A two weight theorem for -fractional singular integrals with an energy side condition

  • Eric T. Sawyer

    McMaster University, Hamilton, Canada
  • Chun-Yen Shen

    National Central University, Jhongli City, Taoyuan County, Taiwan
  • Ignacio Uriarte-Tuero

    Michigan State University, East Lansing, USA

Abstract

Let and be locally finite positive Borel measures on with no common point masses, and let be a standard -fractional Calderón–Zygmund operator on with . Furthermore, assume as side conditions the conditions and certain -energy conditions. Then we show that is bounded from to if the cube testing conditions hold for and its dual, and if the weak boundedness property holds for .

Conversely, if is bounded from to , then the testing conditions hold, and the weak boundedness condition holds. If the vector of -fractional Riesz transforms (or more generally a strongly elliptic vector of transforms) is bounded from to , then the conditions hold. We do not know if our energy conditions are necessary when .

The innovations in this higher dimensional setting are the control of functional energy by energy modulo , the necessity of the conditions for elliptic vectors, the extension of certain one-dimensional arguments to higher dimensions in light of the differing Poisson integrals used in and energy conditions, and the treatment of certain complications arising from the Lacey–Wick monotonicity lemma. The main obstacle in higher dimensions is thus identified as the pair of energy conditions.

Cite this article

Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero, A two weight theorem for -fractional singular integrals with an energy side condition. Rev. Mat. Iberoam. 32 (2016), no. 1, pp. 79–174

DOI 10.4171/RMI/882