A T(P) theorem for Sobolev spaces on domains

作者：Prats Marti^*; Tolsa Xavier

来源：Journal of Functional Analysis, 2015, 268(10): 2946-2989.

DOI：10.1016/j.jfa.2015.01.007

• Harmonic analysis
• Sobolev spaces
• Calderon-Zygmund operators
• Carleson measures

Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given 0 < s <= 1 < p < infinity with sp > 2 and a Lipschitz domain Omega subset of C, the Beurling transform Bf = -p.v.1/pi z(2) * f is bounded in the Sobolev space W-s,W-p (Omega) if and only if B-chi Omega is an element of W-s,W-p(Omega).
In this paper we obtain a generalized version of the former result valid for any s is an element of N and for a larger family of Calderon-Zygmund operators in any ambient space R-d as long as p > d. In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally we find a sufficient condition in terms of Carleson measures for p <= d. In the particular case s = 1, this condition is in fact necessary, which yields a complete characterization.

• 出版日期2015-5-15