Alejandra Perini
Boundedness of one-sided fractional integrals in the one-sided Calder\'on-Hardy spaces

Comment.Math.Univ.Carolin. 52,1 (2011) 57-75.

Abstract:In this paper we study the mapping properties of the one-sided fractional integrals in the Calder\'on-Hardy spaces $\mathcal{H}_{q,\alpha}^{p,+}(\omega)$ for $0< p\leq 1$, $0< \alpha < \infty $ and $1< q< \infty $. Specifically, we show that, for suitable values of $p,q,\gamma, \alpha$ and $s$, if $\omega \in A_s^+$ (Sawyer's classes of weights) then the one-sided fractional integral $I_{\gamma }^+$ can be extended to a bounded operator from $\mathcal{H}_{q,\alpha}^{p,+}(\omega)$ to $\mathcal{H}_{q,\alpha + \gamma}^{p,+}(\omega)$. The result is a consequence of the pointwise inequality $$ N_{q, \alpha +\gamma}^+\left( I_{\gamma }^+ F;x\right) \leq C_{\alpha,\gamma } N_{q, \alpha}^+ \left( F;x\right), $$ where $N_{q, \alpha}^+ (F;x)$ denotes the Calder\'on maximal function.

Keywords: fractional integral, maximal, one-sided Calder\'on-Hardy, one-sided weights spaces
AMS Subject Classification: 42B20 42B35