T1 theorem for Campanato spaces on domains

Abstract

Given a Lipschitz domain $D\subset {\mathbb{R}}^d,$ a Calderón-Zygmund operator $T$ and a modulus of continuity $\omega (x),$ we solve a problem when the restricted operator $T_{D}f=T(f{\chi }_D){\chi }_D$ sends the Campanato space ${\mathcal{C}}_{\omega }(D)$ into itself. The solution is a T1 type sufficient and necessary condition for the characteristic function ${\chi }_D$ of $D$: $$(T{\chi }_D){\chi }_D\in {\mathcal{C}}_{\tilde{\omega }}(D),$$ assumed $\tilde{\omega }(x)=\omega (x)/{\int }_x^1\omega (t)dt/t.$ To check the hypotheses of T1 theorem we need extra restrictions on both the boundary of $D$ and the operator $T.$ It is proved that the restricted Calderón-Zygmund operator $T_D$ with the even kernel is bounded on ${\mathcal{C}}_{\omega }(D),$ provided $D$ be $C^{1,\tilde{\omega }}-$smooth domain. This result is sharp.

20 pages

References in corpus (1)

• A T(P) theorem for Sobolev spaces on domains