Anisotropic Hardy-Lorentz spaces and their applications

Abstract

Let p ∈ (0, 1], q ∈ (0, ∞] and A be a general expansive matrix on R n . The authors 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 realvariable characterizations in terms of the atomic or the molecular decompositions, the radial or the non-tangential maximal functions, or the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on R n . As applications, the authors firstA (R n ) and H p2,q2A (R n ) with 0 < p 1 < p < p 2 < ∞ and q 1 , q, q 2 ∈ (0, ∞], and also between H p,q1A (R n ) andA (R n ) with p ∈ (0, ∞) and 0 < q 1 < q < q 2 ≤ ∞ in the real method of interpolation. The authors then establish a criterion on the boundedness of sublinear operators from H p,q A (R n ) into a quasi-Banach space; moreover, the authors obtain the boundedness of δ-type Calderón-Zygmund operators from, 1] and q ∈ (0, ∞], as well as the boundedness of some Calderón-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.

Real-variable characterizations of new anisotropic mixed-norm Hardy spaces

“…From this and an argument similar to that used in the proof of [41, (5.13)], it follows that there exists a positive constant C 5 such that, for any x ∈ (B k 0 +4ω ) ∁ , By this and an argument similar to that used in Step 1 of the proof of [27, Theorem 5.9] (see also [41]), we further conclude that there exists a positive constant C 6 , independent of f , such that h/C 6 is a ( p, ∞, s)-atom and also a ( p, r, s)-atom for any p ∈ (0, ∞) n , r ∈ (max{p + , 1}, ∞] and s as in (3.1).…”

“…For I, by the boundedness of M N on L q (R n ) with q ∈ (1, ∞] (see [41,Remark 2.10]) and the fact that r ∈ (max{p + , 1}, ∞], we conclude that…”

“…On another hand, for any r ∈ (max{p + , 1}, ∞), from an argument similar to that used in Step 2 of the proof of [27,Theorem 5.9] (see also [40,41]), it follows that ℓ (K) converges to ℓ in L r (R n ) as K → ∞. This further implies that, for any given η ∈ (0, 1), there exists a K ∈ [ k + 1, ∞) ∩ Z large enough, depending on η, such that (ℓ − ℓ (K) )/η is a ( p, r, s)-atom and hence…”

“…An operator T from Y to B γ is said to be B γ -sublinear if there exists a positive constant C such that, for any K ∈ N, [36,60,62] Next, using the obtained boundedness criterion, Theorem 8.2, we establish the boundedness of anisotropic Calderón-Zygmund operators from H p A (R n ) to itself [or to L p (R n )]. We begin with recalling the notion of anisotropic convolutional δ-type Calderón-Zygmund operators from [41]. Definition 8.5.…”

“…Later on, Bownik et al [7] further extended the anisotropic Hardy space to the weighted setting. For more progresses about the real-variable theory of function spaces in this anisotropic setting, we refer the reader to [1,37,41,42,43,55]. Nowadays, the anisotropic dilation on Euclidean space R n has proved useful not only in developing the theory of various function spaces, but also in some other fields such as partial differential equations (see, for instance, [30]) and wavelet theory (see, for instance, [6,63]).…”