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
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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
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Año: | 2015
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Volumen: | 22
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Número: | 1
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Página de inicio: | 183
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Página de fin: | 201
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DOI: |
http://dx.doi.org/10.4310/MRL.2015.v22.n1.a10 |
Título revista: | Mathematical Research Letters
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Título revista abreviado: | Math. Res. Lett.
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ISSN: | 10732780
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10732780_v22_n1_p183_Luque |
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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