Illinois Journal of Mathematics

One and two weight norm inequalities for Riesz potentials

David Cruz-Uribe, SFO and Kabe Moen

Full-text: Open access


We consider weighted norm inequalities for the Riesz potentials $I_{\alpha}$, also referred to as fractional integral operators. First, we prove mixed $A_{p}\mbox{-}A_{\infty}$ type estimates in the spirit of (Indiana Univ. Math. J. 61 (2012) 2041–2052, Anal. PDE 6 (2013) 777–818, Houston J. Math. 38 (2012) 799–814). Then we prove strong and weak type inequalities in the case $p<q$ using the so-called log bump conditions. These results complement the strong type inequalities of Pérez (Indiana Univ. Math. J. 43 (1994) 663–683) and answer a conjecture from (Weights, extrapolation and the theory of Rubio de Francia (2011) Birkhäuser). For both sets of results, our main tool is a corona decomposition adapted to fractional averages.

Article information

Illinois J. Math., Volume 57, Number 1 (2013), 295-323.

First available in Project Euclid: 23 June 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory 42B30: $H^p$-spaces 42B35: Function spaces arising in harmonic analysis


Cruz-Uribe, SFO, David; Moen, Kabe. One and two weight norm inequalities for Riesz potentials. Illinois J. Math. 57 (2013), no. 1, 295--323. doi:10.1215/ijm/1403534497.

Export citation


  • D. Cruz-Uribe, A new proof of weighted weak-type inequalities for fractional integrals, Comment. Math. Univ. Carolin. 42 (2001), no. 3, 481–485.
  • D. Cruz-Uribe, J. M. Martell and C. Pérez, Sharp weighted estimates for classical operators, Adv. Math. 229 (2011), 408–441.
  • D. Cruz-Uribe, J. M. Martell and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser, Basel, 2011.
  • D. Cruz-Uribe and K. Moen, Sharp norm inequalities for commutators of classical operators, Publ. Mat. 56 (2012), 147–190.
  • D. Cruz-Uribe and C. Pérez, Sharp two-weight, weak-type norm inequalities for singular integral operators, Math. Res. Lett. 6 (1999), no. 3–4, 417–427.
  • D. Cruz-Uribe and C. Pérez, Two-weight, weak-type norm inequalities for fractional integrals, Calderón–Zygmund operators and commutators, Indiana Univ. Math. J. 49 (2000), no. 2, 697–721.
  • D. Cruz-Uribe, A. Reznikov and A. Volberg, Logarithmic bump conditions and the two-weight boundedness of Calderón–Zygmund operators, Adv. Math. 255 (2014), 706–729.
  • N. Fujii, Weighted bounded mean oscillation and singular integrals, Math. Japon. 22 (1977/78), no. 5, 529–534.
  • N. Fujii, A condition for a two-weight norm inequality for singular integral operators, Studia Math. 98 (1991), no. 3, 175–190.
  • J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland, Amsterdam, 1985.
  • P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, \bnumberx+101.
  • T. Hytönen, The sharp weighted bound for general Calderón–Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506.
  • T. Hytönen and M. Lacey, The $A_p$-$A_{\infty}$ inequality for general Calderón–Zygmund operators, Indiana Univ. Math. J. 61 (2012), 2041–2052.
  • T. Hytönen and F. Nazarov, The local $Tb$ theorem with rough test functions, preprint, 2012.
  • T. Hytönen and C. Pérez, Sharp weighted bounds involving $A_{\infty}$, Anal. PDE 6 (2013), 777–818.
  • T. Hytönen, C. Pérez, S. Treil and A. Volberg, Sharp weighted estimated for dyadic shifts and the ${A}_2$ conjecture, J. Reine Angew. Math. 687 (2014), 43–86.
  • M. Lacey, The $A_p$-$A_{\infty}$ inequality for the Hilbert transform, Houston J. Math. 38 (2012), 799–814.
  • M. Lacey, S. Petermichl and M. C. Reguera, Sharp $A_2$ inequality for Haar shift operators, Math. Ann. 348 (2010), no. 1, 127–141.
  • M. Lacey, E. T. Sawyer and I. Uriarte-Tuero, Two weight inequalities for discrete positive operators, preprint, 2012.
  • M. T. Lacey, K. Moen, C. Pérez and R. H. Torres, Sharp weighted bounds for fractional integral operators, J. Funct. Anal. 259 (2010), no. 5, 1073–1097.
  • M. A. Leckband, Structure results on the maximal Hilbert transform and two-weight norm inequalities, Indiana Univ. Math. J. 34 (1985), no. 2, 259–275.
  • A. Lerner, On an estimate of Calderón–Zygmund operators by dyadic positive operators, J. Anal. Math. 121 (2013), 141–161.
  • A. Lerner, Mixed $A_p$-$A_r$ inequalities for classical singular integrals and Littlewood–Paley operators, J. Geom. Anal. 23 (2013), 1343–1354.
  • A. Lerner, A simple proof of the $A_2$ conjecture, to appear in Int. Math. Res. Not.
  • R. L. Long and F. S. Nie, Weighted Sobolev inequality and eigenvalue estimates of Schrödinger operators, Harmonic analysis (Tianjin, 1988), Lecture Notes in Math., vol. 1494, Springer, Berlin, 1991, pp. 131–141.
  • V. Maz'ya, Sobolev spaces with applications to elliptic partial differential equations, augmented ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011.
  • K. Moen, Sharp one-weight and two-weight bounds for maximal operators, Studia Math. 194 (2009), no. 2, 163–180.
  • B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274.
  • C. J. Neugebauer, Inserting $A_{p}$-weights, Proc. Amer. Math. Soc. 87 (1983), no. 4, 644–648.
  • C. Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J. 43 (1994), no. 2, 663–683.
  • C. Pérez, 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 (1995), no. 1, 135–157.
  • C. Pérez, S. Treil and A. Volberg, On ${A}_2$ conjecture and corona decomposition of weights, preprint, 2010.
  • Y. Rakotondratsimba, Two-weight norm inequality for Calderón–Zygmund operators, Acta Math. Hungar. 80 (1998), no. 1–2, 39–54.
  • M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Dekker, New York, 1991.
  • E. T. Sawyer, A two weight weak type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), no. 1, 339–345.
  • E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), no. 2, 533–545.
  • E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874.
  • C. Sbordone and I. Wik, Maximal functions and related weight classes, Publ. Mat. 38 (1994), no. 1, 127–155.
  • E. M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy.
  • I. E. Verbitsky, Weighted norm inequalities for maximal operators and Pisier's theorem on factorization through $L^{p\infty}$, Integral Equations Operator Theory 15 (1992), no. 1, 124–153.
  • J. M. Wilson, Weighted inequalities for the dyadic square function without dyadic $A_\infty$, Duke Math. J. 55 (1987), no. 1, 19–50.
  • J. M. Wilson, Weighted norm inequalities for the continuous square function, Trans. Amer. Math. Soc. 314 (1989), no. 2, 661–692.
  • J. M. Wilson, Weighted Littlewood–Paley theory and exponential-square integrability, Lecture Notes in Math., vol. 1924, Springer, Berlin, 2007.