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Parabolic Equations Involving Bessel Operators and Singular Integrals

  • J. J. Betancor
  • M. De León-Contreras
Article

Abstract

In this paper we consider the evolution equation \(\partial _t u=\Delta _\mu u+f\) and the corresponding Cauchy problem, where \(\Delta _\mu \) represents the Bessel operator \(\partial _x^2+(\frac{1}{4}-\mu ^2)x^{-2}\), for every \(\mu >-1\). We establish weighted and mixed weighted Sobolev type inequalities for solutions of Bessel parabolic equations. We use singular integrals techniques in a parabolic setting.

Keywords

Parabolic equations Riesz transforms Weighted inequalities Bessel operators 

Mathematics Subject Classification

Primary 42B25 42B15 Secondary 42B20 46B20 46E40 

Notes

Acknowledgements

The authors thank Professor José Luis Torrea (UAM, Madrid) for posing the problems studied in this paper and for reading a first version of the manuscript. His comments have allowed us to improve Theorem 1.5 and its proof.

References

  1. 1.
    Betancor, J.J., Buracewski, D., Fariña, J.C., Martínez, M.T., Torrea, J.L.: Riesz transform related to Bessel operators. Proc. R. Soc. Edinb. Sect. A 137(4), 701–725 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Betancor, J.J., Castro, A.J., Curbelo, J.: Spectral multipliers for multidimensional Bessel operators. J. Fourier Anal. Appl. 17, 932–975 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Betancor, J.J., Castro, A.J., Stinga, P.R.: The fractional Bessel equation in Hölder spaces. J. Approx. Theory 184, 55–99 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Betancor, J.J., Dziubanski, J., Torrea, J.L.: On Hardy spaces associated with Bessel operators. J. Anal. Math. 107, 195–219 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Betancor, J.J., Harboure, E., Nowak, A., Viviani, B.: Mapping properties of functional operators in harmonic analysis related to Bessel operators. Studia Math. 197, 101–140 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Betancor, J.J., Fariña, J.C., Martínez, M.T., Rodríguez-Mesa, L.: Higher order Riesz transforms associated to Bessel operators. Ark. Mat. 56, 219–250 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Betancor, J.J., Fariña, J.C., Martínez, M.T., Torrea, J.L.: Riesz transform and \(g\)-function associated with Bessel operators and their appropriate Banach spaces. Israel J. Math. 157, 259–282 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Betancor, J.J., Fariña, J.C., Rodríguez-Mesa, L., Testoni, R., Torrea, J.L.: A choice of Sobolev spaces associated with ultraspherical expansions. Pub. Math. 54(1), 221–242 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Betancor, J.J., Stempak, K.: Relating multipliers and transplantation for Fourier–Bessel expansions and Hankel transform. Tohoku Math. J. 53(1), 109–129 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Calderón, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Calderón, A.P., Zygmund, A.: Singular integral operators and differential equations. Am. J. Math. 79, 901–921 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Coulhon, T., Duong, X.T.: Maximal regularity and kernel bounds: observations on a theorem by Hieber and Pruss. Adv. Part. Differ. Equ. 5(1–3), 343–368 (2000)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Carbonaro, A., Matafume, G., Spina, C.: Parabolic Schrödinger operators. J. Math. Anal. Appl. 343, 965–974 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Castro, A.J., Nystrom, K., Sande, O.: Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients. Calc. Var. Part. Diff. Equ. 55, 124 (2016).  https://doi.org/10.1007/s00526-016-1058-8 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Castro, A.J., Rodríguez-López, S., Staubach, W.: \(L^2\)-solvability of the Dirichlet, Neumann and regularity problems for parabolic equations with time-independent Hölder-continuous coefficients. Trans. Am. Math. Soc. 370(1), 265–319 (2018)CrossRefzbMATHGoogle Scholar
  16. 16.
    Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative Sur Certains Espaces Homogenes, Lecture Notes in Math., 242. Springer, Berlin (1971)CrossRefGoogle Scholar
  17. 17.
    Duong, X.T., Li, J., Mao, S., Wu, H., Yang, D.: Compactness of Riesz transforms commutator associated with Bessel operators. J. Anal. Math. (to appear)Google Scholar
  18. 18.
    Duong, X.T., Li, J., Wick, B.D., Yang, D.: Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting. Indiana Univ. J. Math. 66(4), 1081–1106 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fabes, E.B.: Singular integrals and partial differential equations of parabolic type. Studia Math. 28, 81–131 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fabes, E.B., Sadosky, C.: Pointwise convergence for parabolic singular integrals. Studia Math. 26, 225–232 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gao, N., Jiang, Y.: \(L^p\) estimates for parabolic Schrödinger operator with certain potentials. J. Math. Anal. Appl. 310, 128–143 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hieber, M., Pruss, J.: Heat kernels and maximal \(L^p -L^q\) estimates for parabolic evolution equations. Commun. Part. Differ. Equ. 22, 1647–1669 (1997)CrossRefzbMATHGoogle Scholar
  23. 23.
    Harboure, E., Segovia, C., Torrea, J.L., Viviani, B.: Power weighted \(L^p\)-inequalities for Laguerre–Riesz transforms. Arkiv Mat. 46, 285–313 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jones, B.F.: Singular integrals and parabolic equations. Bull. Am. Math. Soc. 69(4), 501–503 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jones, B.F.: A class of singular integrals. Am. J. Math. 86, 441–462 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kim, I., Lim, S., Kim, K.-H.: An \(L_q(L_p)\)-theory for parabolic pseudodifferential equations: Calderón–Zygmund approach. Potential Anal. 45, 463 (2016).  https://doi.org/10.1007/s11118-016-9552-3 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Krylov, N.V.: On the Calderón–Zygmund theorem with applications to parabolic equations. Algebra i Analiz 13, 1–25 (2001). in Russian: English translation in St. Petersburg Math. J. 13, 509–526 (2002)Google Scholar
  28. 28.
    Krylov, N.V.: The Calderón–Zygmund theorem and parabolic equations in \(L_p(\mathbb{R}, C^{2+d})\)-space. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), I, 799–820 (2002)Google Scholar
  29. 29.
    Lebedev, N.N.: Special Functions and Their Applications, Selected Russian Publications in the Mathematical Sciences. Prentice-Hall Inc., Englewood Cliffs (1965)Google Scholar
  30. 30.
    Liu, Y., Huang, J., Dong, J.: \(L^p\) estimates for higher-order parabolic Schrödinger operators with certain nonnegative potentials. Bull. Malays. Math. Sci. Doc. 37(2), 153–164 (2014)zbMATHGoogle Scholar
  31. 31.
    Macías, R., Segovia, C., Torrea, J.L.: Heat diffusion maximal operators for Laguerre semigroups with negative parameters. J. Funct. Anal. 229(2), 300–316 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Macías, R., Segovia, C., Torrea, J.L.: Weighted norm estimates for the maximal operator of the Laguerre functions heat diffusion semigroup. Studia Math. 172(2), 149–167 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Miao, C., Zhang, J., Zheng, J.: Maximal estimates for Schrödinger equations with inverse-square potential. Pac. J. Math. 273(1), 1–19 (2015)CrossRefzbMATHGoogle Scholar
  34. 34.
    Monniaux, S.: Maximal regularity and applications to PDE’s, Lectures given at the TU Berlin in May (2009)Google Scholar
  35. 35.
    Muckenhoupt, B., Stein, E.: Classical expansions and their relationship to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)CrossRefzbMATHGoogle Scholar
  36. 36.
    Nowak, A., Sjögren, P.: The multidimensional pencil phenomenon for Laguerre heat-diffusion maximal operators. Math. Ann. 344(1), 231–248 (2009)CrossRefzbMATHGoogle Scholar
  37. 37.
    Nowak, A., Stempak, K.: Weighted estimates for the Hankel transform transplantation operator. Tohoku Math. J. 58(2), 277–301 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Nystrom, K.: \(L^2\) solvability of boundary value problems for divergence form parabolic equations with complex coefficients. J. Differ. Equ. 262(3), 2808–2939 (2017)CrossRefzbMATHGoogle Scholar
  39. 39.
    Ouhabaz, E.M., Spina, C.: Riesz transforms of some parabolic operators. In: AMSI International Conference of Harmonic Analysis and Applications, vol. 45, pp. 115–123. Proc. Centre Math Appl. Austral. Nat. Univ., Austral. Nat. Univ., Canberra (2013)Google Scholar
  40. 40.
    Ping, L., Stinga, P.R., Torrea, J.L.: On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Commun. Pure Appl. Anal. 16, 855–882 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rubio de Francia, J.L., Ruiz, F.J., Torrea, J.L.: Calderón Zygmund theory for operator valued kernels. Adv. Math. 62(1), 7–48 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ruiz, F.J., Torrea, J.L.: Parabolic differential equations and vector-valued Fourier analysis. Colloq. Math. 58, 61–75 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ruiz, F.J., Torrea, J.L.: Vector-valued Calderón–Zygmund theory and Carleson measures on spaces of homogeneous nature. Studia Math. 88, 221–243 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Sadosky, C.: A note on parabolic fractional and singular integrals. Studia Math. 26, 328–335 (1966)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood–Paley Theory, Annals of Mathematical Studies. Princeton Univ. Press, Princeton (1970)Google Scholar
  46. 46.
    Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals, 3rd edn. Chelsea Publishing Co., New York (1986)zbMATHGoogle Scholar
  47. 47.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1958)Google Scholar
  48. 48.
    Zemanian, A.H.: Generalized Integral Transformations. Interscience Publishers, New York (1968)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversidad de La LagunaSan Cristóbal de La LagunaSpain
  2. 2.Departamento de Matemáticas Facultad de CienciasUniversidad Autónoma de MadridMadridSpain

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