×

The generalized localization for multiple Fourier integrals. (English) Zbl 1195.42053

Authors’ abstract: We investigate almost-everywhere convergence properties of the Bochner-Riesz means of \(N\)-fold Fourier integrals under summation over domains bounded by the level surfaces of the elliptic polynomials. It is proved that if the order of the Bochner-Riesz means \(s \geqslant (N - 1)(1/p - 1/2)\), then the Bochner-Riesz means of a function \(f \in L_p(\mathbb{R}^N), 1 \leqslant p \leqslant 2\) converge to zero almost-everywhere on \(\mathbb{R}^N \setminus \mathrm{supp}(f)\).

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Alimov, Sh.; Ashurov, R.; Pulatov, A., Multiple Fourier series and Fourier integrals, (), 1-97 · Zbl 0782.42012
[2] Ahmedov, Anvarjon, The principle of general localization on unit sphere, J. math. anal. appl., 356, 1, 310-321, (2009) · Zbl 1172.42012
[3] Ashurov, R., Summability almost everywhere of Fourier series in \(L_p\) with respect to eigenfunctions, Mat. zametki, 34, 6, 837-843, (1983)
[4] Bastys, A., Generalized localization of Fourier series with respect to the eigenfunctions of the Laplace operator in the classes \(L_p\), Litovskii mat. sb., 31, 3, 387-405, (1983)
[5] Carbery, A.; Soria, F., Almost everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an \(L_2\)-localisation principle, Rev. mat. iberoamericana, 4, 2, 319-337, (1988) · Zbl 0692.42001
[6] Carbery, A.; Soria, F., Pointwise Fourier inversion and localisation in \(R^n\), J. Fourier anal. appl., 3, 847-858, (1997) · Zbl 0896.42007
[7] Il’in, V., On a generalized interpretation of the principle of localization for Fourier series with respect to fundamental systems of functions, Sibirsk. mat. zh., 9, 5, 1093-1106, (1968) · Zbl 0191.07503
[8] Randol, B., On the Fourier transform of the indicator function of a planar set, Trans. amer. math. soc., 139, 271-279, (1969) · Zbl 0183.26904
[9] Sjölin, P., Regularity and integrability of spherical means, Monatsh. math., 96, 4, 277-291, (1983)
[10] Stein, E.; Weiss, G., An introduction to Fourier analysis on Euclidean spaces, (1971), Princeton Univ. Press
[11] Tao, T., On the maximal Bochner-Bochner-Riesz conjecture in the plane for \(p < 2\), Trans. amer. math. soc., 354, 5, 1947-1959, (2002) · Zbl 0992.42003
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.