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arXiv · 1711.09303

T1 theorem for Campanato spaces on domains

Andrei V. Vasin
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Abstract

Given a Lipschitz domain $D\subset \mathbb{R}^d,$ a Calderón-Zygmund operator $T$ and a modulus of continuity $ω(x),$ we solve a problem when the restricted operator $T_Df=T(fχ_D)χ_D$ sends the Campanato space $\mathcal{C}_ω(D)$ into itself. The solution is a T1 type sufficient and necessary condition for the characteristic function $χ_D$ of $D$: $$(Tχ_D)χ_D \in \mathcal{C}_{\tildeω}(D),$$ assumed $\tildeω(x)= ω(x)/\int_x^1 ω(t)dt/t.$ 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 $\mathcal{C}_ω(D),$ provided $D$ be $C^{1,\tildeω}-$smooth domain. This result is sharp.

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Andrei V. Vasin. 2017-11-25. T1 theorem for Campanato spaces on domains. https://arxiv.org/abs/1711.09303

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