Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces

By The Anh Bui, Xuan Thinh Duong. 2010

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One defines a non-homogeneous space (X,μ) as a metric space equipped with a non-doubling measure μ so that the volume of the ball with center x, radius r has an upper bound of the form rn for some n>0. The aim of this paper is to study the boundedness of Calderón–Zygmund singular integral operators T on various function spaces on (X,μ) such as the Hardy spaces, the Lp spaces, and the regularized BMO spaces. This article thus extends the work of X. Tolsa (Math. Ann. 319:89–149, 2011) on the non-homogeneous space (ℝn,μ) to the setting of a general non-homogeneous space (X,μ). Our framework of the non-homogeneous space (X,μ) is similar to that of Hytönen (2011) and we are able to obtain quite a few properties similar to those of Calderón–Zygmund operators on doubling spaces such as the weak type (1,1) estimate, boundedness from Hardy space into L1, boundedness from L∞ into the regularized BMO, and an interpolation theorem. Furthermore, we prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderón–Zygmund decomposition on the non-homogeneous space (X,μ), and use this decomposition to show the boundedness of the maximal operators in the form of a Cotlar inequality as well as the boundedness of commutators of Calderón–Zygmund operators and BMO functions.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/fd167db35c2e8418b3670b3434f50ad49fda4542

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In Applied Mathematics

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2. Dachun Yang
3. Xing Fu
4. Yoshihiro Sawano
5. Jun Liu
6. Ciqiang Zhuo
7. Yangyang Zhang
8. Yuan Zhou

Citations in the corpus

• GENERALIZED FRACTIONAL INTEGRALS AND THEIR COMMUTATORS OVER NON-HOMOGENEOUS METRIC MEASURE SPACES (2013, methodology): “-homogeneous metric measure spaces. Recently, many classical results have been proved still valid if the underlying spaces are replaced by the non-homogeneous metric measure spaces (see, for example, [18, 22, 2, 19, 20, 21, 25, 8, 24]). It is now also known that the theory of the singular integral o”