Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator T satisfies a bound like ∥ T ∥ Lp(w) ≤ c [w] β A p w ε A p , then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted L p norm ∥ T ∥ L p ( R n ) as p goes to 1 and +∞. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximaltype, Caldeŕon-Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner- Riesz multipliers.We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases. © 2015 International Press.

Registro:

Documento: Artículo
Título:Optimal exponents in weighted estimates without examples
Autor:Luque, T.; Pérez, C.; Rela, E.
Filiación:School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Department of Mathematics, University of Basque Country UPV/EHU, Leioa and IKERBASQUE, P.O. Box 644, Bilbao, 48080, Spain
Departamento de Matemática, Facultad de Ciencias Exactas Y Naturales, Universidad de Buenos Aires, Pabellón I, Buenos Aires, Capital Federal, 1428, Argentina
Año:2015
Volumen:22
Número:1
Página de inicio:183
Página de fin:201
DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n1.a10
Título revista:Mathematical Research Letters
Título revista abreviado:Math. Res. Lett.
ISSN:10732780
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10732780_v22_n1_p183_Luque

Referencias:

  • Buckley, S.M., Estimates for operator norms on weighted spaces and reverse Jensen inequalities (1993) Trans. Amer. Math. Soc., 340 (1), pp. 253-272
  • Chung, D., Pereyra, M.C., Pérez, C., Sharp bounds for general commutators on weighted Lebesgue spaces (2012) Trans. Amer. Math. Soc., 364 (3), pp. 1163-1177
  • Coifman, R.R., Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals (1974) Studia Math, 51, pp. 241-250
  • Cruz-Uribe, D., Martell, J.M., Pérez, C., Sharp weighted estimates for classical operators (2012) Adv. Math, 229, pp. 408-441
  • Cruz-Uribe, D.V., Martell, J.M., Pérez, C., Weights, extrapolation and the theory of Rubio de Francia (2011) Operator Theory: Advances and Applications, 215. , Birkhäuser/Springer Basel AG, Basel
  • Duoandikoetxea, J., Extrapolation of weights revisited: New proofs and sharp bounds (2011) J. Funct. Anal., 260 (6), pp. 1886-1901
  • Duoandikoetxea, J., Martín-Reyes, F., Ombrosi, S., Calderón weights as Muckenhoupt weights (2013) Indiana Univ. Math. J., 62 (3), pp. 891-910
  • Fefferman, R., Pipher, J., Multiparameter operators and sharp weighted inequalities (1997) Amer. J. Math., 119 (2), pp. 337-369
  • Grafakos, L., Modern fourier analysis (2009) Graduate Texts in Mathematics, 250. , Springer, New York, 2nd ed., ISBN 978-0-387-09433-5
  • Hunt, R., Muckenhoupt, B., Wheeden, R., Weighted norm inequalities for the conjugate function and Hilbert transform (1973) Trans. Amer. Math. Soc., 176, pp. 227-251
  • Hytönen, T., The sharp weighted bound for general Calderón-Zygmund operators (2012) Ann. of Math., 175 (3), pp. 1476-1506
  • Hytönen, T., Pérez, C., Sharp weighted bounds involving A∞ (2013) Anal. PDE, 6 (4), pp. 777-818
  • Hytönen, T., Pérez, C., Rela, E., Sharp reverse hölder property for A∞ weights on spaces of homogeneous type (2012) J. Funct. Anal., 263 (12), pp. 3883-3899
  • Lacey, M.T., Moen, K., Pérez, C., Torres, R.H., Sharp weighted bounds for fractional integral operators (2010) J. Funct. Anal., 259 (5), pp. 1073-1097
  • Li, K., Sun, W., Sharp bound of the maximal Bochner-Riesz operator in weighted Lebesgue spaces (2012) J. Math. Anal. Appl., 395 (1), pp. 385-392
  • Li, K., Sun, W., Corrigendum to 'Sharp bound of the maximal Bochner-Riesz operator in weighted Lebesgue space' (2013) J. Math. Anal. Appl., 405 (2), p. 746. , J. Math. Anal. Appl. 395 (2012), 385-392
  • Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function (1972) Trans. Amer. Math. Soc., 165, pp. 207-226
  • Muckenhoupt, B., Wheeden, R., Weighted norm inequalities for fractional integrals (1974) Trans. Amer. Math. Soc., 192, pp. 261-274
  • Pérez, C., A course on singular integrals and weights, in (2013) Advanced Courses in Mathematics-CRM Barcelona, 229, pp. 303-321. , Birkhäuser OT series
  • Pérez, C., Weighted norm inequalities for general maximal operators (1991) Publ. Mat., 35 (1), pp. 169-186. , Conference on Mathematical Analysis (El Escorial, 1989)
  • Pérez, C., Endpoint estimates for commutators of singular integral operators (1995) J. Funct. Anal., 128 (1), pp. 163-185
  • Pérez, C., On sufficient conditions for the boundedness of the Hardy- Littlewood maximal operator between weighted Lp-spaces with different weights (1995) Proc. London Math. Soc., 71 (1), pp. 135-157. , 3
  • Pérez, C., Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function (1997) J. Fourier Anal. Appl., 3 (6), pp. 743-756
  • Petermichl, S., The sharp weighted bound for the Riesz transforms (2008) Proc. Amer. Math. Soc., 136 (4), pp. 1237-1249
  • Stein, E.M., (1993) Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, 43. , Princeton Mathematical Series, Princeton University Press, Princeton, NJ. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III

Citas:

---------- APA ----------
Luque, T., Pérez, C. & Rela, E. (2015) . Optimal exponents in weighted estimates without examples. Mathematical Research Letters, 22(1), 183-201.
http://dx.doi.org/10.4310/MRL.2015.v22.n1.a10
---------- CHICAGO ----------
Luque, T., Pérez, C., Rela, E. "Optimal exponents in weighted estimates without examples" . Mathematical Research Letters 22, no. 1 (2015) : 183-201.
http://dx.doi.org/10.4310/MRL.2015.v22.n1.a10
---------- MLA ----------
Luque, T., Pérez, C., Rela, E. "Optimal exponents in weighted estimates without examples" . Mathematical Research Letters, vol. 22, no. 1, 2015, pp. 183-201.
http://dx.doi.org/10.4310/MRL.2015.v22.n1.a10
---------- VANCOUVER ----------
Luque, T., Pérez, C., Rela, E. Optimal exponents in weighted estimates without examples. Math. Res. Lett. 2015;22(1):183-201.
http://dx.doi.org/10.4310/MRL.2015.v22.n1.a10