Analysis & PDE

A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure

Michael Lacey, Eric Sawyer, and Ignacio Uriarte-Tuero

Full-text: Open access


Let σ and ω be positive Borel measures on with σ doubling. Suppose first that 1<p2. We characterize boundedness of certain maximal truncations of the Hilbert transform T from Lp(σ) to Lp(ω) in terms of the strengthened Ap condition

( s Q ( x ) p d ω ( x ) ) 1 p ( s Q ( x ) p d σ ( x ) ) 1 p C | Q | ,

where sQ(x)=|Q|(|Q|+|xxQ|), and two testing conditions. The first applies to a restricted class of functions and is a strong-type testing condition,

Q T ( χ E σ ) ( x ) p d ω ( x ) C 1 Q d σ ( x )  for all  E Q ,

and the second is a weak-type or dual interval testing condition,

Q T ( χ Q f σ ) ( x ) d ω ( x ) C 2 ( Q | f ( x ) | p d σ ( x ) ) 1 p ( Q d ω ( x ) ) 1 p

for all intervals Q in and all functions fLp(σ). In the case p>2 the same result holds if we include an additional necessary condition, the Poisson condition

( r = 1 | I r | σ | I r | p 1 = 0 2 | ( I r ) ( ) | χ ( I r ) ( ) ( y ) ) p d ω ( y ) C r = 1 | I r | σ | I r | p ,

for all pairwise disjoint decompositions Q=r=1Ir of the dyadic interval Q into dyadic intervals Ir. We prove that analogues of these conditions are sufficient for boundedness of certain maximal singular integrals in n when σ is doubling and 1<p<. Finally, we characterize the weak-type two weight inequality for certain maximal singular integrals T in n when 1<p<, without the doubling assumption on σ, in terms of analogues of the second testing condition and the Ap condition.

Article information

Anal. PDE, Volume 5, Number 1 (2012), 1-60.

Received: 7 October 2009
Revised: 2 February 2011
Accepted: 2 March 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

two weight singular integral maximal function maximal truncation


Lacey, Michael; Sawyer, Eric; Uriarte-Tuero, Ignacio. A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure. Anal. PDE 5 (2012), no. 1, 1--60. doi:10.2140/apde.2012.5.1.

Export citation


  • M. Cotlar and C. Sadosky, “On the Helson–Szegő theorem and a related class of modified Toeplitz kernels”, pp. 383–407 in Harmonic analysis in Euclidean spaces (Williamstown, MA 1978), vol. 1, edited by G. Weiss and S. Wainger, Proc. Sympos. Pure Math. 35, Amer. Math. Soc., Providence, R.I., 1979.
  • M. Cotlar and C. Sadosky, “On some $L\sp{p}$ versions of the Helson–Szegő theorem”, pp. 306–317 in Conference on harmonic analysis in honor of Antoni Zygmund (Chicago, 1981), vol. 1, edited by W. Beckner et al., Wadsworth, Belmont, CA, 1983.
  • D. Cruz-Uribe, J. M. Martell, and C. Pérez, “Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture”, Adv. Math. 216:2 (2007), 647–676.
  • M. T. Lacey, E. T. Sawyer, and I. Uriarte-Tuero, “A two weight inequality for the Hilbert transform assuming an energy hypothesis”, preprint, version 7, 2011.
  • J. Mateu, P. Mattila, A. Nicolau, and J. Orobitg, “BMO for nondoubling measures”, Duke Math. J. 102:3 (2000), 533–565.
  • B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function”, Trans. Amer. Math. Soc. 165 (1972), 207–226.
  • C. Muscalu, T. Tao, and C. Thiele, “Multi-linear operators given by singular multipliers”, J. Amer. Math. Soc. 15:2 (2002), 469–496.
  • F. Nazarov, S. Treil, and A. Volberg, “Cauchy integral and Calderón–Zygmund operators on nonhomogeneous spaces”, Internat. Math. Res. Notices 15 (1997), 703–726.
  • F. Nazarov, S. Treil, and A. Volberg, “The Bellman functions and two-weight inequalities for Haar multipliers”, J. Amer. Math. Soc. 12:4 (1999), 909–928.
  • F. Nazarov, S. Treil, and A. Volberg, “The $Tb$-theorem on non-homogeneous spaces”, Acta Math. 190:2 (2003), 151–239.
  • F. Nazarov, S. Treil, and A. Volberg, “Two weight inequalities for individual Haar multipliers and other well localized operators”, Math. Res. Lett. 15:3 (2008), 583–597.
  • F. Nazarov, S. Treil, and A. Volberg, “Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures”, preprint, 2005 and arXiv, 2010.
  • F. Peherstorfer, A. Volberg, and P. Yuditskii, “Two-weight Hilbert transform and Lipschitz property of Jacobi matrices associated to hyperbolic polynomials”, J. Funct. Anal. 246:1 (2007), 1–30.
  • S. Petermichl, “Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol”, C. R. Acad. Sci. Paris Sér. I Math. 330:6 (2000), 455–460.
  • S. Petermichl, S. Treil, and A. Volberg, “Why the Riesz transforms are averages of the dyadic shifts?”, pp. 209–228 in Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), vol. extra, 2002.
  • W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987.
  • E. T. Sawyer, “A characterization of a two-weight norm inequality for maximal operators”, Studia Math. 75:1 (1982), 1–11.
  • E. Sawyer, “A two weight weak type inequality for fractional integrals”, Trans. Amer. Math. Soc. 281:1 (1984), 339–345.
  • E. T. Sawyer, “A characterization of two weight norm inequalities for fractional and Poisson integrals”, Trans. Amer. Math. Soc. 308:2 (1988), 533–545.
  • E. Sawyer and R. L. Wheeden, “Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces”, Amer. J. Math. 114:4 (1992), 813–874.
  • E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, 1993.
  • E. M. Stein and R. Shakarchi, Real analysis: Measure theory, integration, and Hilbert spaces, Princeton Lectures in Analysis 3, Princeton University Press, 2005.
  • A. Volberg, Calderón–Zygmund capacities and operators on nonhomogeneous spaces, CBMS Regional Conference Series in Mathematics 100, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2003.
  • D. Zheng, “The distribution function inequality and products of Toeplitz operators and Hankel operators”, J. Funct. Anal. 138:2 (1996), 477–501.