Majorization of Singular Integral Operators with Cauchy Kernel on $L^2$

Abstract

Let $a, b, c$ and $d$ be functions in $L^2 = L^2(\mathbb{T}, d\theta/2\pi)$, where $\mathbb{T}$ denotes the unit circle. Let $\mathcal{P}$ denote the set of all trigonometric polynomials. Suppose the singular integral operators $A$ and $B$ are defined by $A=aP+bQ$ and $B = cP+dQ$ on $\mathcal{P}$, where $P$ is an analytic projection and $Q = I-P$ is a co-analytic projection. In this paper, we use the Helson--Szegő type set $(HS)(r)$ to establish the condition of $a, b, c$ and $d$ satisfying $\|Af\|_2 \geq \|Bf\|_2$ for all $f$ in $\mathcal{P}$. If $a, b, c$ and $d$ are bounded measurable functions, then $A$ and $B$ are bounded operators, and this is equivalent to that $B$ is majorized by $A$ on $L^2$, i.e., $A^*A \geq B^*B$ on $L^2$. Applications are then presented for the majorization of singular integral operators on weighted $L^2$ spaces, and for the normal singular integral operators $aP + bQ$ on $L^2$ when $a-b$ is a complex constant.