Revista Matemática Iberoamericana

Endpoint estimates from restricted rearrangement inequalities

María J. Carro and Joaquim Martín

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Abstract

Let $T$ be a sublinear operator such that $(Tf)^*(t)\le h(t, \|f\|_1)$ for some positive function $h(t,s)$ and every function $f$ such that $\|f\|_{\infty}\le 1$. Then, we show that $T$ can be extended continuously from a logarithmic type space into a weighted weak Lorentz space. This type of result is connected with the theory of restricted weak type extrapolation and extends a recent result of Arias-de-Reyna concerning the pointwise convergence of Fourier series to a much more general context.

Article information

Source
Rev. Mat. Iberoamericana Volume 20, Number 1 (2004), 131-150.

Dates
First available in Project Euclid: 2 April 2004

Permanent link to this document
http://projecteuclid.org/euclid.rmi/1080928423

Mathematical Reviews number (MathSciNet)
MR2076775

Zentralblatt MATH identifier
1063.46061

Subjects
Primary: 46M35: Abstract interpolation of topological vector spaces [See also 46B70] 47A30: Norms (inequalities, more than one norm, etc.)

Keywords
rearrangement inequality real interpolation Banach couples extrapolation theory Carleson's operator

Citation

Carro, María J.; Martín, Joaquim. Endpoint estimates from restricted rearrangement inequalities. Rev. Mat. Iberoamericana 20 (2004), no. 1, 131--150. http://projecteuclid.org/euclid.rmi/1080928423.


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