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Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes

  • Deniz Karlı EMAIL logo
Veröffentlicht/Copyright: 9. Juni 2018
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 21 Heft 2

Abstract

In this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between fractional derivatives and stable processes and prove a version of Mikhlin theorem under a condition given in terms of the infinitesimal generator of symmetric stable process. The classical Mikhlin theorem is shown to be a corollary of this new generalized version in this paper.

MSC 2010: Primary 60J45; Secondary 42A61, 60G52, 26A33
Key Words and Phrases: fractional derivatives; generator form; Mikhlin multiplier theorem; stable process; bounded operator; stochastic process

References

[1] D. Applebaum, Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004).10.1017/CBO9780511755323Suche in Google Scholar

[2] R.F. Bass, A probabilistic approach to the boundedness of singular integral operators. Séminaire de probabilités24 (1990), 15–40.10.1007/BFb0083754Suche in Google Scholar

[3] J.F. Colombeau, Elementary Introduction to New Generalized Functions. North Holland, Amsterdam (1985).Suche in Google Scholar

[4] R. Gorenflo, Y. Luchko, M. Stojanovic, Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density. Fract. Calc. Appl. Anal. 16, No 2 (2013), 297–316; 10.2478/s13540-013-0019-6; https://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.Suche in Google Scholar

[5] L. Grafakos, Classical and Modern Fourier Analysis. Prentice Hall, New Jersey (2004).Suche in Google Scholar

[6] M.E. Hernandez-Hernandez, V.N. Kolokoltsov, On the solution of two-sided fractional ordinary differential equations of Caputo type. Fract. Calc. Appl. Anal. 19, No 6 (2016), 1393–1413; 10.1515/fca-2016-0072; https://www.degruyter.com/view/j/fca.2016.19.issue-6/issue-files/fca.2016.19.issue-6.xml.Suche in Google Scholar

[7] K. Hu, N. Jacob, C. Yuan, Existence and uniqueness for a class of stochastic time fractional space pseudo-differential equations. Fract. Calc. Appl. Anal. 19, No 1 (2016), 56–68; 10.1515/fca-2016-0004; https://www.degruyter.com/view/j/fca.2016.19.issue-1/issue-files/fca.2016.19.issue-1.xml.Suche in Google Scholar

[8] D. Karli, Harnack inequality and regularity for a product of symmetric stable process and Brownian motion. Potential Analysis38, No 1 (2013), 95–117; 10.1007/s11118-011-9265-6.Suche in Google Scholar

[9] D. Karli, An extension of a boundedness result for singular integral operators. Colloquium Mathematicum145 (2016), 15–33; 10.4064/cm6722-1-2016.Suche in Google Scholar

[10] D. Karli, A multipler related to symmetric stable processes. Hacettepe J. of Mathematics and Statistics46, No 2 (2017), 217–228; 10.15672/HJMS.20164517212.Suche in Google Scholar

[11] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Application of Fractional Differential Equations. Mathematics Studies 204, Elsevier - North-Holland, Amsterdam (2006).Suche in Google Scholar

[12] V.S. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics 301, Longman, Harlow (1994).Suche in Google Scholar

[13] V.N. Kolokoltsov, Markov Processes, Semigroups and Generators. De Gruyter, New York (2011).10.1515/9783110250114Suche in Google Scholar

[14] V.N. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions. Proc. London Math. Society80 (2000), 725–768.10.1112/S0024611500012314Suche in Google Scholar

[15] M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus. De Gruyter, Berlin (2011).10.1515/9783110258165Suche in Google Scholar

[16] G. Pagnini, The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes. Fract. Calc. Appl. Anal. 16, No 2 (2013), 436–453; 10.2478/s13540-013-0027-6; https://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.Suche in Google Scholar

[17] G. Pagnini, P. Paradisi, A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation, Fract. Calc. Appl. Anal. 19, No 2 (2016), 408–440; 10.1515/fca-2016-0022; https://www.degruyter.com/view/j/fca.2016.19.issue-2/issue-files/fca.2016.19.issue-2.xml.Suche in Google Scholar

[18] S.M. Ross, Stochastic Processes. Wiley, New Jersey (1996).Suche in Google Scholar

[19] S. Samko, A note on Riesz fractional integrals in the limiting case a(x)p(x) = n, Fract. Calc. Appl. Anal. 16, No 2 (2013), 370–377; 10.2478/s13540-013-0023-x; https://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.Suche in Google Scholar

[20] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications. Gordon and Breach Science Publishers S.T., Amsterdam (1993).Suche in Google Scholar

[21] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, New Jersey (1993).Suche in Google Scholar

[22] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, New Jersey (1970).10.1515/9781400883882Suche in Google Scholar

[23] E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton University Press, New Jersey (1970).10.1515/9781400881871Suche in Google Scholar

[24] C.A. Tudor, Analysis of Variations for Self-Similar Processes (A Stochastic Calculus Approach). Springer, Berlin (2013).10.1007/978-3-319-00936-0Suche in Google Scholar

[25] C.A. Tudor, Recent developments on stochastic heat equation with additive fractional-colored noise, Fract. Calc. Appl. Anal. 17, No 1 (2014), 224–246; 10.2478/s13540-014-0164-6; https://www.degruyter.com/view/j/fca.2014.17.issue-1/issue-files/fca.2014.17.issue-1.xml.Suche in Google Scholar

Received: 2017-2-3
Published Online: 2018-6-9
Published in Print: 2018-4-25

© 2018 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 21–2–2018)
  4. Research Paper
  5. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients
  6. Research Paper
  7. Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms
  8. Research Paper
  9. Fractional generalizations of Zakai equation and some solution methods
  10. Research Paper
  11. Stability analysis of impulsive fractional difference equations
  12. Research Paper
  13. Mellin convolutions, statistical distributions and fractional calculus
  14. Research Paper
  15. Fractional wavelet frames in L2(ℝ)
  16. Research Paper
  17. Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions
  18. Research Paper
  19. Two point fractional boundary value problems with a fractional boundary condition
  20. Research Paper
  21. Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
  22. Research Paper
  23. Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes
  24. Research Paper
  25. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications
  26. Research Paper
  27. Asymptotic behavior of mild solutions for nonlinear fractional difference equations
  28. Research Paper
  29. Positive solutions to nonlinear systems involving fully nonlinear fractional operators
https://doi.org/10.1515/fca-2018-0027
Schlagwörter für diesen Artikel
fractional derivatives; generator form; Mikhlin multiplier theorem; stable process; bounded operator; stochastic process

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 21–2–2018)
  4. Research Paper
  5. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients
  6. Research Paper
  7. Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms
  8. Research Paper
  9. Fractional generalizations of Zakai equation and some solution methods
  10. Research Paper
  11. Stability analysis of impulsive fractional difference equations
  12. Research Paper
  13. Mellin convolutions, statistical distributions and fractional calculus
  14. Research Paper
  15. Fractional wavelet frames in L2(ℝ)
  16. Research Paper
  17. Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions
  18. Research Paper
  19. Two point fractional boundary value problems with a fractional boundary condition
  20. Research Paper
  21. Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
  22. Research Paper
  23. Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes
  24. Research Paper
  25. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications
  26. Research Paper
  27. Asymptotic behavior of mild solutions for nonlinear fractional difference equations
  28. Research Paper
  29. Positive solutions to nonlinear systems involving fully nonlinear fractional operators
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