In this paper we define Besov-Lipschitz and Triebel-Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein-Uhlenbeck and the Poisson-Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces $L^p_α(γ_d)$ are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusions semigroups.
We prove inequalities of Hardy type for functions in Triebel-Lizorkin spaces $F^s_{pq}(G)$ on a domain $G\subset\mathbb{R}^n$, whose boundary has the Aikawa dimension strictly less than $n-sp$.
In this article, the authors establish the pointwise characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type via clarifying the relationship among Hajłasz-Sobolev spaces, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces, grand Besov and Triebel-Lizorkin spaces, and Besov and Triebel-Lizorkin spaces. A major novelty of this article is that all results presented in this article get rid of both the dependence on the reverse doubling condition of the measure and the metric condition of the quasi-metric under consideration. Moreover, the pointwise characterization of th…
In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.
In this article, the authors establish the wavelet characterization of Besov and Triebel--Lizorkin spaces on a given space $(X,d,μ)$ of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel--Lizorkin sequence spaces on $X$, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel--Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood--Paley…
We prove an inequality of Hardy type for functions in Triebel-Lizorkin spaces. The distance involved is being measured to a given Ahlfors d-regular set in R^n, with n-1<d<n. As an application of the Hardy inequality, we consider boundedness of pointwise multiplication operators, and extension problems.
In this paper, we obtain the continuity for some multilinear operators related to certain non-convolution operators on the Triebel--Lizorkin space. The operators include Littlewood--Paley operator and Marcinkiewicz operator.
In this paper, the authors construct some counterexamples to show that the generalized Carleson measure space and the Triebel-Lizorkin-type space are not equivalent for certain parameters, which was claimed to be true in [Taiwanese J. Math. 15 (2011), 919-926]. Moreover, the authors show that for some special parameters, the generalized Carleson measure space, the Triebel-Lizorkin-type space and the Besov-type space coincide with certain Triebel-Lizorkin space, which answers a question posed in Remark 6.11(i) of [Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010, xi+281 pp.].
In this work we give some maximal inequalities in Triebel-Lizorkin spaces, which are "$\dot{F}_{\infty}^{s,q}$-variants" of Fefferman-Stein vector-valued maximal inequality and Peetre's maximal inequality. We will give some applications of the new maximal inequalities and discuss sharpness of some results.
Let $α\in\mathbb R$, $q\in(0,\infty]$, $p\in(0,\infty)$, and $W$ be an $A_p(\mathbb{R}^n,\mathbb{C}^m)$-matrix weight. In this article, the authors characterize the matrix-weighted Triebel-Lizorkin space $\dot{F}_{p}^{α,q}(W)$ via the Peetre maximal function, the Lusin area function, and the Littlewood-Paley $g_λ^{*}$-function. As applications, the authors establish the boundedness of Fourier multipliers on matrix-weighted Triebel-Lizorkin spaces under the generalized Hörmander condition. The main novelty of these results exists in that their proofs need to fully use both the doubling property…
In this paper we will give a simple stopping time proof in the $\mathbb{R}^d$ setting of the matrix weighted Triebel-Lizorkin bounds proved by F. Nazarov/S. Treil and A. Volberg, respectively. Furthermore, we provide explicit matrix A${}_p$ characteristic dependence and also discuss some interesting open questions.
On a general Lie group $G$ endowed with a sub-Riemannian structure and of local dimension $d$, we characterize the pointwise multipliers of Triebel--Lizorkin spaces $F^{p,q}_α$ for $p,q\in (1,\infty)$ and $α>d/p$, and those of Besov spaces $B^{p,q}_α$ for $q\in [1,\infty]$, $p>d$ and $d/p< α<1$. When $G$ is stratified, we extend the latter characterization to all $p,q\in [1,\infty]$ and $α>d/p$.
In this article, the authors introduce Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss, prove that these (in)homogeneous Besov and Triebel-Lizorkin spaces are independent of the choices of both exp-ATIs (or exp-IATIs) and underlying spaces of distributions, and give some basic properties of these spaces. As applications, the authors show that some known function spaces coincide with certain special cases of Besov and Triebel-Lizorkin spaces and, moreover, obtain the boundedness of Calderón-Zygmund operators on these Besov and Triebel-Lizorkin s…
We prove one generalization of the Littlewood--Paley characterization of the $\mathrm{BMO}$ space where the dilations of a Schwartz function are replaced by a family of functions with suitable conditions imposed on them. We also prove that a certain family of Triebel--Lizorkin spaces can be characterized in a similar way.
We obtain new molecular decompositions and molecular synthesis estimates for Hermite Besov and Hermite Triebel--Lizorkin spaces and use such tools to prove boundedness properties of Hermite pseudo-multipliers on those spaces. The notion of molecule we develop leads to boundedness of pseudo-multipliers associated to symbols of Hörmander-type adapted to the Hermite setting on spaces with non-positive smoothness; in particular, we obtain continuity results for such operators on Lebesgue and Hermite local Hardy spaces. As a byproduct of our results on boundedness properties of pseudo-multipliers, …
In this paper, we obtain the sharp conditions of the inclusion relations between modulation spaces $M_{p,q}^s$ and Triebel-Lizorkin spaces $F_{p,r}$ for $p\leq 1$, which greatly improve and extend the results for the embedding relations between local Hardy spaces and modulation spaces obtained by Kobayashi, Miyachi and Tomita in [Studia Math. 192 (2009), 79-96].
Weighted Triebel-Lizorkin and Besov spaces on the unit ball $B^d$ in $\Rd$ with weights $\W(x)= (1-|x|^2)^{μ-1/2}$, $μ\ge 0$, are introduced and explored. A decomposition scheme is developed in terms of almost exponentially localized polynomial elements (needlets) $\{ϕ_ξ\}$, $\{ψ_ξ\}$ and it is shown that the membership of a distribution to the weighted Triebel-Lizorkin or Besov spaces can be determined by the size of the needlet coefficients $\{\ip{f,ϕ_ξ}\}$ in appropriate sequence spaces.
In a previous paper two of the authors introduced and study Gaussian Besov-Lipschitz spaces $B_{p,q}^α(γ_{d})$ and Gaussian Triebel-Lizorkin spaces $F_{p,q}^α(γ_{d})$. Now, in this paper we introduce the variable Gaussian Besov-Lipschitz spaces $B_{p(\cdot),q(\cdot)}^α(γ_{d})$ and the variable Gaussian Triebel-Lizorkin spaces $F_{p(\cdot),q(\cdot)}^α(γ_{d}),$ that is to say, Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces with variable exponents, under certain additional regularity conditions on the exponents $p(\cdot)$ and $q(\cdot)$ introduced by Dalmasso and Scotto. Trivially, they inc…
Homogeneous Besov and Triebel-Lizorkin spaces associated with multi-dimensional Laguerre function expansions of Hermite type with index $α\in [-1/2,\infty)^d\backslash (-1/2,1/2)^d$, $d\geq 1$, are defined and investigated. To achieve expected goals Schwartz type spaces on $R^d_+$ are introduced and then tempered type distributions are constructed. Also, ideas from a recent paper of Bui and Duong on Besov and Triebel-Lizorkin spaces associated with Hermite functions expansions are used. This means, in particular, using molecular decomposition and an appropriate form of a Calderón reproducing f…
On a metric measure space satisfying the doubling property, we establish several optimal characterizations of Besov and Triebel-Lizorkin spaces, including a pointwise characterization. Moreover, we discuss their (non)triviality under a Poincaré inequality.
We study the continuity and compactness of embeddings for radial Besov and Triebel-Lizorkin spaces with weights in the Muckenhoupt class $A_\infty$. The main tool is a discretization in terms of an almost orthogonal wavelet expansion adapted to the radial situation.
In this article, the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss, and discuss their relations with Besov and Triebel-Lizorkin spaces. As an application, the authors establish the difference characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. A major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space ${\mathcal X}$ via using the geometrical property of ${\mathcal X}$ e…
The Littlewood-Paley theory is extended to weighted spaces of distributions on $[-1,1]$ with Jacobi weights $ \w(t)=(1-t)^α(1+t)^β. $ Almost exponentially localized polynomial elements (needlets) $\{ϕ_ξ\}$, $\{ψ_ξ\}$ are constructed and, in complete analogy with the classical case on $\RR^n$, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $\{\ip{f,ϕ_ξ}\}$ in respective sequence spaces.
In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces $\dot B^s_{p,\,q}$ and Triebel-Lizorkin spaces $\dot F^s_{p,\,q}$ for all $s\in(0,\,1)$ and $p,\,q\in(n/(n+s),\,\infty],$ both in ${\mathbb R}^n$ and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve $\dot F^s_{n/s,\,q}$ on $\rn$ for all $s\in(0,\,1)$ and $q\in(n/(n+s),\,\infty]$. A metric measure space version of the above morphism property is also established.
Matrix-Weighted Besov-Triebel-Lizorkin Spaces of Optimal Scale: Real-Variable Characterizations, Invariance on Integrable Index, and Sobolev-Type Embedding
In this paper, we establish the equivalence between the Hajłasz-Sobolev spaces or classical Triebel-Lizorkin spaces and a class of grand Triebel-Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when $p\in(n/(n+1),\infty)$, we give a new characterization of the Hajłasz-Sobolev spaces $\dot M^{1, p}({\mathbb R}^n)$ via a grand Littlewood-Paley function.
A pair of dual frames with almost exponentially localized elements (needlets) are constructed on $\RR_+^d$ based on Laguerre functions. It is shown that the Triebel-Lizorkin and Besov spaces induced by Laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients.
An RD-space $\mathcal X$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal X$. In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on $\mathcal X$ are independent of the choice of the regularity $ε\in (0,1)$; as a result of this, the Besov and Triebel-Lizorkin spaces on $\mathcal X$ are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogene…
In a previous paper the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail on Gaussian Besov-Lipschitz spaces $B_{p,q}^α(γ_d)$. In this paper we will continue our study proving the boundedness of those operators on Gaussian Triebel-Lizorkin spaces $F_{p,q}^α(γ_d)$. Also these results can be extended to the case of Laguerre or Jacobi expansions and even further to the general framework of diffusions semigroups.
Let $(M,ρ,μ)$ be a metric measure space satisfying the doubling, reverse doubling and non-collapsing conditions, and $\mathscr{L}$ be a self-adjoint operator on $L^2 (M, dμ)$ whose heat kernel $p_t (x,y)$ satisfy the small-time Gaussian upper bound, Hölder continuity and Markov property. In this paper, we give characterizations of inhomogeneous "classical" and "non-classical" Besov and Triebel-Lizorkin spaces associated to $\mathscr{L}$ in terms of continuous Littlewood-Paley and Lusin area functions defined by the heat semigroup, for complete range of indices. This extends related classical r…
In this note, we establish characterizations for the homogeneous Besov-type spaces $\dot{B}^{s,τ}_{p,q}(\mathbb{R}^n)$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,τ}_{p,q}(\mathbb{R}^n)$, introduced by Yang and Yuan, through fractional Hajłasz-type gradients for suitable values of the parameters $p$, $q$ and $τ$ when $0 < s < 1$, and through grand Littlewood-Paley-type maximal functions for all admissible values of the parameters. These characterizations extend the characterizations obtained by Koskela, Yang and Zhou for the standard homogeneous Besov and Triebel-Lizorkin spaces.
We characterize the Schauder and unconditional basis properties for the Haar system in the Triebel-Lizorkin spaces $F^s_{p,q}(\Bbb R^d)$, at the endpoint cases $s=1$, $s=d/p-d$ and $p=\infty$. Together with the earlier results in [10], [4], this completes the picture for such properties in the Triebel-Lizorkin scale, and complements a similar study for the Besov spaces given in [5].
In this note we give equivalent characterizations for a fractional Triebel-Lizorkin space $F^s_{p,q}(Ω)$ in terms of first-order differences in a uniform domain $Ω$. The characterization is valid for any positive, non-integer real smoothness $s\in \mathbb{R}_+\setminus \mathbb{N}$ and {indices $1\leq p<\infty$, $1\leq q \leq \infty$} as long as the fractional part $\{s\}$ is greater than $d/p-d/q$.
In this paper we prove a parabolic Triebel-Lizorkin space estimate for the operator given by \[T^αf(t,x) = \int_0^t \int_{{\mathbb R}^d} P^α(t-s,x-y)f(s,y) dyds,\] where the kernel is \[P^α(t,x) = \int_{{\mathbb R}^d} e^{2πix\cdotξ} e^{-t|ξ|^α} dξ.\] The operator $T^α$ maps from $L^{p}F_{s}^{p,q}$ to $L^{p}F_{s+α/p}^{p,q}$ continuously. It has an application to a class of stochastic integro-differential equations of the type $du = -(-Δ)^{α/2} u dt + f dX_t$.
We give a new characterization of (homogeneous) Triebel-Lizorkin spaces $\dot{\mathcal F}^{s}_{p,q}(Z)$ in the smoothness range $0 < s < 1$ for a fairly general class of metric measure spaces $Z$. The characterization uses Gromov hyperbolic fillings of $Z$. This gives a short proof of the quasisymmetric invariance of these spaces in case $Z$ is $Q$-Ahlfors regular and $sp = Q > 1$. We also obtain first results on complex interpolation for these spaces in the framework of doubling metric measure spaces.
This paper provides equivalence characterizations of homogeneous Triebel-Lizorkin and Besov-Lipschitz spaces, denoted by $\dot{F}^s_{p,q}(\mathbb{R}^n)$ and $\dot{B}^s_{p,q}(\mathbb{R}^n)$ respectively, in terms of maximal functions of the mean values of iterated difference. It also furnishes the reader with inequalities in $\dot{F}^s_{p,q}(\mathbb{R}^n)$ in terms of iterated difference and in terms of iterated difference along coordinate axes. The corresponding inequalities in $\dot{B}^s_{p,q}(\mathbb{R}^n)$ in terms of iterated difference and in terms of iterated difference along coordinate …
We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p} = F^{s}_{p,p}$, and $H^{1,p} = F^{1}_{p,2}$. When $s\to 1$, the $F^{s}_{p,p}$ norm becomes the $F^{1}_{p,p}$ norm but BBM showed that the $W^{s,p}$ norm becomes the $H^{1,p} = F^{1}_{p,2}$ norm. Naively, for $p \neq 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{s,p}$ into $F^{s}_{p,q}$ for $q \in \{p,2\}$ with sharp constants with respect to $s \in (0,1)$. As a consequence we obtain an $\mathbb{R}^N$-version of the B…
Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces III: Characterizations of Molecules and Wavelets, Trace Theorems, and Boundedness of Pseudo-Differential Operators and Calderón--Zygmund Operators
In this article, via certain lower bound conditions on the measures under consideration, the authors fully characterize the Sobolev embeddings for the scales of Hajłasz-Triebel-Lizorkin and Hajłasz-Besov spaces in the general context of quasi-metric measure spaces for an optimal range of the smoothness parameter $s$. An interesting facet of this work is how the range of $s$ for which the above characterizations of these embeddings hold true is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. Moreover, although stated for Hajłasz-Triebel-Lizorkin and…
Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces $\dot{\mathbf{F}}^α_{p,q}$ for the endpoint case of $p = \infty$ and the full scale of parameters $α\in \mathbb{R}$ and $q \in (0,\infty]$. In particular, a Peetre-type characterization of the anisotropic Besov space $\dot{\mathbf{B}}^α_{\infty,\infty} = \dot{\mathbf{F}}^α_{\infty,\infty}$ is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in $\dot{\mathbf{F}}^α_{\infty,q}$.
We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces $Z$. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices $s<1,$ as well as the first order Hajłasz-Sobolev space $M^{1,p}(Z)$. They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset $F \subset Z$ are Besov spaces defined intrinsically on $F$. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.
Let $(M, ρ,μ)$ be an RD-space satisfying the non-collapsing condition. In this paper, the authors introduce Besov-type spaces $B_{p,q}^{s,τ}(M)$ and Triebel--Lizorkin-type spaces $F_{p,q}^{s,τ}(M)$ associated to a non-negative self-adjoint operator $L$ whose heat kernels satisfy some Gaussian upper bound estimate, Hölder continuity, and the stochastic completeness property. Characterizations of these spaces via Peetre maximal functions and heat kernels are established for full range of indices. Also, frame characterizations of these spaces are given. When $L$ is the Laplacian operator on $\mat…
Let $\ell\in\mathbb{N}$ and $p\in(1,\infty]$. In this article, the authors prove that the sequence $\{f-B_{\ell,2^{-k}}f\}_{k\in\mathbb{Z}}$ consisting of the differences between $f$ and the ball average $B_{\ell,2^{-k}}f$ characterizes the Besov space $\dot B^α_{p,q}(\rn)$ with $q\in (0, \infty]$ and the Triebel-Lizorkin space $\dot F^α_{p,q}(\rn)$ with $q\in (1,\infty]$ when the smoothness order $α\in(0,2\ell)$. More precisely, it is proved that $f-B_{\ell,2^{-k}}f$ plays the same role as the approximation to the identity $φ_{2^{-k}}\ast f$ appearing in the definitions of $\dot B^α_{p,q}(\rn…
In this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the Lebesgue spaces $L^p(\Rdst)$, $1<p<+\infty$. The novelty and difficulty of this construction is that we allow for non-lattice translations. We prove that for an arbitrary expansive matrix $A$ and any set $Λ$ - satisfying a certain spreadness condition but otherwise irregular- there exists a smooth window whose translations along the elements of $Λ$ and dilations by powers of $A$ provide an atomic decomposition for the whole range of the anisotropic Tr…
In this article, the authors first introduce the Triebel-Lizorkin-type space $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ with variable exponents, and establish its $φ$-transform characterization in the sense of Frazier and Jawerth, which further implies that this new scale of function spaces is well defined. The smooth molecular and the smooth atomic characterizations of $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ are also obtained, which are used to prove a trace theorem of $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$. The authors also characterize the space $F_{p(\cdot),q(\c…
