Sharp bounds for fractional type operators with $L^{\alpha,s}$-Hörmander conditions

Authors

  • Gonzalo H. Ibañez-Firnkorn FaMAF, Universidad Nacional de Córdoba, CIEM (CONICET), 5000 Córdoba, Argentina
  • María Silvina Riveros FaMAF, Universidad Nacional de Córdoba, CIEM (CONICET), 5000 Córdoba, Argentina
  • Raúl E. Vidal FaMAF, Universidad Nacional de Córdoba, CIEM (CONICET), 5000 Córdoba, Argentina

DOI:

https://doi.org/10.33044/revuma.2211

Abstract

We provide the sharp bound for a fractional type operator given by a kernel satisfying the $L^{\alpha,s}$-Hörmander condition and certain fractional size condition, $0 < \alpha < n$ and $1 < s\leq \infty$. In order to prove this result we use a new appropriate sparse domination. Examples of these operators include the fractional rough operators. For the case $s=\infty$ we recover the sharp bound of the fractional integral, $I_{\alpha}$, proved by Lacey et al. [J. Functional Anal. 259 (2010), no. 5, 1073–1097].

Downloads

Download data is not yet available.

References

N. Accomazzo, J. C. Martínez-Perales, and I. P. Rivera-Ríos, On Bloom-type estimates for iterated commutators of fractional integrals, Indiana Univ. Math. J. 69 (2020), no. 4, 1207–1230. MR 4124126.

K. Astala, T. Iwaniec, and E. Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), no. 1, 27–56. MR 1815249.

A. L. Bernardis, M. Lorente, and M. S. Riveros, Weighted inequalities for fractional integral operators with kernel satisfying Hörmander type conditions, Math. Inequal. Appl. 14 (2011), no. 4, 881–895. MR 2884902.

S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253–272. MR 1124164.

T. A. Bui, J. M. Conde-Alonso, X. T. Duong, and M. Hormozi, A note on weighted bounds for singular operators with nonsmooth kernels, Studia Math. 236 (2017), no. 3, 245–269. MR 3600764.

M. E. Cejas, K. Li, C. Pérez, and I. P. Rivera-Ríos, Vector-valued operators, optimal weighted estimates and the $C_p$ condition, Sci. China Math. 63 (2020), no. 7, 1339–1368. MR 4119561.

D. Cruz-Uribe, Elementary proofs of one weight norm inequalities for fractional integral operators and commutators, in Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory. Vol. 2, 183–198, Assoc. Women Math. Ser., 5, Springer, Cham, 2017. MR 3688143.

E. Dalmasso, G. Pradolini, and W. Ramos, The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces, Fract. Calc. Appl. Anal. 21 (2018), no. 3, 628–653. MR 3827147.

S. Fackler and T. P. Hytönen, Off-diagonal sharp two-weight estimates for sparse operators, New York J. Math. 24 (2018), 21–42. MR 3761937.

Z. Guo, P. Li, and L. Peng, $L^p$ boundedness of commutators of Riesz transforms associated to Schrödinger operator, J. Math. Anal. Appl. 341 (2008), no. 1, 421–432. MR 2394095.

F. Gürbüz, Some estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, Canad. Math. Bull. 60 (2017), no. 1, 131–145. MR 3612105.

T. P. Hytönen, The sharp weighted bound for general Calderón–Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506. MR 2912709.

G. H. Ibañez-Firnkorn and I. P. Rivera-Ríos, Sparse and weighted estimates for generalized Hörmander operators and commutators, Monatsh. Math. 191 (2020), no. 1, 125–173. MR 4050113.

G. H. Ibañez Firnkorn and M. S. Riveros, Certain fractional type operators with Hörmander conditions, Ann. Acad. Sci. Fenn. Math. 43 (2018), no. 2, 913–929. MR 3839843.

D. S. Kurtz, Sharp function estimates for fractional integrals and related operators, J. Austral. Math. Soc. Ser. A 49 (1990), no. 1, 129–137. MR 1054087.

M. T. Lacey, K. Moen, C. Pérez, and R. Torres, Sharp weighted bounds for fractional integral operators, J. Funct. Anal. 259 (2010), no. 5, 1073–1097. MR 2652182.

A. K. Lerner, On pointwise estimates involving sparse operators, New York J. Math. 22 (2016), 341–349. MR 3484688.

A. K. Lerner and F. Nazarov, Intuitive dyadic calculus: the basics, Expo. Math. 37 (2019), no. 3, 225–265. MR 4007575.

A. K. Lerner, S. Ombrosi, and I. P. Rivera-Ríos, On pointwise and weighted estimates for commutators of Calderón–Zygmund operators, Adv. Math. 319 (2017), 153–181. MR 3695871.

K. Li, Sparse domination theorem for multilinear singular integral operators with $L^r$-Hörmander condition, Michigan Math. J. 67 (2018), no. 2, 253–265. MR 3802254.

M. Lorente and M. S. Riveros, Weights for commutators of the one-sided discrete square function, the Weyl fractional integral and other one-sided operators, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 4, 845–862. MR 2173343.

J. M. Martell, C. Pérez, and R. Trujillo-González, Lack of natural weighted estimates for some singular integral operators, Trans. Amer. Math. Soc. 357 (2005), no. 1, 385–396. MR 2098100.

B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. MR 0340523.

S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical $A_p$ characteristic, Amer. J. Math. 129 (2007), no. 5, 1355–1375. MR 2354322.

S. Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1237–1249. MR 2367098.

S. Petermichl and A. Volberg, Heating of the Ahlfors–Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281–305. MR 1894362.

G. Pradolini and O. Salinas, Estimates on the $(L^p(w),L^q(w))$ operator norm of the fractional maximal function, Rev. Un. Mat. Argentina 40 (1996), no. 1-2, 69–74. MR 1450828.

M. S. Riveros and M. Urciuolo, Weighted inequalities for some integral operators with rough kernels, Cent. Eur. J. Math. 12 (2014), no. 4, 636–647. MR 3152177.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. MR 0290095.

Downloads

Published

2022-11-04

Issue

Section

Article