An operator approach to weighted norm inequalities for singular inegrals

S. R. Treil'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIII, Tome 135 (1984), pp. 150-174

Résumé

A new approach to weighted norm inequalities for singular integral operators is developed. This appoach uses Hilbert space methods of Operator Theory. Theorem. Let $R_1$ be a positive operator in $L^2(\mathbb T)$ with domain $\operatorname{Dom}R_1$ such that $\operatorname{Ker} R_1=\{0\}$, $0\inf_n\|R_1z^n\|\leqslant\sup_n\|R_1z^n\|+\infty$, and $\inf_n\operatorname{dist}(\|R_1z^n\|^{-1}\cdot R_1z^n, Z(R_1z^n, k\ne n))>0$. Then there exists an operator $R_2$ satisfying 1. $\|R_2(\sum_{j\leqslant k\leqslant n}\hat f(k)z^k)\|\leqslant c\cdot\|R_1f\|$; 2. $\inf_n\|R_2z^n\|>0$; 3. $\inf_n\operatorname{dist}(\|R_1z^n\|^{-1}\cdot R_1z^n, Z(R_1z^n, |k||n|))>0$. In case the system $\{Z^n\}_{n\in\mathbb Z}$ is fundamental in $\operatorname{Dom}R_1$ with respect to the graph norm $\|f\|^2_\Gamma\overset{\text{def}}{=}\|f\|^2+\|R_1f\|^2$ the conclusion of the above theorem can be strengthened: 4. $R_2$ is a bounded positive operator. If in addition $\sup_{n\geqslant0}\|R_1S^nR_1^{-1}\|\infty$, $S$ being the shift operator, i. e. $Sf=z\cdot f$, then $R_2$ is multiplication by a positive function $v$. This theorem generalizes the well-known Koosis theorem.