Multiple vector-valued, mixed norm estimates for Littlewood-Paley square functions

• Benea, Cristina ^[1] ; Muscalu, Camil ^[2]
  1. [1] University of Nantes

     University of Nantes

     Arrondissement de Nantes, Francia

     
  2. [2] Cornell University

     Cornell University

     City of Ithaca, Estados Unidos

     
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 66, Nº 2, 2022, págs. 631-681
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We prove that for any LQ-valued Schwartz function f defined on Rd, one has the multiple vector-valued, mixed-norm estimate kfkLP (LQ) . kSfkLP (LQ) valid for every d-tuple P and every n-tuple Q satisfying 0 < P, Q < ∞ componentwise. Here S := Sd1 ⊗ · · · ⊗ SdN is a tensor product of several Littlewood–Paley square functions Sdj defined on arbitrary Euclidean spaces R dj for 1 ≤ j ≤ N, with the property that d1 + · · · + dN = d. This answers a question that came up implicitly in our recent works [2], [3], [5] and completes in a natural way classical results of Littlewood–Paley theory. The proof is based on the helicoidal method introduced by the authors in the aforementioned papers.

• Referencias bibliográficas
  • C. Benea and F. Bernicot, Conservation de certaines proprietes à travers un contrôle epars d’un opérateur et applications au projecteur de...
  • C. Benea and C. Muscalu, Multiple vector-valued inequalities via the helicoidal method, Anal. PDE 9(8) (2016), 1931–1988. DOI: 10.2140/apde.2016.9.1931.
  • C. Benea and C. Muscalu, Quasi-Banach valued inequalities via the helicoidal method, J. Funct. Anal. 273(4) (2017), 1295–1353. DOI: 10.1016/j.jfa.2017...
  • C. Benea and C. Muscalu, The helicoidal method, in: “Operator Theory: Themes and Variations”, Theta Ser. Adv. Math. 20, Theta, Bucharest,...
  • C. Benea and C. Muscalu, Sparse domination via the helicoidal method, Rev. Mat. Iberoam. 37(6) (2021), 2037–2118. DOI: 10.4171/rmi/1266
  • D. Cruz-Uribe and J. M. Martell, Limited range multilinear extrapolation with applications to the bilinear Hilbert transform, Math. Ann. 371(1-2)...
  • D. Cruz-Uribe, J. M. Martell, and C. Perez, Extrapolation from A∞ weights and applications, J. Funct. Anal. 213(2) (2004), 412–439. DOI: 10.1016/j.jfa.2003.09.002.
  • D. Cruz-Uribe, J. M. Martell, and C. Perez ´ , “Weights, Extrapolation and the Theory of Rubio de Francia”, Operator Theory: Advances and...
  • Y. Ding, Y. Han, G. Lu, and X. Wu, Boundedness of singular integrals on multiparameter weighted Hardy spaces H p w(Rn × Rm), Potential Anal....
  • J. Duoandikoetxea, “Fourier Analysis”, Translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics...
  • M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93(1) (1990), 34–170. DOI: 10.1016/0022-1236(90)90137-A
  • R. F. Gundy and E. M. Stein, Hp theory for the poly-disc, Proc. Nat. Acad. Sci. U.S.A. 76(3) (1979), 1026–1029. DOI: 10.1073/pnas.76.3.1026
  • J. Hart, R. H. Torres, and X. Wu, Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces,...
  • L. Huang, J. Liu, D. Yang, and W. Yuan, Atomic and Littlewood–Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications,...
  • D. S. Kurtz, Classical operators on mixed-normed spaces with product weights, Rocky Mountain J. Math. 37(1) (2007), 269–283. DOI: 10.1216/rmjm/1181069331
  • K. Li, J. M. Martell, and S. Ombrosi, Extrapolation for multilinear Muckenhoupt classes and applications, Adv. Math. 373 (2020), 107286, 43...
  • J. Marcinkiewicz and A. Zygmund, Quelques inégalités pour les opérations linéaires, Fund. Math. 32 (1939), 115–121. DOI: 10.4064/fm-32-1-115-121
  • C. Muscalu, J. Pipher, T. Tao, and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam. 22(3) (2006), 963–976. DOI: 10.4171/RMI/480.
  • C. Muscalu and W. Schlag, “Classical and Multilinear Harmonic Analysis”, Vol II, Cambridge Studies in Advanced Mathematics 138, Cambridge...
  • E. M. Stein, “Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals”, With the assistance of Timothy S. Murphy,...