Endpoint estimates from restricted rearrangement inequalities

Abstract

Let $T$ be a sublinear operator such that $(Tf{)}^{\ast }(t)\le h(t,\Vert f{\Vert }_1)$ for some positive function $h(t,s)$ and every function $f$ such that $\Vert f{\Vert }_{\infty }\le 1$. 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.