Weighted norm inequalities for the bilinear maximal operator on variable Lebesgue spaces

• Autores: David Cruz-Uribe , O. M. Guzmán
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 64, Nº 2, 2020, págs. 453-498
• Idioma: inglés
• DOI: 10.5565/publmat6422004
• Enlaces
  • Texto completo  1    2  
• Resumen

  • We extend the theory of weighted norm inequalities on variable Lebesgue spaces to the case of bilinear operators. We introduce a bilinear version of the variable Ap(·) condition and show that it is necessary and sufficient for the bilinear maximal operator to satisfy a weighted norm inequality. Our work generalizes the linear results of the first author, Fiorenza, and Neugebauer [7] in the variable Lebesgue spaces and the bilinear results of Lerner et al. [22] in the classical Lebesgue spaces. As an application we prove weighted norm inequalities for bilinear singular integral operators in the variable Lebesgue spaces.

• Referencias bibliográficas
  • E. I. Berezhnoi, Two-weighted estimations for the Hardy–Littlewood maximal function in ideal Banach spaces, Proc. Amer. Math. Soc. 127(1)...
  • C. Capone, D. Cruz-Uribe, SFO, and A. Fiorenza, The fractional maximal operator and fractional integrals on variable Lp spaces, Rev. Mat....
  • D. Cruz-Uribe, Two weight inequalities for fractional integral operators and commutators, in: “Advanced Courses of Mathematical Analysis VI”,...
  • D. Cruz-Uribe, L. Diening, and P. Hast ¨ o¨, The maximal operator on weighted variable Lebesgue spaces, Fract. Calc. Appl. Anal. 14(3) (2011),...
  • D. V. Cruz-Uribe and A. Fiorenza, “Variable Lebesgue Spaces. Foundations and Harmonic Analysis”, Applied and Numerical Harmonic Analysis,...
  • D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, The maximal function on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 28(1) (2003), 223–238.
  • D. Cruz-Uribe, SFO, A. Fiorenza, and C. J. Neugebauer, Weighted norm inequalities for the maximal operator on variable Lebesgue spaces, J....
  • D. Cruz-Uribe, J. M. Martell, and C. Perez ´ , Extrapolation from A∞ weights and applications, J. Funct. Anal. 213(2) (2004), 412–439. DOI:...
  • D. V. Cruz-Uribe, J. M. Martell, and C. Perez ´ , “Weights, Extrapolation and the Theory of Rubio de Francia”, Operator Theory: Advances and...
  • D. Cruz-Uribe, OFS, K. Moen, and H. V. Nguyen, The boundedness of multilinear Calder´on–Zygmund operators on weighted and variable Hardy spaces,...
  • D. Cruz-Uribe, OFS and V. Naibo, Kato–Ponce inequalities on weighted and variable Lebesgue spaces, Differential Integral Equations 29(9–10)...
  • D. Cruz-Uribe, SFO and L.-A. D. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63(2) (2014), 447–493. DOI: 10.1512/iumj.2014.63.5232.
  • D. Cruz-Uribe, SFO and L.-A. D. Wang, Extrapolation and weighted norm inequalities in the variable Lebesgue spaces, Trans. Amer. Math. Soc....
  • W. Damian, A. K. Lerner, and C. P ´ erez ´ , Sharp weighted bounds for multilinear maximal functions and Calder´on–Zygmund operators, J. Fourier...
  • L. Diening, Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129(8) (2005), 657–700. DOI: 10.1016/j....
  • L. Diening, P. Harjulehto, P. Hast ¨ o, and M. R ¨ u ◦ ˇzicka ˇ , “Lebesgue and Sobolev Spaces with Variable Exponents”, Lecture Notes in...
  • L. Diening and P. Hast ¨ o¨, Muckenhoupt weights in variable exponent spaces, Unpublished manuscript.
  • J. Garc´ıa-Cuerva and J. L. Rubio de Francia, “Weighted Norm Inequalities and Related Topics”, North-Holland Mathematics Studies 116, Notas...
  • L. Grafakos, “Classical Fourier Analysis”, Second edition, Graduate Texts in Mathematics 249, Springer, New York, 2008. DOI: 10.1007/978-0-387-09432-8.
  • J.-L. Journe´, “Calder´on–Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calder´on”, Lecture Notes in Mathematics...
  • V. Kokilashvili, M. Masty lo, and A. Meskhi, The multisublinear maximal type operators in Banach function lattices, J. Math. Anal. Appl. 421(1)...
  • ] A. K. Lerner, S. Ombrosi, C. Perez, R. H. Torres, and R. Trujillo-Gonz ´ a- ´ lez, New maximal functions and multiple weights for the multilinear...
  • B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. DOI: 10.2307/1995882.