Skip to main content
SpringerLink
Log in

The Rate of Convergence on Fractional Power Dissipative Operator on Compact Manifolds

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

On a compact connected manifold M, we concern the fractional power dissipative operator e−t{ℒ{α, and obtain the almost-everywhere convergence rate (as t → 0+) of e−t{ℒ{α (f) when f is in some Sobolev type Hardy spaces. The main result is a non-trivial extension of a recent result on ℝn by Cao and Wang in [2].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. P. Calderón, A. Torchinsky, Parabolic maximal functions associated with a distribution. II. Adv. Math. 24, No 2 (1977), 101–171.

    Article  MathSciNet  Google Scholar 

  2. Z. Cao, M. Wang, The rate of convergence on fractional power dissipative operator on some Sobolev type spaces. To appear in: Appl. Math. J. Chinese Univ. Ser. B.

  3. J. Chen, Q. Deng, Y. Ding, D. Fan, Estimates on fractional power dissipative equations in function spaces. Nonlinear Anal. 75, No 5 (2012), 2959–2974.

    Article  MathSciNet  Google Scholar 

  4. J. Chen, D. Fan, F. Zhao, On the rate of almost everywhere convergence of combinations and multivariate averages. Potential Anal. 51, No 3 (2019), 397–423.

    Article  MathSciNet  Google Scholar 

  5. X. Chen, D. Fan, Convergence of a wave operator on compact manifolds. To appear.

  6. L. Colzani, Riesz means of eigenfunction expansions of elliptic differential operators on compact manifolds. Rend. Sem. Mat. Fis. Milano 58 (1988), 149–167.

    Article  MathSciNet  Google Scholar 

  7. L. Colzani, G. Travaglini, Hardy-Lorentz spaces and expansions in eigenfunctions of the Laplace-Beltrami operator on compact manifolds. Colloq. Math. 58, No 2 (1990), 305–315.

    Article  MathSciNet  Google Scholar 

  8. D. Fan, Z. Wu, Transference of maximal multipliers on Hardy spaces. Proc. Amer. Math. Soc. 123, No 10 (1995), 3169–3174.

    Article  MathSciNet  Google Scholar 

  9. L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Springer-Verlag, Berlin (1983).

    MATH  Google Scholar 

  10. M. Marias, Lp-boundedness of oscillating spectral multipliers on Riemannian manifolds. Ann. Math. Blaise Pascal 10, No 1 (2003), 133–160.

    Article  MathSciNet  Google Scholar 

  11. C. Miao, B. Yuan, B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Anal. 68, No 3 (2008), 461–484.

    Article  MathSciNet  Google Scholar 

  12. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton, NJ (1993).

    MATH  Google Scholar 

  13. E. Terraneo, Non-uniqueness for a critical non-linear heat equation. Comm. Partial Differential Equations 27, No 1-2 (2002), 185–218.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, P.R. China

    Yali Pan & Junyan Zhao

  2. Department of Mathematics, School of Information Huaibei Normal University, Huaibei, 235000, P.R. China

    Yali Pan

  3. Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201, USA

    Dashan Fan

Authors
  1. Yali Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dashan Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Junyan Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junyan Zhao.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pan, Y., Fan, D. & Zhao, J. The Rate of Convergence on Fractional Power Dissipative Operator on Compact Manifolds. Fract Calc Appl Anal 24, 1130–1159 (2021). https://doi.org/10.1515/fca-2021-0049

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2021-0049

MSC 2010

Key Words and Phrases

Search

Navigation