Optimal exponents in weighted estimates without examples
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 ∥...
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_10732780_v22_n1_p183_Luque |
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todo:paper_10732780_v22_n1_p183_Luque2023-10-03T16:02:52Z Optimal exponents in weighted estimates without examples Luque, T. Pérez, C. Rela, E. 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. Fil:Rela, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_10732780_v22_n1_p183_Luque |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
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R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| description |
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. |
| format |
JOUR |
| author |
Luque, T. Pérez, C. Rela, E. |
| spellingShingle |
Luque, T. Pérez, C. Rela, E. Optimal exponents in weighted estimates without examples |
| author_facet |
Luque, T. Pérez, C. Rela, E. |
| author_sort |
Luque, T. |
| title |
Optimal exponents in weighted estimates without examples |
| title_short |
Optimal exponents in weighted estimates without examples |
| title_full |
Optimal exponents in weighted estimates without examples |
| title_fullStr |
Optimal exponents in weighted estimates without examples |
| title_full_unstemmed |
Optimal exponents in weighted estimates without examples |
| title_sort |
optimal exponents in weighted estimates without examples |
| url |
http://hdl.handle.net/20.500.12110/paper_10732780_v22_n1_p183_Luque |
| work_keys_str_mv |
AT luquet optimalexponentsinweightedestimateswithoutexamples AT perezc optimalexponentsinweightedestimateswithoutexamples AT relae optimalexponentsinweightedestimateswithoutexamples |
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1782023839361794048 |