Hardy spaces for ball quasi-Banach function spaces

Sawano, Yoshihiro; Ho, Kwok Pun Victor; Yang, Dachun; Yang, Sibei

doi:10.4064/dm750-9-2016

Cited by 100 publications

(221 citation statements)

References 0 publications

Supporting

Mentioning

214

Contrasting

Order By: Relevance

“…(We note in passing that extrapolation into the scale of Musielak-Orlicz spaces was considered separately in [6].) Additional examples Hardy spaces where our approach might be applicable are given in [33].…”

Section: Extensions To Other Banach Function Spacesmentioning

confidence: 99%

A new approach to norm inequalities on weighted and variable Hardy spaces

Cruz-Uribe¹,

Moen²,

Nguyen³

2020

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

We give new proofs of Hardy space estimates for fractional and singular integral operators on weighted and variable exponent Hardy spaces. Our proofs consist of several interlocking ideas: finite atomic decompositions in terms of L ∞ atoms, vector-valued inequalities for maximal and other operators, and Rubio de Francia extrapolation. Many of these estimates are not new, but we give new and substantially simpler proofs, which in turn significantly simplifies the proofs of the Hardy spaces inequalities.

show abstract

Section: Extensions To Other Banach Function Spacesmentioning

confidence: 99%

A new approach to norm inequalities on weighted and variable Hardy spaces

Cruz-Uribe¹,

Moen²,

Nguyen³

2020

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

show abstract

“…In this section, we recall some notions and preliminary facts on ball quasi-Banach function spaces defined in [67].…”

Section: Preliminariesmentioning

confidence: 99%

“…where · X ′ is called the associate norm of · X (see, for instance, [ (i) By [67,Proposition 2.3], we know that, if X is a ball Banach function space, then its associate space X ′ is also a ball Banach function space.…”

Section: Preliminariesmentioning

confidence: 99%

“…To seal this gap, we employ a different method from [52]. Via borrowing some ideas from [67], we first introduce the weak tent space associated to X and establish its atomic characterization (see Theorem 3.7 below), which is a key tool to decompose a distribution f into a sequence of molecules. Using the some ideas from Liu [54], we prove that the sum of the sequence of molecules indeed converges in the sense of distribution [see (3.27) below].…”

mentioning

confidence: 99%

“…Using the some ideas from Liu [54], we prove that the sum of the sequence of molecules indeed converges in the sense of distribution [see (3.27) below]. Then, applying some ideas used in the proof of [67,Theorem 3.21], we prove that, under some even weaker assumptions on the Lusin area function, the distribution which the sum of the aforementioned sequence of molecules converges to equals f in the sense of distribution, and hence seal the aforementioned gap existed in the proof of [52,Theorem 4.5]. To obtain the Littlewood-Paley g-function characterization of W H X (R n ), we first establish a discrete Calderón reproducing formula under some weak assumptions (see Lemma 3.19(i) below), and then we apply an approach initiated by Ullrich [71] and further developed by Liang et al [51], which provides one way to control the W X norm of the Lusin area function by the corresponding norm of the Littlewood-Paley g-function, via a key technical lemma (see Lemma 3.21 below) and an auxiliary Peetre type square function g a, * ( f ).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation

et al. 2019

Self Cite

View full text Add to dashboard Cite

Let X be a ball quasi-Banach function space on R n . In this article, assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, the authors establish various Littlewood-Paley function characterizations of WH X (R n ) under some weak assumptions on the Littlewood-Paley functions. The authors also prove that the real interpolation intermediate space (H X (R n ), L ∞ (R n )) θ,∞ , between the Hardy space associated with X, H X (R n ), and the Lebesgue space L ∞ (R n ), is WH X 1/(1−θ) (R n ), where θ ∈ (0, 1). All these results are of wide applications. Particularly, when X := M p q (R n ) (the Morrey space), X := L p (R n ) (the mixed-norm Lebesgue space) and X := (E q Φ ) t (R n ) (the Orlicz-slice space), all these results are even new; when X := L Φ ω (R n ) (the weighted Orlicz space), the result on the real interpolation is new and, when X := L p(·) (R n ) (the variable Lebesgue space) and X := L Φ ω (R n ), the Littlewood-Paley function characterizations of WH X (R n ) obtained in this article improves the existing results via weakening the assumptions on the Littlewood-Paley functions.

show abstract

Real‐variable characterizations of local Hardy spaces on spaces of homogeneous type

Yang

Yuan

2021

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

Let (X,d,μ) be a space of homogeneous type, with upper dimension μ, in the sense of R. R. Coifman and G. Weiss. Let η be the Hölder regularity index of wavelets constructed by P. Auscher and T. Hytönen. In this article, the authors introduce the local Hardy space h∗,pfalse(Xfalse) via local grand maximal functions and also characterize h∗,pfalse(Xfalse) via local radial maximal functions, local non‐tangential maximal functions, local atoms and local Littlewood–Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of h∗,pfalse(Xfalse). As an application, the authors give the dual spaces of h∗,pfalse(Xfalse) when p∈(ω/(ω+η),1), which further completes the result of G. Dafni and H. Yue on the dual space of h∗,1false(Xfalse). This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of Hnormalcw1false(Xfalse) when μ(X)<∞.

show abstract

Hardy spaces for ball quasi-Banach function spaces

Cited by 100 publications

References 0 publications

A new approach to norm inequalities on weighted and variable Hardy spaces

A new approach to norm inequalities on weighted and variable Hardy spaces

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation

Real‐variable characterizations of local Hardy spaces on spaces of homogeneous type

Contact Info

Product

Resources

About