ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 42, 43, 44 Meskhi, Alexander, A. Razmadze Mathematical Institute Georgian Academy of Sciences, Georgia Solution of some weight problems for the Riemann-Liouville and Weyl Operators defined on SHT Let (X, d, μ) be a space of homogeneous type (SHT). Suppose that

R_{α}f(x) = ∫_{\{d(x_{0},y)<d(x_{0},x)\}} \frac{f(y)}{(μB(x, d(x, y)))^{1−α}} dμ

(W_{α}f(x) = ∫_{\{d(x_{0},y)>d(x_{0}.x)\}} \frac{f(y)}{(μB(x, d(x, y)))^{1−α}} dμ)

be Riemann-Liouville (Weyl) transform, where x_{0} ∈ X, α > 0. The necessary and sufficient conditions for the weight function v guaranteeing the boundedness of the operator R_{α} from L^{p}(X) to L_{v}^{q}(X) (1 < p ≤ q < ∞, α > \frac{1}{p}) are found. The appropriate problem for the operator W_{α} is solved.