$BMO, H^1$, and Calderón-Zygmund operators for non doubling measures

By Xavier Tolsa. 2000

196 citations recorded by the source corpus.

Abstract. Given a Radon measure $\mu$ on ${\mathbb R}^d$, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties which hold for the classical space $BMO(\mu)$ when $\mu$ is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming $\mu$ doubling. For instance, Calderón-Zygmund operators which are bounded on $L^2(\mu)$ are also bounded from $L^\infty(\mu)$ into the new BMO space. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space $H^1$. Using a sharp maximal operator it is shown that operators which are bounded from $L^\infty(\mu)$ into the new BMO space and from its predual $H^1$ into $L^1(\mu)$ must be bounded on $L^p(\mu)$, $1< p< infty$. From this result one can obtain a new proof of the T(1) theorem for the Cauchy transform for non doubling measures. Finally, it is proved that commutators of Calderón-Zygmund operators bounded on $L^2(\mu)$ with functions of the new BMO are bounded on $L^p(\mu), 1< p < \infty$.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e8c94fa77c89818195a02f6ed3ecc55e504a35b9

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Citations in the corpus

Littlewood–Paley Theory and the T(1) Theorem with Non-doubling Measures (2000, methodology): “This operator is bounded on Lp(� ), 1 To3 , Section 6]).”
Morrey Spaces for Non–doubling Measures (2005, unknown): “[2, 3]). However, recently it has been shown that many results in this theory also hold without the doubling assumption; see [4, 5, 6 ] and many other literatures.”
Boundedness properties of fractional integral operators associated to non-doubling measures (2002, methodology): “with an n−dimensional measure � , n ≤ d. It is for this context for which X. Tolsa has introduced (in [ To3 ]) the space “regular”BMO, although his definition makes sense, in principle, in our general setting.”
Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces (2010, methodology): “Let us recall that the main techniques used in [15] rely on the Besicovitch covering lemma and the construction of the (α,β)-doubling balls in R.”
A proof of the weak (1,1) inequality for singular integrals with non doubling measures based on a Calderón-Zygmund decomposition (2000, methodology): “The Calder´ on-Zygmund decomposition mentioned above has been obtained in Lemma 7.3 of [ To4 ] and in that paper it has been used to show that if a linear operator is bounded from a suitable space of type H1 into L1(µ) and from L∞(µ) into a space of type BMO, then it is bounded in Lp(µ).”
SHARP MAXIMAL INEQUALITIES AND COMMUTATORS ON MORREY SPACES WITH NON-DOUBLING MEASURES (2007, unknown): “Our BMO here is a RBMO (regular bounded mean oscillation) introduced by X. Tolsa [ 9 ] which are the suitable substitutes for the classical spaces.”
H^1 and BMO for certain locally doubling metric measure spaces of finite measure (2008, background): “Finally, we recall that Hardy spaces and spaces of functions of bounded mean oscillation have recently been studied on various nondoubling metric measure spaces [MMNO, NTV, To , V]. We point out that our spaces are different and that they provide end-point estimates for singular integrals which do ”
The fall of the doubling condition in Calderón-Zygmund theory (2002, background): “The right notion of BMO in the non-homogeneous setting has been introduced in [ T4 ], where the H1-BMO duality is discussed thoroughly and some applications are given.”