Abstract
In 2019, Grafakos and Stockdale introduced an L^q mean Hörmander condition and proved a “limited-range” Calderón–Zygmund theorem. Comparing their theorem with the classical one, it requires weaker assumptions and implies the L^p boundedness for the “limited-range” instead of 1< p < infty . However, in this paper, we show that the L^q mean Hörmander condition is actually enough to obtain the L^p boundedness for all 1< p < infty even in the worst case q=1. We use a similar method to that used by Fefferman (Acta Math 124:9–36, 1970): form the Calderón–Zygmund decomposition with the bounded overlap property and approximate the bad part. Also we give a criterion of the L^2 boundedness for convolution type singular integral operators under the L^1 mean Hörmander condition.
Highlights
The Calderón–Zygmund theorem is a well-known tool to investigate the L p boundedness of singular integral operators. It was originally developed by Calderón and Zygmund [2] and later improved by Hörmander [6]
Suppose that T is bounded from L p0 (Rd ) to L p0,∞(Rd ) for some 1 < p0 < ∞ and its kernel K satisfies the Hörmander condition;
Suppose that T is bounded from L p0 (Rd ) to L p0,∞(Rd ) for some 1 < p0 < ∞ and its kernel K satisfies the Hq condition for some 1 ≤ q < p0 where q denotes the Hölder conjugate of q
Summary
The Calderón–Zygmund theorem is a well-known tool to investigate the L p boundedness of singular integral operators. It was originally developed by Calderón and Zygmund [2] and later improved by Hörmander [6]. Today it is usually stated as follows: Communicated by Rodolfo H.Torres.
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