Some weighted inequalities for general one-sided maximal operators

F. J. Martín-Reyes¹ ; ¹

¹

Studia Mathematica, Tome 122 (1997) no. 1, pp. 1-14

Résumé

We characterize the pairs of weights on ℝ for which the operators $M^{+}_{h,k}f(x) = sup_{c>x}h(x,c) ʃ_{x}^{c} f(s)k(x,s,c)ds$ are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p q ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on ${(x,c): x c}$, while k is defined on ${(x,s,c): x s c}$. If $h(x,c) = (c-x)^{-β}$, $k(x,s,c) = (c-s)^{α-1}$, 0 ≤ β ≤ α ≤ 1, we obtain the operator $M^{+}_{α,β}f = sup_{c>x} 1/(c-x)^{β} ʃ_{x}^{c} f(s)/(c-s)^{1-α} ds$. For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal weights and 1 p ∞, weak and strong type are equivalent. If we take α = β we characterize the strong type weights for the operator $M^{+}_{α,α}$ introduced by W. Jurkat and J. Troutman in the study of $C_α$ differentiation of the integral.

Détail

DOI : 10.4064/sm-122-1-1-14

Keywords: one-sided maximal operators, Cesàro averages, weights