Atomic hardy-type spaces between H (1) and L (1) on metric spaces with non-doubling measures

作者：Liu, Li Guang; Yang, Da Chun^*; Yang, Dong Yong

来源：Acta Mathematica Sinica-English Series, 2011, 27(12): 2445-2468.

DOI：10.1007/s10114-011-9118-7

• Non-doubling measure
• polynomial growth
• Gauss measure
• (ax plus b)-group
• atomic Hardy-type space
• BMO-type space
• operator

Let (Y, d, d lambda) be ( R-n, | . |, mu), where | . | is the Euclidean distance, mu is a nonnegative Radon measure on R-n satisfying the polynomial growth condition, or the Gauss measure metric space (R-n, | . |, d.), or the space (S, d, rho), where S = R-n alpha R+ is the (ax + b)-group, d is the left-invariant Riemannian metric and rho 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 {X-s(Y)}(0 <=epsilon <= 8) and the BMO-type spaces {BMO(Y, s)}(0<s<infinity). Let H-1(Y) be the known atomic Hardy space and L-0(1)(Y) the subspace of integral is an element of L-1(Y) with integral 0. The authors prove that the dual space of X-s(Y) is BMO(Y, s) when s is an element of (0,infinity), X-s(Y) = H-1(Y) when s is an element of (0, 1], and X-infinity(Y) = L-0(1)(Y) (or L-1(Y)). As applications, the authors show that if T is a linear operator bounded from H 1 ( Y) to L 1 ( Y) and from L-1 (Y) to L-1,L-infinity(Y), then for all r is an element of (1,infinity) and s is an element of(r,infinity], T is bounded from X-r(Y) to the Lorentz space L-1,(s) (Y), which applies to the Calderon-Zygmund operator on (R-n, | . |, mu), the imaginary powers of the Ornstein- Uhlenbeck operator on (R-n, | . |, d gamma) and the spectral operator associated with the spectral multiplier on (S, d, rho). All these results generalize the corresponding results of Sweezy, Abu- Shammala and Torchinsky on Euclidean spaces.

• 出版日期2011-12
• 单位北京师范大学