Revista de la
Unión Matemática Argentina
Hermite Besov and Triebel–Lizorkin spaces and applications
Fu Ken Ly and Virginia Naibo
Volume 66, no. 1 (2023), pp. 243–263    

https://doi.org/10.33044/revuma.4360

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Abstract

We present an overview of Besov and Triebel–Lizorkin spaces in the Hermite setting and applications on boundedness properties of Hermite pseudo-multipliers and fractional Leibniz rules in such spaces. We also give a new weighted estimate for Hermite multipliers for weights related to Hermite operators.

References

  1. S. Bagchi and R. Garg, On $L^2$-boundedness of pseudo-multipliers associated to the Grushin operator, 2023. arXiv 2111.10098 [math.AP].
  2. S. Bagchi and S. Thangavelu, On Hermite pseudo-multipliers, J. Funct. Anal. 268 no. 1 (2015), 140–170.  DOI  MR  Zbl
  3. B. Bongioanni, A. Cabral, and E. Harboure, Extrapolation for classes of weights related to a family of operators and applications, Potential Anal. 38 no. 4 (2013), 1207–1232.  DOI  MR  Zbl
  4. B. Bongioanni, A. Cabral, and E. Harboure, Schrödinger type singular integrals: weighted estimates for $p=1$, Math. Nachr. 289 no. 11-12 (2016), 1341–1369.  DOI  MR  Zbl
  5. B. Bongioanni, E. Harboure, and O. Salinas, Classes of weights related to Schrödinger operators, J. Math. Anal. Appl. 373 no. 2 (2011), 563–579.  DOI  MR  Zbl
  6. B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian Acad. Sci. Math. Sci. 116 no. 3 (2006), 337–360.  DOI  MR  Zbl
  7. J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 no. 2 (1981), 209–246.  DOI  MR  Zbl
  8. H.-Q. Bui, T. A. Bui, and X. T. Duong, Weighted Besov and Triebel-Lizorkin spaces associated with operators and applications, Forum Math. Sigma 8 (2020), Paper No. e11, 95 pp.  DOI  MR  Zbl
  9. T. A. Bui, Hermite pseudo-multipliers on new Besov and Triebel-Lizorkin spaces, J. Approx. Theory 252 (2020), Paper No. 105348, 16 pp.  DOI  MR  Zbl
  10. T. A. Bui, T. Q. Bui, and X. T. Duong, Quantitative weighted estimates for some singular integrals related to critical functions, J. Geom. Anal. 31 no. 10 (2021), 10215–10245.  DOI  MR  Zbl
  11. T. A. Bui, T. Q. Bui, and X. T. Duong, Quantitative estimates for square functions with new class of weights, Potential Anal. 57 no. 4 (2022), 545–569.  DOI  MR  Zbl
  12. T. A. Bui and X. T. Duong, Besov and Triebel-Lizorkin spaces associated to Hermite operators, J. Fourier Anal. Appl. 21 no. 2 (2015), 405–448.  DOI  MR  Zbl
  13. T. A. Bui and X. T. Duong, Higher-order Riesz transforms of Hermite operators on new Besov and Triebel-Lizorkin spaces, Constr. Approx. 53 no. 1 (2021), 85–120.  DOI  MR  Zbl
  14. T. A. Bui, J. Li, and F. K. Ly, Weighted embeddings for function spaces associated with Hermite expansions, J. Approx. Theory 264 (2021), Paper No. 105534, 25 pp.  DOI  MR  Zbl
  15. D. Cardona and M. Ruzhansky, Hörmander condition for pseudo-multipliers associated to the harmonic oscillator, 2018. arXiv 1810.01260 [math.FA].
  16. J. Dziubański, Atomic decomposition of $H^p$ spaces associated with some Schrödinger operators, Indiana Univ. Math. J. 47 no. 1 (1998), 75–98.  DOI  MR  Zbl
  17. J. Epperson, Triebel-Lizorkin spaces for Hermite expansions, Studia Math. 114 no. 1 (1995), 87–103.  DOI  MR  Zbl
  18. J. Epperson, Hermite multipliers and pseudo-multipliers, Proc. Amer. Math. Soc. 124 no. 7 (1996), 2061–2068.  DOI  MR  Zbl
  19. J. Epperson, Hermite and Laguerre wave packet expansions, Studia Math. 126 no. 3 (1997), 199–217.  DOI  MR  Zbl
  20. M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 no. 4 (1985), 777–799.  DOI  MR  Zbl
  21. M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 no. 1 (1990), 34–170.  DOI  MR  Zbl
  22. G. Garrigós, E. Harboure, T. Signes, J. L. Torrea, and B. Viviani, A sharp weighted transplantation theorem for Laguerre function expansions, J. Funct. Anal. 244 no. 1 (2007), 247–276.  DOI  MR  Zbl
  23. A. G. Georgiadis and M. Nielsen, Pseudodifferential operators on spaces of distributions associated with non-negative self-adjoint operators, J. Fourier Anal. Appl. 23 no. 2 (2017), 344–378.  DOI  MR  Zbl
  24. S. Gong, B. Ma, and Z. Fu, A continuous characterization of Triebel-Lizorkin spaces associated with Hermite expansions, J. Funct. Spaces (2015), Art. ID 104897, 11 pp.  DOI  MR  Zbl
  25. E. Harboure, J. L. Torrea, and B. E. Viviani, Riesz transforms for Laguerre expansions, Indiana Univ. Math. J. 55 no. 3 (2006), 999–1014.  DOI  MR  Zbl
  26. J. Huang, Hardy-Sobolev spaces associated with Hermite expansions and interpolation, Nonlinear Anal. 157 (2017), 104–122.  DOI  MR  Zbl
  27. A. K. Lerner, Quantitative weighted estimates for the Littlewood-Paley square function and Marcinkiewicz multipliers, Math. Res. Lett. 26 no. 2 (2019), 537–556.  DOI  MR  Zbl
  28. J. Li, R. Rahm, and B. D. Wick, $A_p$ weights and quantitative estimates in the Schrödinger setting, Math. Z. 293 no. 1-2 (2019), 259–283.  DOI  MR  Zbl
  29. F. K. Ly, On the $L^2$ boundedness of pseudo-multipliers for Hermite expansions, 2022. arXiv 2203.09058 [math.CA].
  30. F. K. Ly and V. Naibo, Pseudo-multipliers and smooth molecules on Hermite Besov and Hermite Triebel-Lizorkin spaces, J. Fourier Anal. Appl. 27 no. 3 (2021), Paper No. 57, 59 pp.  DOI  MR  Zbl
  31. F. K. Ly and V. Naibo, Fractional Leibniz rules associated to bilinear Hermite pseudo-multipliers, Int. Math. Res. Not. IMRN 2023 no. 7 (2023), 5401–5437.  DOI  MR  Zbl
  32. G. Mauceri, The Weyl transform and bounded operators on $L^{p}(\mathbf{R}^{n})$, J. Functional Analysis 39 no. 3 (1980), 408–429.  DOI  MR  Zbl
  33. Y. Meyer, Remarques sur un théorème de J.-M. Bony, Rend. Circ. Mat. Palermo (2), suppl. 1 (1981), 1–20.  MR  Zbl
  34. B. Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), 243–260.  DOI  MR  Zbl
  35. B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 139 (1969), 231–242.  DOI  MR  Zbl
  36. B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92.  DOI  MR  Zbl
  37. P. Petrushev and Y. Xu, Decomposition of spaces of distributions induced by Hermite expansions, J. Fourier Anal. Appl. 14 no. 3 (2008), 372–414.  DOI  MR  Zbl
  38. T. Runst, Pseudodifferential operators of the “exotic” class $L^0_{1,1}$ in spaces of Besov and Triebel-Lizorkin type, Ann. Global Anal. Geom. 3 no. 1 (1985), 13–28.  DOI  MR  Zbl
  39. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993.  MR  Zbl
  40. K. Stempak and J. L. Torrea, Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal. 202 no. 2 (2003), 443–472.  DOI  MR  Zbl
  41. S. Thangavelu, Multipliers for Hermite expansions, Rev. Mat. Iberoamericana 3 no. 1 (1987), 1–24.  DOI  MR  Zbl
  42. S. Thangavelu, On regularity of twisted spherical means and special Hermite expansions, Proc. Indian Acad. Sci. Math. Sci. 103 no. 3 (1993), 303–320.  DOI  MR  Zbl
  43. S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Mathematical Notes 42, Princeton University Press, Princeton, NJ, 1993.  MR  Zbl
  44. R. H. Torres, Continuity properties of pseudodifferential operators of type $1,1$, Comm. Partial Differential Equations 15 no. 9 (1990), 1313–1328.  DOI  MR  Zbl
  45. R. H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 90 no. 442 (1991).  DOI  MR  Zbl
  46. J. Zhang and D. Yang, Quantitative boundedness of Littlewood-Paley functions on weighted Lebesgue spaces in the Schrödinger setting, J. Math. Anal. Appl. 484 no. 2 (2020), 123731, 26 pp.  DOI  MR  Zbl