ISSN 1439-8516 CN 11-2039/O1
those 2π-periodic, real-valued functions f on R, which are analytic in Sβ := {z ∈ C : |Im z| < β} and satisfy the restriction |f(r)(z)| ≤ 1, z ∈ Sβ. The optimal quadrature formulae about information composed of the values of a function and its kth (k = 1, . . . , r - 1) derivatives on free knots for the classes are obtained, and the error estimates of the optimal quadrature formulae are exactly determined.
R+ is the (ax + b)-group, d is the left-invariant Riemannian metric and ρ is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces {Xs(Y)}0<s≤∞ and the BMO-type spaces {BMO(Y, s)}0<s<∞. Let H1(Y) be the known atomic Hardy space and L01(Y) the subspace of f ∈ L1(Y) with integral 0. The authors prove that the dual space of Xs(Y) is BMO(Y, s) when s ∈ (0,∞), Xs(Y) = H1(Y) when s ∈ (0, 1], and X∞(Y) = L01(Y) (or L1(Y)). As applications, the authors show that if T is a linear operator bounded from H1(Y) to L1(Y) and from L1(Y) to L1,∞(Y), then for all r ∈ (1,∞) and s ∈ (r,∞], T is bounded from Xr(Y) to the Lorentz space L1,s(Y), which applies to the Calderòn-Zygmund operator on (Rn, |·|, μ), the imaginary powers of the Ornstein-Uhlenbeck operator on (Rn, |·|, dγ) and the spectral operator associated with the spectral multiplier on (S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.
