T1 theorem for Campanato spaces on domains

2017/11/25 by Andrei V. Vasin, Vasin, Andrei V.
Mathematics · #Advanced Harmonic Analysis Research #Nonlinear Partial Differential Equations #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.1711.09303

Abstract

Given a Lipschitz domain D⊂ 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 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 ∈ C_ω(D), assumed ω(x)= ω(x)/∫_x¹ 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 C_ω(D), provided D be C^1,ω-smooth domain. This result is sharp.

Citations

• Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation

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