Let ${\mathcal X}$ be a space of homogeneous type in the sense of Coifman and Weiss and ${\mathcal D}$ a collection of balls in $\cx$. The authors introduce the localized atomic Hardy space $H^{p, q}_{\mathcal D}({\mathcal X})$ with $p\in (0,1]$ and $q\in[1,\infty]\cap(p,\infty]$, the localized Morrey-Campanato space ${\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ and the localized Morrey-Campanato-BLO space $\widetilde{\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ with $\az\in{\mathbb R}$ and $p\in(0, \infty)$ and establish their basic properties including $H^{p, q}_{\mathcal D}({\mathcal X})=H^{p, \infty}_{\mathcal D}({\mathcal X})$ and several equivalent characterizations for ${\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ and $\wz{\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$. Especially, the authors prove that when $p\in(0,1]$, the dual space of $H^{p, \infty}_{\mathcal D}({\mathcal X})$ is ${\mathcal E}^{1/p-1, 1}_{\mathcal D}({\mathcal X})$. Let $ρ$ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces ${\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ and $\widetilde{\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$, respectively, by ${\mathcal E}^{α, p}_ρ({\mathcal X})$ and $\widetilde{\mathcal E}^{α, p}_ρ({\mathcal X})$, when ${\mathcal D}$ is determined by $ρ$. The authors then obtain the boundedness from ${\mathcal E}^{α, p}_ρ({\mathcal X})$ to $\widetilde{\mathcal E}^{α, p}_ρ({\mathcal X})$ of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley $g$-function which are defined via kernels modeled on the semigroup generated by the Schrödinger operator.
The aim of this paper is to get the boundedness of the commutators of multi-sublinear operators generated by local campanato functions and multilinear Calderón-Zygmund operators on the product generalized local Morrey spaces.
In this article, the authors 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 $\pm$-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasize is given to the quasi-Banach case and to the interpolation property.
Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2 \le p < \infty$. In this article, his methods are developed and employed to prove an analogue of such an inequality "beyond the index $p=\infty$", i.e., for spaces of Hölder functions and BMO.
The aim of this paper is to introduce and investigative some new function classes of Morrey-Campanato type. Let $0<p<\infty$ and $0\leq λ<n+p$. We say that $f\in \mathcal{\bar{L}}^{p,λ}(Ω)$ if $$\sup_{x_{0}\in Ω,ρ>0}ρ^{-λ}\int_{Ω(x_{0},ρ)}\big|f(x)-|f|_{Ω(x_{0},ρ)}\big|^pdx<\infty,$$ where $Ω(x_{0},ρ)=Q(x_{0},ρ)\cap Ω$ and $Q(x,ρ)$ is denote the cube of $\mathbb{R}^n$. Some basic properties and characterizations of these classes are presented. If $0\leq λ<n$, the space is equivalent to related Morrey space. If $λ=n$, then $f \in \mathcal{\bar{L}}^{p,n}(Ω)$ if and only if $f\in BMO(Ω)$ with $f^…
We prove well-posedness results for the Dirichlet problem in $\mathbb{R}^{n}_{+}$ for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized Hölder spaces $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)$ and in generalized Morrey-Campanato spaces $\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ under certain assumptions on the growth function $ω$. We also identify a class of growth functions $ω$ for which $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)=\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and for which the aforementioned well-posed…
Let $φ: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $\vz(x,\cdot)$ is nondecreasing, $φ(x,0)=0$, $φ(x,t)>0$ when $t>0$, $\lim_{t\to\infty}φ(x,t)=\infty$ and $\vz(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight uniformly in $t$. Let $ϕ: [0,\infty)\to[0,\infty)$ be nondecreasing. In this article, the authors introduce the Musielak-Orlicz Morrey space $\mathcal M^{φ,ϕ}(\mathbb R^n)$ and obtain the boundedness on $\mathcal M^{φ,ϕ}(\mathbb R^n)$ of the intrinsic Lusin area function $S_α$, the intrinsic $g$-function $g_α$, the intrinsic $g_λ^*$-function $g^\ast_{λ, α}$ and …
Let $\mathcal{L}$ be the Schrödinger operator with potential $V$, that is, $\mathcal L=-Δ+V$, where it is assumed that $V$ satisfies a reverse Hölder inequality. We consider weighted Morrey-Campanato spaces $BMO_{\mathcal L,w}^α(\mathbb R^d)$ and $BLO_{L,w}^α(\mathbb R^d)$ in the Schrödinger setting. We prove that the variation operator $V_σ(\{T_t\}_{t>0})$, $σ>2$, and the oscillation operator $O(\{T_t\}_{t>0}, \{t_j\}_{j\in \mathbb Z})$, where $t_j<t_{j+1}$, $j\in \mathbb Z$, $\lim_{j\rightarrow +\infty}t_j=+\infty$ and $\lim_{j\rightarrow -\infty} t_j=0$, being $T_t=t^k\partial_t^k e^{-t\mat…
In this paper, the authors prove the boundedness of commutators generated by the weighted Hardy operator on weighted $λ$-central Morrey space with the weight $ω$ satisfying the doubling condition. Moreover, the authors give the characterization for the weighted $λ$-central Campanato space by introducing a new kind of operator which is related to the commutator of weighted Hardy operator.
We introduce $B_w^u$-function spaces which unify Lebesgue, Morrey-Campanato, Lipschitz, $B^p$, CMO, local Morrey-type spaces, etc., and investigate the interpolation property of $B_w^u$-function spaces. We also apply it to the boundedness of linear and sublinear operators, for example, the Hardy-Littlewood maximal and fractional maximal operators, singular and fractional integral operators with rough kernel, the Littlewood-Paley operator, Marcinkiewicz operator, and so on.
Let $p,q\in [1,\infty]$, $α\in{\mathbb{R}}$, and $s$ be a non-negative integer. In this article, the authors introduce a new function space $\widetilde{JN}_{(p,q,s)_α}(\mathcal{X})$ of John-Nirenberg-Campanato type, where $\mathcal{X}$ denotes $\mathbb{R}^n$ or any cube $Q_{0}$ of $\mathbb{R}^n$ with finite edge length. The authors give an equivalent characterization of $\widetilde{JN}_{(p,q,s)_α}(\mathcal{X})$ via both the John-Nirenberg-Campanato space and the Riesz-Morrey space. Moreover, for the particular case $s=0$, this new space can be equivalently characterized by both maximal functio…
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying some mild assumptions. In this article, the authors first find a reasonable version $\widetilde{T}$ of the Calderón--Zygmund operator $T$ on the ball Campanato-type function space $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ with $q\in[1,\infty)$, $s\in\mathbb{Z}_+^n$, and $d\in(0,\infty)$. Then the authors prove that $\widetilde{T}$ is bounded on $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ if and only if, for any $γ\in\mathbb{Z}^n_+$ with $|γ|\leq s$, $T^*(x^γ)=0$, which is hence sharp. Moreover, $\widetilde{T}$ is proved to be th…
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying some mild assumptions and let $α\in(0,n)$ and $β\in(1,\infty)$. In this article, when $α\in(0,1)$, the authors first find a reasonable version $\widetilde{I}_α$ of the fractional integral $I_α$ on the ball Campanato-type function space $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ with $q\in[1,\infty)$, $s\in\mathbb{Z}_+^n$, and $d\in(0,\infty)$. Then the authors prove that $\widetilde{I}_α$ is bounded from $\mathcal{L}_{X^β,q,s,d}(\mathbb{R}^n)$ to $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ if and only if there exists a positive c…
The Kato-Yajima smoothing estimate is a smoothing weighted $L^2$ estimate with a singular power weight for the Schrödinger propagator. The weight has been generalized relatively recently to Morrey-Campanato weights. In this paper we make this generalization more sharp in terms of the so-called Kerman-Sawyer weights. Our result is based on a more sharpened Fourier restriction estimate in a weighted $L^2$ space. Obtained results are also extended to the fractional Schrödinger propagator.
In this note we prove the estimate $M^{\sharp}_{0,s}(Tf)(x) \le c\,M_γf(x)$ for general fractional type operators $T$, where $M^{\sharp}_{0,s}$ is the local sharp maximal function and $M_γ$ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of $Tf$ by $M_γf$. Similar estimates hold for $T$ replaced by fractional type operators with kernels satisfying Hörmander-type conditions or integral operators with homogeneous kernels, and $M_γ$ replaced by an appropriate maximal function $M_T$. We also prove two-weight, $L^p_v…
A local median decomposition is used to prove that a weighted local mean of a function is controlled by a weighted local mean of its local sharp maximal function. Together with (a local version of) the estimate $M^{\sharp}_{0,s}(Tf)(x) \le c\,Mf(x)$ for Calderón-Zygmund singular integral operators, this allows us to express the local weighted integral control of $Tf$ by $Mf$. Similar estimates hold for $T$ replaced by singular integrals with kernels satisfying Hörmander-type conditions or integral operators with homogeneous kernels, and $M$ replaced by an appropriate maximal function $M_T$. Us…
We prove that for any homogeneous, second order, constant complex coefficient elliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ with boundary data in BMO is well-posed in the class of functions $u$ with $dμ_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. We establish a Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions $u$ of such systems satisfying the said Carleson measure condition. These imply that BMO can be characterized as the collection of nontangential pointwise traces of smooth null-s…
