nikodým maximal function; Fourier transform; spherical multipliers PDFBibTeX XMLCite \textit{J. Bourgain}, Geom. Funct. Anal. 1, No. 2, 147--187 (1991; Zbl 0756.42014) Full Text: DOI EuDML References: [1] [Bol]J. Bourgain, Remarks on Montgomery’s conjecture on exponential sums, preprint IHES (1990). [2] [Bo2]J. Bourgain, On the restriction and multiplier problem in \(\mathbb{R}\)3, Preprint IHES, M/90/74, to appear in Springer LNM. [3] [Bo3]J. Bourgain,L p -estimates for oscillatory integrals in several variables, to appear in GAFA. [4] [Co]A. Cordoba, A note on Bochner-Riesz operators, Duke Math. J. 46, N3 (1979), 565–572. · Zbl 0438.42013 · doi:10.1215/S0012-7094-79-04625-8 [5] [CS]L. Carleson, P. Sjölin, Oscillatory integrals and multiplier problem of the disc, Studia Math. 44 (1972), 287–299. · Zbl 0215.18303 [6] [Fa1]K.J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Math. 85. [7] [Fa2]K.J. Falconer, Sections of setz of zero Lebesgue measure, Mathematika, 27, (YEAR), 90–96. [8] [Fe1]C. Fefferman, A note on spherical summation multipliers, Israel J. Math., (1973), 44–52. · Zbl 0262.42007 [9] [Fe2]C. Fefferman, The multiplier problem for the ball, Annals Math. 94 (1971), 330–336. · Zbl 0234.42009 · doi:10.2307/1970864 [10] [He]C. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sc. USA (1954). · Zbl 0059.09901 [11] [Ho]L. Hormander, Oscillatory integrals and multipliers onFL P , Arkiv Math. 11 (1973), 1–11. · Zbl 0254.42010 · doi:10.1007/BF02388505 [12] [Pi]G. Pisier, Factorization of operators throughL p,andL p,1 , and noncommutative generalizations, Math. Ann. 276 (1986), 105–136. · Zbl 0619.47016 · doi:10.1007/BF01450929 [13] [St]E. Stein, Limits of sequences of operators, Annals Math. 74 (1961), 140–170. · Zbl 0103.08903 · doi:10.2307/1970308 [14] [St2]E.M. Stein, Some problems in harmonic analysis, Proc. Symposia in Pure Math. 35, I, 3–20 (AMS publications). [15] [St3]E. Stein, beijing Lectures in Harmonic Analysis, Princeton University Press, N. 112. [16] [T]P. Tomas, A restriction theorem for the Fourier transform, Bull. AMS, 81, N2 (1975), 477–478. · Zbl 0298.42011 · doi:10.1090/S0002-9904-1975-13790-6 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.