Nonclassical weighted norm estimates for some Calderón–Zygmund operators on the plane

P. P. Kargaev

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 22, Tome 217 (1994), pp. 74-82

Résumé

Let $\mu$ be a Borel measure with a compact support $F\subset\mathbb C$, $\rho$ be the distance from the set $F$; $$ A_K(f)(z)=\int_FK(\zeta,z)f(\zeta)\,dm(\zeta),\qquad z\in\mathbb C\setminus F, $$ where $K(\zeta,z)=(\zeta-z)^{-2}$ or $K(\zeta,z)=(|\zeta-z|(\zeta-z))^{-1}$ and $m$ is the Lebesque measure. Let $\psi\colon(0,+\infty)\to\mathbb R_+$ be a nondecreasing positive function, $\Phi(z)=\psi(\rho(z))\rho(z)$, $z\in\mathbb C\setminus F$. We prove that under some additional assumptions on p, the operator $A_K$ is bounded from $L^2(\mu)$ to $L^2(\Phi m)$ if and only if $$ \int^1_0\frac{\psi(t)}t\,dt+\int_1^{+\infty}\frac{\psi(t)}{t^2}\,dt<+\infty. $$ This means that the interference effect is not observed in such situations. Bibliography: 4 titles.