Littlewood–Paley Theory and the T(1) Theorem with Non-doubling Measures
✍ Scribed by Xavier Tolsa
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 348 KB
- Volume
- 164
- Category
- Article
- ISSN
- 0001-8708
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✦ Synopsis
Let m be a Radon measure on R d which may be non-doubling. The only condition that m must satisfy is m(B(x, r)) [ Cr n , for all x ¥ R d , r > 0, and for some fixed 0 < n [ d. In this paper, Littlewood-Paley theory for functions in L p (m) is developed. One of the main difficulties to be solved is the construction of ''reasonable'' approximations of the identity in order to obtain a Caldero ´n type reproducing formula. Moreover, it is shown that the T(1) theorem for n-dimensional Caldero ´n-Zygmund operators, without doubling assumptions, can be proved using the Littlewood-Paley type decomposition that is obtained for functions in L 2 (m), as in the classical case of homogeneous spaces.}