We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of f, M f, by an expression involving the nonincreasing rearrangement of f. This estimate is used to obtain necessary and suucient conditions for the boundedness of M between classical Lorentz spaces.

We prove in this paper some sharp weighted inequalities for the vector–valued maximal function Mq of Fefferman and Stein defined by Mqf(x) = ( ∞ ∑ i=1 (Mfi(x)) q )1/q , where M is the Hardy–Littlewood maximal function. As a consequence we derive the main result establishing that in the range 1 < q < p < ∞ there exists a constant C such that ∫ Rn Mqf(x) p w(x)dx ≤ C ∫ Rn |f(x)|qM [ p q ]+1 w(x)d...

Almost everywhere convergence on the solution of Schr\"odinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this was essentially solved building sharp $L^p$-estimate maximal function. Du-Guth-Li \cite{DGL} proved $L^p$-estimates for all $p \geq 2$ $\mathbb{R}^{2+1}$. Du-Zhang \cite{DZ} $L^2$-estimate $\mathbb{R}^{n+1}$ with $n 3$, but $p>2$ function...

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear m...