Revista Matemática Iberoamericana

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Volume 20, Issue 1, 2004, pp. 131–150
DOI: 10.4171/RMI/383

Published online: 2004-04-30

Endpoint estimates from restricted rearrangement inequalities

María Belén J. Carro[1] and Joaquim Martín[2]

(1) Universitat de Barcelona, Spain
(2) Universitat Autònoma de Barcelona, Bellaterra, Spain

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.

Keywords: Rearrangement inequality, real interpolation, Banach couples, extrapolation theory, Carleson’s operator

Carro María Belén, Martín Joaquim: Endpoint estimates from restricted rearrangement inequalities. Rev. Mat. Iberoam. 20 (2004), 131-150. doi: 10.4171/RMI/383