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The Bellman functions and sharp weighted inequalities for square functions. (English) Zbl 0972.42011
Havin, V. P. (ed.) et al., Complex analysis, operators, and related topics. The S. A. Vinogradov memorial volume. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 113, 97-113 (2000).
In previous work, e.g. [F. Nazarov, S. Treil and A. Volberg, J. Am. Math. Soc. 12, No. 4, 909-928 (1999; Zbl 0951.42007)], the authors used a minimization technique based on Bellman functions to prove weighted inequalities for the Hilbert transform and other operators involving pairs of weights. Bellman functions arise naturally in the $$L^{2}$$ case of applying Sawyer’s approach of reducing weighted norm inequalities for operators to the test case of applying the operator to localizations of the weight. In the present work the authors fix a single weight $$w$$ and ask how the norms of certain square functions, as operators on $$L^{2}(w),$$ depend on the $$A_{2}$$-norm of $$w$$. Sharp inequalities follow from the construction of appropriate Bellman functions for the operators, after reduction to a Carleson-type embedding theorem. Both, the classical Littlewood-Paley square function $$S(f)(t)=(\int_{\Gamma (t)}|\nabla f(x,y)|^{2}dxdy)^{1/2}$$ and the discrete or Haar square function $$S_{d}(f)(t)=(\sum_{t\in I}|\langle f,h_{I}\rangle |^{2}/|I|)^{1/2},$$ in which $$h_{I}(x)$$ is the $$L^{2}$$-normalized Haar function equal to $$|I|^{-1/2}$$ on the right side of the dyadic interval $$I$$ and equal to $$-|I|^{-1/2}$$ on its left side, are considered. Defining $$|T|_{w}$$ as the norm of the operator $$T$$ from $$L^{2}(w)$$ to itself, the authors prove the following inequality for the dyadic square function operator: $$|S_{d}|_{w}\leq C|w|_{A_{2}}^{2}$$. Here $$|w|_{A_{2}}^{2}=\sup_{I}\frac{1}{|I|^{2}} \int_{I}w\int_{I}1/w$$ in which the supremum is taken over all intervals $$I$$. The authors show that the square function estimate is sharp in the $$A_{2}$$ -variable in the sense that there is a $$c>0$$ such that for any $$N>1$$ there is a $$w\in A_{2}$$ with $$|w|_{A_{2}}^{2}=N$$ and such that $$|S_{d}|_{w}\geq cN$$. The authors prove a corresponding sharp estimate for the classical square function but in this case the $$A_{2}$$-norm of $$w$$ is replaced by a conformally invariant version $$|w|_{A_{2},\text{ inv}}$$ of the $$A_{2}$$-norm, based on harmonic extensions of $$w$$ and $$1/w$$, which is known to satisfy $$c|w|_{A_{2}}\leq |w|_{A_{2},\text{inv}}\leq C|w|_{A_{2}}^{2}$$ when $$|w|_{A_{2}}\geq 1$$.
For the entire collection see [Zbl 0934.00031].

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42A50 Conjugate functions, conjugate series, singular integrals 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators