Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

New weighted estimates for bilinear fractional integral operators


Author: Kabe Moen
Journal: Trans. Amer. Math. Soc. 366 (2014), 627-646
MSC (2010): Primary 42B20, 26A33
DOI: https://doi.org/10.1090/S0002-9947-2013-06067-9
Published electronically: July 26, 2013
MathSciNet review: 3130311
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a plethora of weighted estimates for bilinear fractional integral operators of the form

$\displaystyle BI_\alpha (f,g)(x)=\int _{\mathbb{R}^n}\frac {f(x-t)g(x+t)}{\vert t\vert^{n-\alpha }}\,dt, \qquad 0<\alpha <n.$

When the target space has an exponent greater than one, many weighted estimates follow trivially from Hölder's inequality and the known linear theory. We address the case where the target Lebesgue space is at most one and prove several interesting one and two weight estimates. As an application we formulate a bilinear version of the Stein-Weiss inequality for fractional integrals.

References [Enhancements On Off] (What's this?)

  • [1] A. Bernardis, O. Gorosito, and G. Pradolini, Weighted inequalities for multilinear potential operators and its commutators, preprint http://arxiv.org/abs/1007.0445.
  • [2] F. Bernicot, D. Maldonado, K. Moen, and V. Naibo, Bilinear Sobolev-Poincaré inequalities and Leibniz-type rules, J. Geom. Anal. (2012), DOI: 10.1007/s12220-012-9367-4.
  • [3] Xi Chen and Qingying Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl. 362 (2010), no. 2, 355-373. MR 2557692 (2010i:42036), https://doi.org/10.1016/j.jmaa.2009.08.022
  • [4] David V. Cruz-Uribe, José Maria Martell, and Carlos Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2797562 (2012f:42001)
  • [5] David Cruz-Uribe and Kabe Moen, Sharp norm inequalities for commutators of classical operators, Publ. Mat. 56 (2012), no. 1, 147-190. MR 2918187, https://doi.org/10.5565/PUBLMAT_56112_06
  • [6] D. Cruz-Uribe and K. Moen, One and two weight norm inequalities for Riesz potentials Illinois J. Math. to appear.
  • [7] Yong Ding and Chin-Cheng Lin, Rough bilinear fractional integrals, Math. Nachr. 246/247 (2002), 47-52. MR 1944548 (2003j:42019), https://doi.org/10.1002/1522-2616(200212)246:1$ \langle $47::AID-MANA47$ \rangle $3.0.CO;2-7
  • [8] Loukas Grafakos, On multilinear fractional integrals, Studia Math. 102 (1992), no. 1, 49-56. MR 1164632 (93d:42021)
  • [9] Loukas Grafakos and Nigel Kalton, Some remarks on multilinear maps and interpolation, Math. Ann. 319 (2001), no. 1, 151-180. MR 1812822 (2002a:46032), https://doi.org/10.1007/PL00004426
  • [10] Takeshi Iida, Yasuo Komori-Furuya, and Enji Sato, A note on multilinear fractional integrals, Anal. Theory Appl. 26 (2010), no. 4, 301-307. MR 2770468 (2011m:42024), https://doi.org/10.1007/s10496-010-0301-y
  • [11] Carlos E. Kenig and Elias M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), no. 1, 1-15. MR 1682725 (2000k:42023a)
  • [12] Michael T. Lacey, The bilinear maximal functions map into $ L^p$ for $ 2/3<p\leq 1$, Ann. of Math. (2) 151 (2000), no. 1, 35-57. MR 1745019 (2001b:42015), https://doi.org/10.2307/121111
  • [13] Michael Lacey and Christoph Thiele, $ L^p$ estimates on the bilinear Hilbert transform for $ 2<p<\infty $, Ann. of Math. (2) 146 (1997), no. 3, 693-724. MR 1491450 (99b:42014), https://doi.org/10.2307/2952458
  • [14] Michael Lacey and Christoph Thiele, On Calderón's conjecture, Ann. of Math. (2) 149 (1999), no. 2, 475-496. MR 1689336 (2000d:42003), https://doi.org/10.2307/120971
  • [15] Andrei K. Lerner, Sheldy Ombrosi, Carlos Pérez, Rodolfo H. Torres, and Rodrigo Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222-1264. MR 2483720 (2010f:42024), https://doi.org/10.1016/j.aim.2008.10.014
  • [16] Diego Maldonado, Kabe Moen, and Virginia Naibo, Weighted multilinear Poincaré inequalities for vector fields of Hörmander type, Indiana Univ. Math. J. 60 (2011), no. 2, 473-506. MR 2963782, https://doi.org/10.1512/iumj.2011.60.4156
  • [17] Kabe Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math. 60 (2009), no. 2, 213-238. MR 2514845 (2010d:42028), https://doi.org/10.1007/BF03191210
  • [18] K. Moen and V. Naibo, Higher-order multilinear Poincaré and Sobolev inequalities in Carnot groups, submitted (2010); http://arxiv.org/abs/1008.0414.
  • [19] Benjamin Muckenhoupt and Richard Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. MR 0340523 (49 #5275)
  • [20] Carlos Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J. 43 (1994), no. 2, 663-683. MR 1291534 (95m:42028), https://doi.org/10.1512/iumj.1994.43.43028
  • [21] C. Pérez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted $ L^p$-spaces with different weights, Proc. London Math. Soc. (3) 71 (1995), no. 1, 135-157. MR 1327936 (96k:42023), https://doi.org/10.1112/plms/s3-71.1.135
  • [22] Gladis Pradolini, Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators, J. Math. Anal. Appl. 367 (2010), no. 2, 640-656. MR 2607287 (2011c:42051), https://doi.org/10.1016/j.jmaa.2010.02.008
  • [23] Eric T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), no. 2, 533-545. MR 930072 (89d:26009), https://doi.org/10.2307/2001090
  • [24] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813-874. MR 1175693 (94i:42024), https://doi.org/10.2307/2374799
  • [25] E. M. Stein and Guido Weiss, Fractional integrals on $ n$-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503-514. MR 0098285 (20 #4746)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 42B20, 26A33

Retrieve articles in all journals with MSC (2010): 42B20, 26A33


Additional Information

Kabe Moen
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487-0350

DOI: https://doi.org/10.1090/S0002-9947-2013-06067-9
Keywords: Bilinear operators, fractional integration, weighted inequalities
Received by editor(s): November 7, 2011
Published electronically: July 26, 2013
Additional Notes: The author was partially supported by NSF Grant 1201504
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society