Revistes Catalanes amb Accés Obert (RACO)

Two weighted inequalities for convolution maximal operators

A. L. Bernardis, F. J. Martín-Reyes

Resum


Let $\varphi\colon \mathbb{R} \to [0,\infty)$ an integrable function such that $\varphi\chi_{(-\infty,0)} = 0$ and $\varphi$ is decreasing in $(0,\infty)$. Let $\tau_h f(x) = f(x-h)$, with $h\in \mathbb{R} \setminus \{0\}$ and $f_R(x) = \frac{1}{R}f(\frac{x}{R})$, with $R>0$. In this paper we characterize the pair of weights $(u,v)$ such that the operators $M_{\tau_h\varphi}f(x) = \sup_{R>0}|f|\ast [\tau_h\varphi]_R(x)$ are of weak type $(p,p)$ with respect to $(u,v)$, $1 < p < \infty$.

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