A proof of the weak (1,1) inequality for singular integrals with non doubling measures based on a Calderón-Zygmund decomposition

By Xavier Tolsa. 2000

91 citations recorded by the source corpus.

Given a doubling measure $\mu$ on ${\mathbb R}^d$, it is a classical result of harmonic analysis that Calderon-Zygmund operators which are bounded in $L^2(\mu)$ are also of weak type (1,1). Recently it has been shown that the same result holds if one substitutes the doubling condition on $\mu$ by a mild growth condition on $\mu$. In this paper another proof of this result is given. The proof is very close in spirit to the classical argument for doubling measures and it is based on a new Calderon-Zygmund decomposition adapted to the non doubling situation.

View Paper: https://www.semanticscholar.org/paper/300e57177342abd3179f64088fce92ebd63fc9ed

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

1. Tao Mei
2. Javier Parcet
3. Dachun Yang
4. Guoen Hu
5. Joan Verdera
6. Yan Meng
7. Xavier Tolsa
8. Tuomas Hytönen

Citations in the corpus

• Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, background): “The Calderón-Zygmund decomposition in [37] looks like the most difficult step and an interesting problem.”
• The fall of the doubling condition in Calderón-Zygmund theory (2002, background): “and [ T2 ]), Cotlar’s inequality for the maximal singular integral ([NTV1] and [T3]) and many others ([MMNO], [OP] and [GM]).”
• Multilinear Commutators of Singular Integrals with Non Doubling Measures (2005, methodology): “applying the Calder´ on-Zygmund decomposition to f at level λ (see Lemma 2.4 in [ 13 ]), we can obtain a sequence of cubes {Qj} with bounded overlaps, that is, �”
• Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces (2010, unknown): “See, for example, [10–12, 15], and [16].”
• Littlewood–Paley Theory and the T(1) Theorem with Non-doubling Measures (2000, background): “1 and the fact that HCZO’s are bounded on L(m) (see [10], and also [18] for a different proof).”
• The semiadditivity of continuous analytic capacity and the inner boundary conjecture (2004, background): “Next lemma deals with a now quite well known dualization of the usual weak (1;1) inequality for the Cauchy transform, which is a consequence of its L2(„) boundedness (see either [To2], [NTV1] or [ To3 ], for example).”