Correction to: Weighted Estimates for Vector-Valued Intrinsic Square Functions and Commutators in the Morrey-Type Spaces

In this paper, the boundedness properties of vector-valued intrinsic square functions and their vector-valued commutators with \(BMO(\mathbb {R}^{n})\) functions are discussed. We first show the weighted strong-type and weak-type estimates of vector-valued intrinsic square functions in the Morrey-type spaces. Then, we obtain weighted strong-type estimates of vector-valued analogues of commutators in Morrey-type spaces. In the endpoint case, we establish the weighted weak \(L\log L\)-type estimates for these vector-valued commutators in the setting of weighted Lebesgue spaces. Furthermore, we prove weighted endpoint estimates of these commutator operators in Morrey-type spaces. In particular, we can obtain strong-type and endpoint estimates of vector-valued intrinsic square functions and their commutators in the weighted Morrey spaces and the generalized Morrey spaces.

In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we deal with the vector-valued intrinsic square functions, which were introduced recently by Wilson. In his fundamental work, Wilson established strong-type and weak-type estimates for vector-valued intrinsic square functions on weighted Lebesgue spaces. The goal of this paper is to extend his results to these weighted amalgam spaces. Moreover, we define vector-valued analogues of commutators with \(\mathrm {BMO}({\mathbb {R}}^n)\) functions, and obtain the mapping properties of vector-valued commutators on the weighted amalgam spaces as well. In the endpoint case, we also establish the weighted weak \(L\log L\)-type estimates for vector-valued commutators in the setting of weighted Lebesgue spaces.

We will obtain the strong type and weak type estimates for vector-valued analogues of intrinsic square functions in the weighted Morrey spacesLp,κ(w)when1≤p<∞,0<κ<1, and in the generalized Morrey spacesLp,Φfor1≤p<∞, whereΦis a growth function on(0,∞)satisfying the doubling condition.

In this article, the authors study the boundedness of the vector-valued inequality for the intrinsic square function and the boundedness of the scalar-valued intrinsic square function on variable exponents Herz spaces K˙ρ(·)α,q(·)(Rn). In addition, the boundedness of commutators generated by the scalar-valued intrinsic square function and BMO class is also studied on K˙ρ(·)α,q(·)(Rn).

Let φ : R n × [ 0 , ∞ ) → [ 0 , ∞ ) \varphi : \mathbb R^n\times [0,\infty )\to [0,\infty ) be such that φ ( x , ⋅ ) \varphi (x,\cdot ) is an Orlicz function and φ ( ⋅ , t ) \varphi (\cdot ,t) is a Muckenhoupt A ∞ ( R n ) A_\infty (\mathbb R^n) weight uniformly in t t . In this article, for any α ∈ ( 0 , 1 ] \alpha \in (0,1] and s ∈ Z + s\in \mathbb {Z}_+ , the authors establish the s s -order intrinsic square function characterizations of H φ ( R n ) H^{\varphi }(\mathbb R^n) in terms of the intrinsic Lusin area function S α , s S_{\alpha ,s} , the intrinsic g g -function g α , s g_{\alpha ,s} and the intrinsic g λ ∗ g_{\lambda }^* -function g λ , α , s ∗ g^\ast _{\lambda , \alpha ,s} with the best known range λ ∈ ( 2 + 2 ( α + s ) / n , ∞ ) \lambda \in (2+2(\alpha +s)/n,\infty ) , which are defined via L i p α ( R n ) \mathrm {Lip}_\alpha ({\mathbb R}^n) functions supporting in the unit ball. A φ \varphi -Carleson measure characterization of the Musielak-Orlicz Campanato space L φ , 1 , s ( R n ) {\mathcal L}_{\varphi ,1,s}({\mathbb R}^n) is also established via the intrinsic function. To obtain these characterizations, the authors first show that these s s -order intrinsic square functions are pointwise comparable with those similar-looking s s -order intrinsic square functions defined via L i p α ( R n ) \mathrm {Lip}_\alpha ({\mathbb R}^n) functions without compact supports, which when s = 0 s=0 was obtained by M. Wilson. All these characterizations of H φ ( R n ) H^{\varphi }(\mathbb R^n) , even when s = 0 s=0 , \[ φ ( x , t ) := w ( x ) t p for\ all t ∈ [ 0 , ∞ ) and x ∈ R n \varphi (x,t):=w(x)t^p\ \textrm {for\ all}\ t\in [0,\infty )\ \textrm {and}\ x\in {\mathbb R}^n \] with p ∈ ( n / ( n + α ) , 1 ] p\in (n/(n+\alpha ), 1] and w ∈ A p ( 1 + α / n ) ( R n ) w\in A_{p(1+\alpha /n)}(\mathbb R^n) , also essentially improve the known results.

In this chapter, for any α ∈ (0, 1] and \(s \in \mathbb{Z}_{+}\), we establish the s-order intrinsic square function characterizations of \(H^{\varphi }(\mathbb{R}^{n})\) by means of the intrinsic Lusin area function S α, s , the intrinsic g-function g α, s or the intrinsic g λ ∗-function g λ, α, s ∗ with the best known range λ ∈ (2 + 2(α + s)∕n, ∞), which are defined via \(\mathrm{Lip}_{\alpha }(\mathbb{R}^{n})\) functions supporting in the unit ball. A φ-Carleson measure characterization of the Musielak-Orlicz Campanato space \(\mathcal{L}_{\varphi,1,s}(\mathbb{R}^{n})\) is also established via the intrinsic function. To obtain these characterizations, we first show that these s-order intrinsic square functions are pointwisely comparable with those similar-looking s-order intrinsic square functions defined via \(\mathrm{Lip}_{\alpha }(\mathbb{R}^{n})\) functions without compact supports.

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogs of intrinsic square functions in the generalized weighted Morrey spaces M w p , φ ( l 2 ). We study the boundedness of intrinsic square functions including the Lusin area integral, the Littlewood-Paley g-function and g λ ∗ -function, and their multilinear commutators on vector-valued generalized weighted Morrey spaces M w p , φ ( l 2 ). In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on φ(x,r) without assuming any monotonicity property of φ(x,r) on r.MSC:42B25, 42B35.

We will obtain the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley𝒢-function, and𝒢λ*-function on the weighted Herz spacesK˙qα,p(w1,w2)(Kqα,p(w1,w2))with general weights.

We study the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood–Paley g-function and g λ * $g^*_\lambda $ -function on the generalized Morrey spaces L p , Φ $L^{p,\Phi }$ for 1 ≤ p < ∞ $1\le p<\infty $ , where Φ is a growth function on ( 0 , ∞ ) $(0,\infty )$ satisfying the doubling condition. The boundedness of commutators generated by BMO ( ℝ n ) $\operatorname{BMO}(\mathbb {R}^n)$ functions and intrinsic square functions is also obtained.

We establish the boundedness of vector-valued intrinsic square function on Morrey and block spaces with variable exponents.

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a measurable function satisfying some decay condition and some locally log-H\"older continuity. In this article, via first establishing characterizations of the variable exponent Hardy space $H^{p(\cdot)}(\mathbb R^n)$ in terms of the Littlewood-Paley $g$-function, the Lusin area function and the $g_\lambda^\ast$-function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_\lambda^\ast$-function. The $p(\cdot)$-Carleson measure characterization for the dual space of $H^{p(\cdot)}(\mathbb R^n)$, the variable exponent Campanato space $\mathcal{L}_{1,p(\cdot),s}(\mathbb R^n)$, in terms of the intrinsic function is also presented.

Intrinsic square function and wavelet characterizations of variable Hardy–Lorentz spaces

The intrinsic square functions including the Lusin area function, Littlewood-Paley g g -function and g λ ∗ g^\ast _\lambda -function dominate pointwisely the classical Littlewood-Paley functions and can be used to characterize the weighted Hardy spaces and more general Musielak-Orlicz Hardy spaces etc. This paper shows that for b ∈ B M O ( R n ) b\in \mathrm {BMO(\mathbb {R}^n)} , the commutators generated by these intrinsic square functions with b b are bounded from H ω p ( R n ) H^p_\omega (\mathbb {R}^n) to L ω p ( R n ) L^p_\omega (\mathbb {R}^n) for some 0 > p ≤ 1 0>p\le 1 and ω ∈ A ∞ \omega \in A_\infty if and only if b ∈ B M O ω , p ( R n ) b\in \mathcal {BMO}_{\omega ,p}(\mathbb {R}^n) , which are a class of non-trivial subspaces of B M O ( R n ) \mathrm {BMO(\mathbb {R}^n)} .

The aim of this paper is to establish the intrinsic square function characterizations in terms of the intrinsic Littlewood–Paleyg-function, the intrinsic Lusin area function, and the intrinsicgλ∗-function of the variable Hardy–Lorentz spaceHp⋅,qℝn, forp⋅being a measurable function onℝnsatisfying0<p−≔ess infx∈ℝnpx≤ess supx∈ℝnpx≕p+<∞and the globally log-Hölder continuity condition andq∈0,∞, via its atomic and Littlewood–Paley function characterizations.

Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, via using the atomic and Littlewood-Paley function characterizations of variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_{\lambda}^{\ast}$-function.