Pointwise convergence of fractional powers of Hermite type operators

• Guillermo Flores ^[1] ; Gustavo Garrigós ^[2] ; Teresa Signes ^[2] ; Beatriz Viviani ^[3]
  1. [1] Universidad Nacional de Córdoba

     Universidad Nacional de Córdoba

     Argentina

     
  2. [2] Universidad de Murcia

     Universidad de Murcia

     Murcia, España

     
  3. [3] Universidad Nacional del Litoral

     Universidad Nacional del Litoral

     Argentina

     

  Mostrar afiliaciones +
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 66, Nº. 1, 2023, págs. 187-205
• Idioma: inglés
• DOI: 10.33044/revuma.4357
• Enlaces
  • Texto completo (pdf)
• Resumen

  • When L is the Hermite or the Ornstein–Uhlenbeck operator, we find minimal integrability and smoothness conditions on a function f so that the fractional power Lσf(x0) is well-defined at a given point x0. We illustrate the optimality of the conditions with various examples. Finally, we obtain similar results for the fractional operators (−∆ + R) σ, with R > 0.

• Referencias bibliográficas
  • I. Abu-Falahah and J. L. Torrea, Hermite function expansions versus Hermite polynomial expansions, Glasg. Math. J. 48 no. 2 (2006), 203–215....
  • L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 no. 8 (2007),...
  • I. Cardoso, On the pointwise convergence to initial data of heat and Poisson problems for the Bessel operator, J. Evol. Equ. 17 no. 3 (2017),...
  • G. Flores, G. Garrigós, T. Signes, and B. Viviani, Pointwise convergence of fractional powers of Laguerre and Bessel operators, Work in progress.
  • G. Flores and B. Viviani, A Calderón theorem for the Poisson semigroups associated with the Ornstein-Uhlenbeck and Hermite operators, Math....
  • G. Garrigós, A weak 2-weight problem for the Poisson-Hermite semigroup, in Advanced Courses of Mathematical Analysis VI (Málaga, 2014), World...
  • G. Garrigós, S. Hartzstein, T. Signes, and B. Viviani, A.e. convergence and 2-weight inequalities for Poisson-Laguerre semigroups, Ann. Mat....
  • G. Garrigós, S. Hartzstein, T. Signes, J. L. Torrea, and B. Viviani, Pointwise convergence to initial data of heat and Laplace equations,...
  • M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal. 20 no. 1 (2017), 7–51. DOI MR Zbl
  • N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Englewood Cliffs, NJ, 1965. MR Zbl
  • E. H. Lieb and M. Loss, Analysis, second ed., Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001. DOI...
  • L. Roncal and P. R. Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math. 18 no. 3 (2016), 1550033, 26 pp. DOI MR Zbl
  • W. Rudin, Functional Analysis, second ed., International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991. MR Zbl
  • P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations...
  • P. R. Stinga and J. L. Torrea, Regularity theory for the fractional harmonic oscillator, J. Funct. Anal. 260 no. 10 (2011), 3097–3131. DOI...
  • S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Mathematical Notes 42, Princeton University Press, Princeton, NJ, 1993. MR Zbl
  • K. Yosida, Functional Analysis, sixth ed., Grundlehren der Mathematischen Wissenschaften 123, Springer-Verlag, Berlin-New York, 1980. MR...