Hp continuity properties of Calderón-Zygmund-type operators

Álvarez, Josefina; Milman, Mario

doi:10.1016/0022-247x(86)90290-8

Cited by 71 publications

(49 citation statements)

References 12 publications

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“…Similar considerations apply to the Marcinkiewicz integral (see [8] and [6]). Finally, when p < 1, our results cover, for instance, the δ-CZ operators satisfying T * (1) = 0 discussed by Alvarez and Milman (see [2]). These operators, as well as a more general related class introduced in [14], preserve H p and H p,∞ for n/(n + δ) < p ≤ 1, and consequently, by Theorem 2.5, they also preserve H p,q for p in that same range, and q > p.…”

Section: Thussupporting

confidence: 71%

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The Hardy–Lorentz spaces H^p,q(Rⁿ)

Abu-Shammala

Torchinsky

2007

Studia Math.

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Section: Thussupporting

confidence: 71%

“…First, we construct an essentially optimal atomic decomposition; Parilov has obtained independently this result for H 1,q when 1 ≤ q (see [13]). Also, R. Fefferman and Soria gave the atomic decomposition of Weak H 1 (see [10]), and Alvarez the atomic decomposition of Weak H p , 0 < p < 1 (see [1]). …”

mentioning

confidence: 99%

The Hardy–Lorentz spaces H^p,q(Rⁿ)

Abu-Shammala

Torchinsky

2007

Studia Math.

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“…of the classical δ-type Calderón-Zygmund operator, respectively, from H p (R n ) to H p (R n ) and from H p (R n ) to L p (R n ) for δ ∈ (0, 1] and p ∈ ( n n+δ , 1], which is a well-known result (see, for example, [5,65,49]). …”

Section: ])mentioning

confidence: 73%

“…In 1994,Álvarez [3] considered the Calderón-Zygmund theory related to H p,∞ (R n ) with p ∈ (0, 1], while Liu [45] studied the weak Hardy spaces H p,∞ (R n ) with p ∈ (0, 1] on homogeneous groups. Nowadays, it is well known that the weak Hardy spaces H p,∞ (R n ), with p ∈ (0, 1], play a key role when studying the boundedness of operators in the critical case; see, for example, [3,4,25,33,26,72,27]. Moreover, it is known that the weak Hardy spaces H p,∞ (R n ) are special cases of the Hardy-Lorentz spaces H p,q (R n ) which, when p = 1 and q ∈ (1, ∞), were introduced and investigated by Parilov [56].…”

Section: Introductionmentioning

confidence: 99%

Anisotropic Hardy-Lorentz spaces and their applications

Yang

Yuan

2016

Sci. China Math.

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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.

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“…When T is a CalderonZygmund singular integral operator, Perez [21], Pluszynski [20] and Alvarez [1] showed that [b, T] does not map H p (R n ) into L P (R"). However, Perez [21] proved that 7i, m maps the modified spaces //^'"(R") to L\W), Alvarez [1] obtained that [b, T] maps H£\°°0(R n ) to //(IT), and Long and Wang [15] proved that T b , m maps Hg£ s (R") to V (R") and to H" (R n ). In this paper, we will extend these results to the higher order commutators [5,9,11,12,[16][17][18][19]).…”

Section: Obviously T Bi0 = T T B J = [B T]mentioning

confidence: 99%