A T1 theorem and Calderón–Zygmund operators in Campanato spaces on domains

Abstract

Given a Lipschitz domain D⊂double-struckRd, a Calderón–Zygmund operator T and a modulus of continuity ω(x), we solve the problem when the truncated operator TDf=Tfalse(fχDfalse)χD sends the Campanato space scriptCωfalse(Dfalse) into itself. The solution is a T1 type sufficient and necessary condition for the characteristic function χD of D: false(TχDfalse)χD∈scriptCtrueω∼false(Dfalse),whereω∼false(xfalse)=ω(x)1+∫x1ωfalse(tfalse)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 truncated Calderón–Zygmund operator TD with an even kernel is bounded on scriptCωfalse(Dfalse), provided D is a C1,trueω∼‐smooth domain.

A T(P) theorem for Zygmund spaces on domains

“…A related T(1) theorem for Hermit-Calderón-Zygmund operators is proven in [2]. An extension of Theorem 1.1 to weakly smooth spaces between Lip α (D) and BMO(D), that is, for the integer order n = 0, is obtained in [15].…”

“…Let ω be a modulus of continuity of order n ∈ N. To prove the desired equivalence for different values of the parameter p, 1 ≤ p ≤ ∞, one may apply ideas from [3,11]; see also [15,Proposition A.1].…”

“…Let P Q be a polynomial of near best approximation for f in the cube Q. Consider the following auxiliary functions (see, for example, [5,6,15] for similar arguments):…”

Calderón-Zygmund operators on Zygmund spaces on domains

“…We have already extended this theorem on the range of spaces of weak smoothness between Lip α (D) and BMO(D) in [17]. Our main purpose is to extend Theorem 1 on the Zygmund spaces with higher orders of smoothness.…”

Singular Integral Operators in Zygmund Spaces on Domains

“…Theorem 1.1 is extended in [11] to certain spaces of zero smoothness between Lip α (D) and BMO(D). Here we extend Theorem 1.1 for the higher smoothness general Zygmund spaces.…”