Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H^(p,q)_A(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 ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H^(p1,q1)_A(Rn) and H^(p2,q2)_A(R^n) with 0 < p1 < p < p2 < ∞ and q1, q, q2 ∈(0, ∞], and also between H^(p,q1)_A(Rn) and H^(p,q2)_A(R^n) with p ∈(0, ∞)and 0 < q1 < q < q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H^(p,q)_A(R^n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H^(p,∞)_A(R^n) to the weak Lebesgue space L^(p,∞)(R^n)(or to H^p_A(R^n)) in the ln λcritical case, from H^(p,q)_A(R^n) to L^(p,q)(R^n)(or to H^(p,q)_A(R^n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H^(p,q)_A(R^n) to L^(p,∞)(R^n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.
In this article, the authors establish several equivalent characterizations of fractional Haj lasz-Morrey-Sobolev spaces on spaces of homogeneous type in the sense of Coifman and Weiss.
In this article, the authors characterize pointwise multipliers for localized MorreyCampanato spaces, associated with some admissible functions on RD-spaces, which include localized BMO spaces as a special case. The results obtained are applied to Schrdinger operators and some Laguerre operators.
We study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e. g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the ±-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasis is given to the quasi-Banach case and to the interpolation property.
Let(X,d,μ)be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of Hyto¨nen.We prove that the L p(μ)-boundedness with p∈(1,∞)of the Marcinkiewicz integral is equivalent to either of its boundedness from L1(μ)into L1,∞(μ)or from the atomic Hardy space H1(μ)into L1(μ).Moreover,we show that,if the Marcinkiewicz integral is bounded from H1(μ)into L1(μ),then it is also bounded from L∞(μ)into the space RBLO(μ)(the regularized BLO),which is a proper subset of RBMO(μ)(the regularized BMO)and,conversely,if the Marcinkiewicz integral is bounded from L∞b(μ)(the set of all L∞(μ)functions with bounded support)into the space RBMO(μ),then it is also bounded from the finite atomic Hardy space H1,∞fin(μ)into L1(μ).These results essentially improve the known results even for non-doubling measures.
