Abstract
In the 1970s Muckenhoupt and Wheeden made several conjectures relating two weight norm inequalities for the Hardy-Littlewood maximal operator to such inequalities for singular integrals. Using techniques developed for the recent proof of the A 2 conjecture we prove a related pair of conjectures linking the Riesz potential and the fractional maximal operator. As a consequence we are able to prove a number of sharp one and two weight norm inequalities for the Riesz potential.
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Beznosova, O., Reznikov, A.: Equivalent definitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic L log L and A ∞ constants (preprint, 2012)
Cruz-Uribe D., Martell J.M., Pérez C.: Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture. Adv. Math. 216(2), 647–676 (2007)
Cruz-Uribe, D., Martell, J.M., Pérez, C.: Weights, extrapolation and the theory of Rubio de Francia, vol. 215. In: Operator Theory: Advances and Applications. Birkhäuser/Springer Basel AG Basel (2011)
Cruz-Uribe, D., Martell, J.M., Pérez, C.: A note on the off-diagonal muckenhoupt–wheeden conjecture. In: Advanced Courses in Mathematical Analysis V. World Scientific, Singapore (to appear, 2013)
Cruz-Uribe, D., Moen, K.: One and two weight norm inequalities for Riesz potentials. Illinois J. Math. (to appear, 2013)
Cruz-Uribe D., Neugebauer C.J.: Weighted norm inequalities for the geometric maximal operator. Publ. Math. 42(1), 239–263 (1998)
Cruz-Uribe, D., Reznikov, A., Volberg, A.: Logarithmic bump conditions and the two-weight boundedness of Calderón-Zygmund operators, Preprint (2012)
Fujii, N.: Weighted bounded mean oscillation and singular integrals. Math. Japan. 22(5), 529–534 (1977/78)
García-Cuerva, J., Rubio de Francia, J.L., Weighted norm inequalities and related topics. North-Holland Mathematics Studies, vol. 116. North-Holland Publishing Co., Amsterdam (1985)
Grafakos, L.: Classical Fourier analysis. In: Graduate Texts in Mathematics, 2nd edn, vol 249. Springer, New York (2008)
Hytönen, T.: The A 2 theorem: remarks and complements (2012), preprint
Hytönen, T., Lacey, M.: The A p − A ∞ inequality for general Calderón-zygmund operators. Indiana Univ. Math. J. (to appear, 2013)
Hytönen, T., Pérez, C.: Sharp weighted bounds involving A ∞ . Anal. PDE (to appear, 2013)
Lacey M., Moen K., Pérez C., Torres R.H.: Sharp weighted bounds for fractional integral operators. J. Funct. Anal. 259(5), 1073–1097 (2010)
Lacey, M., Sawyer, E.T., Uriarte-Tuero, I.: Two weight inequalities for discrete positive operators (2012), preprint
Lerner, A.: Mixed A p − A r inequalities for classical singular integrals and Littlewood-Paley operators. J. Geom. Anal. (2013). doi:10.1007/s12220-011-9290-0
Lerner, A.: A simple proof of the A 2 conjecture. Int. Math. Res. Not. (2013). doi:10.1093/imrn/rns145
Lerner, A., Moen, K.: Mixed A p − A ∞ estimates with one supremum. preprint, (2012)
Moen K.: Sharp one-weight and two-weight bounds for maximal operators. Studia Math. 194(2), 163–180 (2009)
Moen K.: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 60(2), 213–238 (2009)
Moen K.: Sharp weighted bounds without testing or extrapolation. Arch. Math. 99, 457–466 (2012)
Muckenhoupt, B.: Private communication
Muckenhoupt B., Wheeden R.L.: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261–274 (1974)
Muckenhoupt B., Wheeden R.L.: Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform. Studia Math. 55(3), 279–294 (1976)
Pérez C.: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43(2), 663–683 (1994)
Pérez C.: 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(1), 135–157 (1995)
Pérez C.: On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted L p-spaces with different weights. Proc. Lond. Math. Soc. (3) 71(1), 135–157 (1995)
Recchi, J.: Mixed A 1 − A ∞ bounds for fractional integrals (2012), preprint
Reguera, M., Scurry, J.: On joint estimates for maximal functions and singular integrals in weighted spaces, preprint (2011)
Sawyer E.T.: A characterization of a two-weight norm inequality for maximal operators. Studia Math. 75(1), 1–11 (1982)
Sawyer E.T., Wheeden R.L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114(4), 813–874 (1992)
Stein E.M., Weiss G.: Interpolation of operators with change of measures. Trans. Am. Math. Soc. 87, 159–172 (1958)
Wilson J.M.: Weighted inequalities for the dyadic square function without dyadic A ∞ . Duke Math. J. 55(1), 19–50 (1987)
Wilson J.M.: Weighted norm inequalities for the continuous square function. Trans. Am. Math. Soc. 314(2), 661–692 (1989)
Wilson, J.M.: Weighted Littlewood-Paley theory and exponential-square integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2007)
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D. Cruz-Uribe was supported by the Stewart-Dorwart faculty development fund at Trinity College and by grant MTM2009-08934 from the Spanish Ministry of Science and Innovation. K. Moen was supported by NSF Grant 1201504.
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Cruz-Uribe, D., SFO. & Moen, K. A Fractional Muckenhoupt–Wheeden Theorem and its Consequences. Integr. Equ. Oper. Theory 76, 421–446 (2013). https://doi.org/10.1007/s00020-013-2059-z
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DOI: https://doi.org/10.1007/s00020-013-2059-z