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fractional integral; maximal; one-sided Calderón-Hardy; one-sided weights spaces
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.
[1] Calderón A.P.: Estimates for singular integral operators in terms of maximal functions. Studia Math. 44 (1972), 563–582. MR 0348555
[2] Gatto A., Jiménez J.G., Segovia C.: On the solution of the equation $\Delta^m F = f$ for $f\in H^p$. Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. II (Chicago, 1981), Wadsworth Math. Ser., Wadworth, Belmont, CA, 1983. MR 0730054
[3] Grafakos L.: Classical and Modern Fourier Analysis. Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR 2449250 | Zbl 1148.42001
[4] Harboure E., Salinas O., Viviani B.: Acotación de la integral fraccionaria en espacios de Orlicz y de oscilación media $\phi$ acotada. Actas del 2do. Congreso Dr. A. Monteiro, Bahía Blanca, 1997, pp. 41–50. MR 1253076
[5] Martín-Reyes F.J.: New proof of weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Proc. Amer. Math. Soc. 117 (1993), 691–698. MR 1111435
[6] Martín-Reyes F.J., Ortega P., de la Torre A.: Weighted inequalities for the one-sided maximal functions. Trans. Amer. Math. Soc. 319 (1990), 517–534. MR 0986694
[7] Ombrosi S.: On spaces associated with primitives of distributions in one-sided Hardy spaces. Rev. Un. Mat. Argentina 42 (2001), no. 2, 81–102. MR 1969626 | Zbl 1196.42023
[8] Ombrosi S.: Sobre espacios asociados a primitivas de distribuciones en espacios de Hardy laterales. Ph.D. Thesis, Universidad Nacional de Buenos Aires, 2002.
[9] Ombrosi S., de Rosa L.: Boundeness of the Weyl fractional integral on the one-sided weighted Lebesque and Lipchitz spaces. Publ. Mat. 47 (2003), no. 1, 71–102. MR 1970895
[10] Ombrosi S., Segovia C.: One-sided singular integral operators on Calderón-Hardy spaces. Rev. Un. Mat. Argentina 44 (2003), no. 1, 17–32. MR 2051035 | Zbl 1078.42008
[11] de Rosa L., Segovia C.: Weighted $H^{p}$ spaces for one sided maximal functions. Contemp. Math., 189, American Mathematical Society, Providence, RI, 1995, pp. 161–183. DOI 10.1090/conm/189/02262
[12] Sawyer E.: Weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Trans. Amer. Math. Soc. 297 (1986), 53–61. DOI 10.1090/S0002-9947-1986-0849466-0 | MR 0849466 | Zbl 0627.42009
[13] Stein E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, N.J., 1970. MR 0290095 | Zbl 0281.44003
[14] Zygmund A.: Trigonometric Series. Cambridge University Press, Cambridge, 1959. MR 0107776 | Zbl 1084.42003
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