Anisotropic Hardy-Lorentz spaces and their applications

作者：Liu Jun; Yang DaChun^*; Yuan Wen

来源：Science China Mathematics, 2016, 59(9): 1669-1720.

DOI：10.1007/s11425-016-5157-y

• Lorentz space
• anisotropic Hardy-Lorentz space
• expansive matrix
• Calderon reproducing formula
• grand maximal function
• atom
• molecule
• Calderon-Zygmund operator

Let p is an element of (0, 1], q is an element of (0, infinity] and A be a general expansive matrix on R-n. We introduce the anisotropic Hardy-Lorentz space H-A(p, q) (R-n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the infinity-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on R-n. As applications, we first prove that H-A(p, q) (R-n) is an intermediate space between H-A(p1, q1) (R-n) and H-A(p2, q2) (R-n) with 0 < p(1) < p < p(2) < infinity and q(1), q, q(2) is an element of (0,infinity], and also between H-A(p, q1) (R-n) and H-A(p, q2) (R-n) with p is an element of (0,infinity] and 0 < q(1) < q < q(2) <= infinity in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H-A(p, q) (R-n) into a quasi-Banach space; moreover, we obtain the boundedness of delta-type Calderon-Zygmund operators from H-A(p A) (R-n) to the weak Lebesgue space L-p,L-infinity(R-n) (or to H-A(p,infinity) (R-n)) in the critical case, from H-A(p, q) (R-n) to L-p,L- q (R-n) (or to H-A(p, q) (R-n)) with delta is an element of (0, ln lambda/ln b], p is an element of (1/1 +delta] and q is an element of (0, infinity], as well as the boundedness of some Calderon-Zygmund operators from H-A(p, q) (R-n) to L-p,L-infinity(R-n), where b := vertical bar det A vertical bar, lambda(-):= min{vertical bar lambda vertical bar lambda is an element of sigma(A)} and sigma(A) denotes the set of all eigenvalues of A.

• 出版日期2016-9
• 单位北京师范大学