ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 42, 53 Chen, Jiecheng, Hangzhou University, P. R. China A distribution inequality on maximal and square functions on manifolds Suppose M be a complete connected Riemannian manifold with nonnegative Ricci curvatures, 0 < α, β < ∞, u(x, t) = P_{t}(f)(x) be the Poisson integral integral of f. The nontangential maximal function and the square function of f is defined by

S_{α}(f)(x) = (∫_{Γ_{α}(x)} |∇u(y, t)|^{2} tV_{x}^{−1}(t)dtdy)^{1/2}

N_{β}(f)(x) = sup {|u(y, t)| : (y, t) ∈ Γ_{β}(x)} .

We proved that there is a positive number c_{0} = c(α, β, n) ∈ (0, 1) and C = C(α, β, n) > 1, such that for f ∈ C_{c}(M), γ̇_{1} and γ_{2}> 2, λ > 0, we have

|{x : S_{α}(f) > γ_{1}λ, N_{α}(χ_{N_{β}(f)>λ/γ_{2}} ) < c_{0}}| ≤ Ce^{−(γ_{1}γ_{2})^{2}C^{−1}} |{x : S_{α}(f) > λ}| .

REFERENCES: C. Fefferman and E. M. Stein, H^{p} Spaces of Several Variables, Acta Math, 129(1972), 137-193.