Geodesic
Boundedness of one-sided fractional integrals in the one-sided Calderón-Hardy spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 1, pp. 57-75

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In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-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 Calderón maximal function.
Classification : 42B20, 42B35
Keywords: fractional integral; maximal; one-sided Calderón-Hardy; one-sided weights spaces 
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     title = {Boundedness of one-sided fractional integrals in the one-sided {Calder\'on-Hardy} spaces},
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Perini, Alejandra. Boundedness of one-sided fractional integrals in the one-sided Calderón-Hardy spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 1, pp. 57-75. http://geodesic.mathdoc.fr/item/CMUC_2011__52_1_a4/