Annals of Functional Analysis

The rate of almost-everywhere convergence of Bochner–Riesz means on Sobolev spaces

Junyan Zhao and Dashan Fan

Abstract

We investigate the convergence rate of the generalized Bochner–Riesz means SRδ,γ on Lp-Sobolev spaces in the sharp range of δ and p (p2). We give the relation between the smoothness imposed on functions and the rate of almost-everywhere convergence of SRδ,γ. As an application, the corresponding results can be extended to the n-torus Tn by using some transference theorems. Also, we consider the following two generalized Bochner–Riesz multipliers, (1|ξ|γ1)+δ and (1|ξ|γ2)+δ, where γ1, γ2, δ are positive real numbers. We prove that, as the maximal operators of the multiplier operators with respect to the two functions, their L2(|x|β)-boundedness is equivalent for any γ1, γ2 and fixed δ.

Article information

Source
Ann. Funct. Anal., Volume 10, Number 1 (2019), 29-45.

Dates
Received: 3 November 2017
Accepted: 11 February 2018
First available in Project Euclid: 16 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.afa/1547629221

Digital Object Identifier
doi:10.1215/20088752-2018-0006

Mathematical Reviews number (MathSciNet)
MR3899954

Zentralblatt MATH identifier
07045483

Subjects
Primary: 42B15: Multipliers
Secondary: 41A35: Approximation by operators (in particular, by integral operators) 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47B38: Operators on function spaces (general)

Keywords
Bochner–Riesz means Sobolev spaces almost-everywhere convergence saturation of approximation maximal functions Fourier series

Citation

Zhao, Junyan; Fan, Dashan. The rate of almost-everywhere convergence of Bochner–Riesz means on Sobolev spaces. Ann. Funct. Anal. 10 (2019), no. 1, 29--45. doi:10.1215/20088752-2018-0006. https://projecteuclid.org/euclid.afa/1547629221


Export citation

References

  • [1] J. Bourgain, $L^{p}$-estimates for oscillatory integrals in several variables, Geom. Funct. Anal. 1 (1991), no. 4, 321–374.
  • [2] A. Carbery, J. L. Rubio de Francia, and L. Vega, Almost everywhere summability of Fourier integrals, J. Lond. Math. Soc. (2) 38 (1988), no. 3, 513–524.
  • [3] A. Carbery and F. Soria, Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an $L^{2}$-localisation principle, Rev. Mat. Iberoam. 4 (1988), no. 2, 319–337.
  • [4] T. Chen, “Generalized Bochner-Riesz means of Fourier integrals” in Multivariate Approximation Theory, IV (Oberwolfach, 1989), Internat. Ser. Numer. Math. 90, Birkhäuser, Basel, 1989, 87–94.
  • [5] M. Christ, On almost everywhere convergence of Bochner-Riesz means in higher dimensions, Proc. Amer. Math. Soc. 95 (1985), no. 1, 16–20.
  • [6] L. Colzani and S. Volpi, “Pointwise convergence of Bochner-Riesz means in Sobolev spaces” in Trends in Harmonic Analysis, Springer INdAM Ser. 3, Springer, Milan, 2013, 135–146.
  • [7] D. Fan and F. Zhao, Block-Sobolev spaces and the rate of almost everywhere convergence of Bochner-Riesz means, Constr. Approx. 45 (2017), no. 3, 391–405. Erratum, Constr. Approx. 45 (2017), no. 3, 407.
  • [8] D. Fan and F. Zhao, Almost everywhere convergence of Bochner-Riesz means on some Sobolev type spaces, preprint, arXiv:1608.01575v1 [math.FA].
  • [9] R. A. Fefferman, A theory of entropy in Fourier analysis, Adv. Math. 30 (1978), no. 3, 171–201.
  • [10] L. Grafakos, Classical Fourier Analysis, 3rd ed., Grad. Texts in Math. 249, Springer, New York, 2014.
  • [11] L. Grafakos, Modern Fourier Analysis, 3rd ed., Grad. Texts in Math. 250, Springer, New York, 2014.
  • [12] C. E. Kenig and P. A. Tomas, Maximal operators defined by Fourier multipliers, Studia Math. 68 (1980), no. 1, 79–83.
  • [13] S. Lee, Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators, Duke Math. J. 122 (2004), no. 1, 205–232.
  • [14] S. Lee and A. Seeger, On radial Fourier multipliers and almost everywhere convergence, J. Lond. Math. Soc. (2) 91 (2015), no. 1, 105–126.
  • [15] S. Lu, Conjectures and problems on Bochner-Riesz means, Front. Math. China 8 (2013), no. 6, 1237–1251.
  • [16] S. Lu, M. H. Taibleson, and G. Weiss, “On the almost everywhere convergence of Bochner-Riesz means of multiple Fourier series” in Harmonic Analysis (Minneapolis, 1981), Lecture Notes in Math. 908, Springer, Berlin, 1982, 311–318.
  • [17] S. Lu and S. Wang, Spaces generated by smooth blocks, Constr. Approx. 8 (1992), no. 3, 331–341.
  • [18] Z. Shi, X. Nie, D. Wu, and D. Yan, The equivalence of a class of generalized Bochner-Riesz multipliers and its applications, Science in China Series A-Math. 44 (2014), 535–544.
  • [19] E. M. Stein, On limits of sequences of operators, Ann. of Math. (2) 74 (1961), no. 1, 140–170.
  • [20] E. M. Stein, “An $H^{1}$ function with nonsummable Fourier expansion” in Harmonic Analysis (Cortona, 1982), Lecture Notes in Math. 992, Springer, Berlin, 1983, 193–200.
  • [21] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993.
  • [22] E. M. Stein, M. H. Taibleson, and G. Weiss, “Weak type estimates for maximal operators on certain $H^{p}$ classes” in Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), Rend. Circ. Mat. Palermo (2) 1981, suppl. 1, 81–97.
  • [23] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton Univ. Press, Princeton, 1971.
  • [24] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031–1060.
  • [25] R. S. Strichartz, $H^{p}$ Sobolev spaces, Colloq. Math. 60/61 (1990), no. 1, 129–139.
  • [26] T. Tao, On the maximal Bochner-Riesz conjecture in the plane for $p<2$, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1947–1959.