A two weight theorem for α-fractional singular integrals with an energy side condition
Journal
Revista Matematica Iberoamericana
Journal Volume
32
Journal Issue
1
Pages
79-124
Date Issued
2016
Author(s)
Abstract
Let σ and omega; be locally finite positive Borel measures on ℝn with no common point masses, and let Tα be a standard α-fractional Calderón.Zygmund operator on ℝn with 0 ≤ α < n. Furthermore, assume as side conditions the Aα2 conditions and certain α-energy conditions. Then we show that Tα is bounded from L2(σ) to L2(omega;) if the cube testing conditions hold for Tα and its dual, and if the weak boundedness property holds for Tα. Conversely, if Tα is bounded from L2(σ) to L2(omega;), then the testing conditions hold, and the weak boundedness condition holds. If the vector of α-fractional Riesz transforms Rασ (or more generally a strongly elliptic vector of transforms) is bounded from L2(σ) to L2(omega;), then the Aα2 conditions hold. We do not know if our energy conditions are necessary when n ≥ 2. The innovations in this higher dimensional setting are the control of functional energy by energy modulo Aα2 , the necessity of the Aα2 conditions for elliptic vectors, the extension of certain one-dimensional arguments to higher dimensions in light of the differing Poisson integrals used in A2 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.
SDGs
Type
journal article