A Variant of the Notion of a Space of Homogeneous Type
Document Type
Article
Publication Date
8-15-1995
Identifier/URL
40901148 (Pure)
Abstract
The theory of spaces of homogeneous type, introduced by R. R.Coifman and G. Weiss [{\it Analyseharmonique noncommutative sur certains espaces homogènes},Lecture Notes in Math., 242, Springer, Berlin, 1971; MR0499948 (58 \#17690) ], provides an important extension of the classicalCalderón-Zygmund theory of singular integrals onRn. It applies, for instance, to compact Riemannianmanifolds, and to rather restricted classes of embedded noncompactmanifolds. In general, the theory does not apply to maximalfunctions and singular integral operators on curves and surfaces: inthose circumstances, there is usually a natural family of open balls,and a natural measure that fails one or more of the conditions for thespace to be of homogeneous type; one would like to adapt the theoryin an effective way to treat such cases. \par In the present paper, the authors develop an extension of the notionof a space of homogeneous type. This is particularlywell suited to the kind of harmonic analysis and operator theoryon curves and surfaces and nilpotent Lie groups that the authorshave been undertaking, severally and jointly, over recent years [see,e.g., A. Carbery et al., Duke Math. J. {\bf 59} (1989), no.~3,675--700; MR1046743 (91m:42017) ; A. Carbery, S. Wainger and J. R. Wright, J. Amer. Math.Soc. {\bf 8} (1995), no.~1, 141--179; MR1273412 (95g:43010) ]. It is assumed that thereexist a family of continuous ``distance'' functionsρj (j∈Z), and a regular Borel measure μ, on the locallycompact spaceX. There is a corresponding family of open balls B(x,j,r)={y∈X: ρj(x,y)0, there is a number A(R) such that if ρj(x,z)≤Rand ρj(z,y)≤R, then ρj(x,y)≤A(R)[ρj(x,z)+ρj(z,y)]. The distance functions are assumed to be decreasing(ρj≥ρj+1) and to satisfy a condition of localmeasure equivalence: for each R>0, there exists A(R) suchthat if r≤R, thenμ(B(x,j+1,r))≤A(R)μ(B,j,r) for all x and all j∈Z.\par The authors establish boundedness of the associated maximaloperator MR in which averages are taken over ballsof size R. Extensions of singular integral theory are given forkernels that satisfy a variant of the Hörmander condition. Applications to homogeneous groups are given: in the simplestinstance, the ball B(x,j,r) is the image of the ball B(x,1,r) undera dilation of size 2j. More general circumstances are alsotreated, in which a given pseudo-metricρ in Rn is subjected to invertible lineartransformations A(t) of the underlying space, and thetransformations satisfy the Rivière condition: for someε>0 and all s≥t, ρ(A−1(s)A(t)x,A−1(s)A(t)y)≤(t/s)ερ(x,y).
Repository Citation
Carbery, A.,
Vance, J.,
Wainger, S.,
& Wright, J.
(1995). A Variant of the Notion of a Space of Homogeneous Type. Journal of Functional Analysis, 132 (1), 119-140.
https://corescholar.libraries.wright.edu/math/514
DOI
10.1006/jfan.1995.1102
