Square function estimates and the T(b) theorem. (English) Zbl 0719.42023

Semmes, Stephen

Proc. Am. Math. Soc. 110, No. 3, 721-726 (1990).

The author devises a simple proof on square function estimates provided by the so-called T(b) theorem. As applications the main result enables us to establish easily the boundedness of the Cauchy integral on Lipschitz graphs, a fundamental estimate in the theory of wavelets, etc.
Let \(\theta_ t(x,y)\), \(t>0\), \(x,y\in {\mathbb{R}}^ n\), be a family of functions such that for some positive constants C and \(\alpha\theta_ t(x,y)=0\quad if\quad | x-y| \geq Ct,\) and \[ | \theta_ t(x,y)-\theta_ t(x,y')| \leq Ct^{-n-\alpha}| y- y'|^{\alpha}. \] (That \(\theta_ t's\) are compactly supported is assumed only for notational convenience. An adequate decay of \(\theta_ t(x,y)\) as \(| x-y| \to \infty\) is sufficient.) Set \(\Theta_ tf(x)=\int \theta_ t(x,y)f(y)dx.\) The author shows that the estimate \[ \int^{\infty}_{0}\| \Theta_ tf\|^ 2_ 2(dt/t)\leq C\| f\|^ 2_ 2 \] holds if \(\theta_ t(b)=0\) for all t and if b is para-accretive, generalizing the cancellation condition \(\int \theta_ t(x,y)dy=0\) for all t,x.

Reviewer: H.Tanabe (Toyonaka)

MSC:

42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B05	Fourier series and coefficients in several variables
42C10	Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Keywords:

square function estimates; T(b) theorem; Cauchy integral on Lipschitz graphs; wavelets; para-accretive