Pseudo-localisation of singular integrals in \(L^p\). (English) Zbl 1223.42010

Hytönen, Tuomas P.

Rev. Mat. Iberoam. 27, No. 2, 557-584 (2011).

One of the keywords in this paper is “localization”. An elementary localization result was already implicit in the Calderón-Zygmund decomposition. Let \(b\) denote the bad part of \(f\) associated to a fixed \(\lambda>0\) and let \(\Sigma_{\lambda}\) be the level set where the dyadic Hardy-Littlewood maximal function \(M_d f > \lambda\). Then
\[ \int_{{\mathbb R}^n \setminus 2 \Sigma_{\lambda}} | Tb (x) | \,dx \leq C_n \| f \|_{L^1}, \]
where \(T\) is a standard Calderón-Zygmund operator.
In a study of non-commutative Calderón-Zygmund theory, J. Parcet [J. Funct. Anal. 256, No. 2, 509–593 (2009; Zbl 1179.46051)] established a new pseudo-localization principle for classical singular integrals: For any \(\lambda >0\), there exists a set \(\Sigma_{\lambda,f}\) such that
\[ \int_{{\mathbb R}^n \setminus \Sigma_{\lambda,f}} | Tf (x) | \,dx \lesssim 2^{-\gamma \lambda} \| f \|_{L^1} \]
and
\[ \bigg( \int_{{\mathbb R}^n \setminus \Sigma_{\lambda,f}} | Tf (x) |^2 \,dx \bigg)^{1/2} \lesssim (1+\lambda) 2^{-\gamma \lambda/4} \| f \|_{L^2}, \]
where \(\gamma \in (0,1]\) is the Hölder exponent from the standard estimates of the kernel of \(T\).
Parcet asked whether a pseudo-localization principle might hold in \(L^p (\mathbb{R}^n)\), and also pointed out the difficulty of this problem since no usual form of interpolation is directly applicable. The author proves the following: If \(1<p<\infty\), then
\[ \bigg( \int_{{\mathbb R}^n \setminus \Sigma_{\lambda,f}} | Tf (x) |^p\, dx \bigg)^{1/p} \lesssim (1+\lambda) 2^{-\lambda \min ( \gamma, 1/2, 1/p' )} \| f \|_{L^p}. \]
The proof deals with the full range of \(p \in (0,\infty)\) at once in a unified manner, without resorting to interpolation or duality arguments. The author uses various martingale techniques, which were originally developed to handle the difficulties arising in harmonic analysis of Banach space valued functions by T. Figiel [Geometry of Banach spaces, Proc. Conf., Strobl/Austria 1989, Lond. Math. Soc. Lect. Note Ser. 158, 95–110 (1991; Zbl 0746.47026)], T. R. McConnell [Probab. Math. Stat. 10, No. 2, 283–295 (1989; Zbl 0718.60050)] and himself. He says that their successful application also to the problem at hand displays the power of these methods even in the context of classical analysis.

Reviewer: Yasuo Komori (Numazu)

MSC:

42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
60G46	Martingales and classical analysis

Keywords:

Calderón–Zygmund operator; \(T1\) theorem; operations on the Haar basis; martingale

Citations:

Zbl 1179.46051; Zbl 0746.47026; Zbl 0718.60050

References:

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[3]	Hytönen, T.: The vector-valued non-homogeneous \(Tb\) theorem. Preprint available at · Zbl 1305.42015
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