Endpoint estimates from restricted rearrangement inequalities

Let T be a sublinear operator such that (Tf) ∗ (t) ≤ h(t, f 1) for some positive function h(t,s) and every function f such that f ∞ ≤ 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.