The Bellman functions and two-weight inequalities for Haar multipliers. (English) Zbl 0951.42007

The main result concerns the two weight problem for Haar multiplier operators. In the one variable case such an operator is the same as a martingale transform \[ T_{\varepsilon,{ a}}: f\mapsto \sum_{I\in {\mathcal D}[0,1]} \varepsilon_I{ a_I} \langle f,h_I\rangle h_I, \] in which \({\mathcal D}[0,1] \) denotes the dyadic intervals in \([0,1]\), \(\{\varepsilon_I\}\) is a sequence of \(\pm 1\)’s indexed by dyadic intervals, \( \{ a_I\}\) is a sequence of nonnegative numbers and the Haar functions are \(h_I= (\chi_{I_l}-\chi_{I_r})/\sqrt{|I|}\) which form an orthonormal basis for \(L^2[0,1]\). Such multipliers are generally thought of as discrete analogues of Calderón-Zygmund singular integral operators such as the Hilbert transform. Establishing boundedness of such operators by means of Carleson type estimates is now considered as major step toward proving boundedness properties of singular integrals. The main result establishes that an operator essentially of the form \( T_{\varepsilon,a}\) is bounded from \( L^2_u\) to \( L^2_v\) if and only if one has Sawyer type estimates \( \|T_{\varepsilon,a} (u\chi_I)\|_{L^2_v}\leq C\|\chi_I\|_{L^2_u}\) and \( \|T_{\varepsilon,a} (v\chi_I)\|_{L^2_u}\leq C\|\chi_I\|_{L^2_v}\) in which the constants do not depend on \(\{\varepsilon_I\}\). Here \(\|f\|_{L^2_v}^2 = \int |f|^2 v\) where \(v\) is a nonnegative weight function. The uniformity of the estimates plays a crucial role. Equally important is the role of weighted Carleson embedding theorems which give a necessary and sufficient Carleson measure type condition for the continuity of the mapping \(f\mapsto \{\langle f\sqrt u\rangle_I\}\) from \(L^2([0,1])\) to a weighted space \(\ell^2_a\) of sequences indexed by dyadic intervals. Here \( \langle \cdot\rangle_I\) denotes the average value over \(I\). Contrary to comments made in the introduction, the authors now appear to have extended their techniques to prove the Sawyer type conditions are necessary and sufficient for boundedness of the Hilbert transform from \( L^2_u\) to \( L^2_v\), at least when the weights satisfy a doubling condition.


42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42A50 Conjugate functions, conjugate series, singular integrals
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
42B15 Multipliers for harmonic analysis in several variables
60G46 Martingales and classical analysis
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