Triebel-Lizorkin Spaces Research Articles

Quarklet characterizations for Triebel–Lizorkin spaces

In this paper we prove that under some conditions on the parameters the univariate Triebel–Lizorkin spaces Fr,qs(R) can be characterized in terms of quarklets. So for functions from Triebel–Lizorkin spaces we obtain a quarkonial decomposition as well as a new equivalent quasi-norm. For that purpose we use quarklets that are constructed by means of biorthogonal compactly supported Cohen–Daubechies–Feauveau spline wavelets, where the primal generator is a cardinal B-spline. Moreover we introduce some sequence spaces apposite to our quarklet system and study their properties. Finally we also obtain a quarklet characterization for the Triebel–Lizorkin–Morrey spaces Eu,r,qs(R).

Lipschitz and Triebel–Lizorkin spaces, commutators in Dunkl setting

We first study the Lipschitz spaces Λdβ associated with the Dunkl metric, β∈(0,1), and prove that it is a proper subspace of the classical Lipschitz spaces Λβ on RN, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces Λβ connects to the Triebel–Lizorkin spaces Ḟp,Dα,q associated with the Dunkl Laplacian △D in RN and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian △D−α/2, 0<α<N (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup e−t△D. The key steps in this paper are a finer decomposition of the underlying space via Dunkl metric and Euclidean metric to bypass the use of Fourier analysis, and a discrete weak-type Calderón reproducing formula in these new Triebel–Lizorkin spaces Ḟp,Dα,q.

Lp-Sampling recovery for non-compact subclasses of L∞

In this article, we study the sampling recovery problem for certain relevant multivariate function classes on the cube [0, 1]d, which are not compactly embedded into L∞([0,1]d). Recent tools relating the sampling widths to the Kolmogorov or best m-term trigonometric widths in the uniform norm are therefore not applicable. In a sense, we continue the research on the small smoothness problem by considering limiting smoothness in the context of Besov and Triebel-Lizorkin spaces with dominating mixed regularity such that the sampling recovery problem is still relevant. There is not much information available on the recovery of such functions except for a previous result by Oswald in the univariate case and Dinh Dũng in the multivariate case. As a first step, we prove the uniform boundedness of the ℓp-norm of the Faber coefficients at a fixed level by Fourier analytic means. Using this, we can control the error made by a (Smolyak) truncated Faber series in Lq([0,1]d) with q &amp;lt;∞. It turns out that the main rate of convergence is sharp. Thus, we obtain results also for S1,∞1F([0,1]d), a space “close” to S11W([0,1]d), which is important in numerical analysis, especially numerical integration, but has rather poor Fourier analytical properties.

Stable decomposition of homogeneous Mixed-norm Triebel–Lizorkin spaces

We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on Rd. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel–Lizorkin spaces.In the second part of the paper we study nonlinear m-term approximation with the constructed basis in the mixed-norm setting, where the behavior, in general, for d≥2, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for m-term approximation can still be derived.

Pointwise multipliers for Triebel–Lizorkin and Besov spaces on Lie groups

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 Fαp,q for p,q∈(1,∞) and α>d/p, and those of Besov spaces Bαp,q for q∈[1,∞], p>d and d/p<α<1. When G is stratified, we extend the latter characterization to all p,q∈[1,∞] and α>d/p.

Real-variable characterizations and their applications of matrix-weighted Triebel–Lizorkin spaces

Let α∈R, q∈(0,∞], p∈(0,∞), and W be an Ap(Rn,Cm)-matrix weight. In this article, the authors characterize the matrix-weighted Triebel–Lizorkin space F˙pα,q(W) via the Peetre maximal function, the Lusin area function, and the Littlewood–Paley gλ⁎-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 of matrix weights and the reducing operator associated to matrix weights, which are essentially different from those proofs of the corresponding cases of classical Triebel–Lizorkin spaces that strongly depend on the Fefferman–Stein vector-valued maximal inequality on Lebesgue spaces.

Local theory for 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type

AbstractIn this paper, we introduce new 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type, which contain many classical functions. We establish local theory of these function spaces by the φ‐transform and the wavelet decomposition coefficients. Moreover, we give a characterization of these local spaces by a local polynomial approximation.

Haar Frame Characterizations of Besov–Sobolev Spaces and Optimal Embeddings into Their Dyadic Counterparts

We study the behavior of Haar coefficients in Besov and Triebel–Lizorkin spaces on {mathbb R}, for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness s<1, in which the spaces F^s_{p,q} and B^s_{p,q} are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that 1/p<s<1 and fin B^s_{p,q}, we actually prove that the usual Haar coefficient norm, Vert {2^jlangle f, h_{j,mu }rangle }_{j,mu }Vert _{b^s_{p,q}} remains equivalent to Vert fVert _{B^s_{p,q}}, i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case s=1 and q=infty , we show that such an expression gives an equivalent norm for the Sobolev space W^{1}_p({mathbb R}), 1<p<infty , which is related to a classical result by Bočkarev. Finally, in several endpoint cases we give optimal inclusions between B^s_{p,q}, F^s_{p,q}, and their dyadic counterparts.

Estimates of fundamental solution for Kohn Laplacian in Besov and Triebel-Lizorkin spaces

ABSTRACT We introduce Besov space B ˙ p α , q ( ∂ Ω k ) and Triebel-Lizorkin space F ˙ p α , q ( ∂ Ω k ) on a family of model domains ∂ Ω k = { ( z , z n + 1 ) = ( z 1 , z 2 , … , z n + 1 ) : Im ( z n + 1 ) = ϕ ( | z | 2 ) } with ϕ ( x ) = x k in C n + 1 which can be considered as a space of homogeneous type in the sense of Coifman RR, Weiss G.[Analyse Harmonique Non-commutative and Certains Espaces Homogǹes. Berlin: Springer; 1971. (Lecture Notes in Math.; 242).], Coifman R, Weiss G. [Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc. 1977;83:569–645.]. We study the sharp estimates on the fundamental solution for the Kohn Laplacian and Cauchy-Szegö projection on ∂ Ω k in these spaces.

Bilinear pseudodifferential operators with symbol in BS1,1m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$BS_{1,1}^m$$\\end{document} on Triebel–Lizorkin spaces with critical Sobolev index

In this paper we obtain new estimates for bilinear pseudodifferential operators with symbol in the class BS1,1m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$BS_{1,1}^m$$\\end{document}, when both arguments belong to Triebel-Lizorkin spaces of the type Fp,qn/p(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{p,q}^{n/p}({\\mathbb {R}}^n)$$\\end{document}. The inequalities are obtained as a consequence of a refinement of the classical Sobolev embedding Fp,qn/p(Rn)↪bmo(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F^{n/p}_{p,q}({\\mathbb {R}}^n)\\hookrightarrow \ extrm{bmo}({\\mathbb {R}}^n)$$\\end{document}, where we replace bmo(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{bmo}({\\mathbb {R}}^n)$$\\end{document} by an appropriate subspace which contains L∞(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty ({\\mathbb {R}}^n)$$\\end{document}. As an application, we study the product of functions on Fp,qn/p(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{p,q}^{n/p}({\\mathbb {R}}^n)$$\\end{document} when 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty $$\\end{document}, where those spaces fail to be multiplicative algebras.

The maximal regularity property of abstract integrodifferential equations

We provide a convenient framework for the study of the well-posedness of a variety of abstract (integro)differential equations in general Banach function spaces. It allows us to extend and complement the known theory on the maximal regularity of such equations. More precisely, by methods of harmonic analysis, we identify large classes of Banach spaces which are invariant with respect to distributional Fourier multipliers. Such classes include general vector-valued Banach function spaces varPhi and/or the scales of Besov and Triebel–Lizorkin spaces defined by varPhi . We apply this result to the study of the well-posedness and maximal regularity property of abstract second-order integrodifferential equations, which model various types of elliptic and parabolic problems arising in different areas of applied mathematics.

Anisotropic Triebel–Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I

This paper provides maximal function characterizations of anisotropic Triebel–Lizorkin spaces associated to general expansive matrices for the full range of parameters p in (0,infty ), q in (0,infty ] and alpha in {mathbb {R}}. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. As an application, the existence of dual molecular frames and Riesz sequences is obtained; the wavelet systems are generated by translations and anisotropic dilations of a single function, where neither the translation nor dilation parameters are required to belong to a discrete subgroup. Explicit criteria for molecules are given in terms of mild decay, moment, and smoothness conditions.

The Boundedness of Commutators of Sublinear Operators on Herz Triebel–Lizorkin Spaces with Variable Exponent

In this paper, the authors first discuss the characterization of Herz Triebel–Lizorkin spaces with variable exponent via two families of operators. By this characterization, the authors prove that the Lipschitz commutators of sublinear operators is bounded from Herz spaces with variable exponent to Herz Triebel–Lizorkin spaces with variable exponent. As applications, the corresponding boundedness estimates for the commutators of maximal operator, Riesz potential operator and Calderón–Zygmund operator are established.

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, II

Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces dot{textbf{F}}^{alpha }_{p,q} for the endpoint case of p = infty and the full scale of parameters alpha in mathbb {R} and q in (0,infty ]. In particular, a Peetre-type characterization of the anisotropic Besov space dot{textbf{B}}^{alpha }_{infty ,infty } = dot{textbf{F}}^{alpha }_{infty ,infty } is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in dot{textbf{F}}^{alpha }_{infty ,q}.

Estimates for evolutionary partial differential equations in classical function spaces

Abstract We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.

Spline Representations of Lizorkin–Triebel Spaces with General Weights

Spline Representations of Lizorkin–Triebel Spaces with General Weights

Wavelet characterization of Triebel–Lizorkin spaces for p=∞ on spaces of homogeneous type and its applications

Let ( X , d , μ ) be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, η ∈ ( 0 , 1 ) the Hölder regularity of wavelets on X constructed by P. Auscher and T. Hytönen, s ∈ ( − η , η ) , and q ∈ ( max { ω ω + η , ω ω + η + s } , ∞ ] . In this article, the authors establish the wavelet characterization of the Triebel–Lizorkin space F ̇ ∞ , q s ( X ) . Moreover, the authors introduce almost diagonal operators on the Triebel–Lizorkin sequence space f ̇ ∞ , q s ( X ) and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors establish the molecular characterization of F ̇ ∞ , q s ( X ) . The authors also obtain both the Lusin area function and the Littlewood–Paley g λ ∗ -function characterizations of F ̇ ∞ , q s ( X ) . Besides, the inhomogeneous counterparts are also given. The main novelty of this article is that all these results get rid of their dependence on the reverse doubling property of μ and also the triangle inequality of d , by fully using the geometrical properties of X expressed via its equipped quasi-metric d , dyadic reference points, dyadic cubes, and wavelets.

CHARACTERIZATIONS OF COMMUTATORS OF SINGULAR INTEGRAL OPERATORS ON MORREY TRIEBEL-LIZORKIN

In this paper, we obtain the characterizations of Morrey Triebel-Lizorkin spaces by two families of operators. Applying the characterizations of Morrey TriebelLizorkin spaces, it is proved that b is a Lipschitz function if and only if the commutator [b, T] is bounded from Morrey spaces to Morrey Triebel-Lizorkin spaces, where T is singular integral operator or Riesz potential operator.

Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae for fractional Sobolev spaces via interpolation and extrapolation

The real interpolation spaces between \(L^{p}({{\mathbb {R}}}^{n})\) and \({\dot{H}}^{t,p}({{\mathbb {R}}}^{n})\) (resp. \(H^{t,p}({{\mathbb {R}}}^{n})\)), \(t>0,\) are characterized in terms of fractional moduli of smoothness, and the underlying seminorms are shown to be “the correct” fractional generalization of the classical Gagliardo seminorms. This is confirmed by the fact that, using the new spaces combined with interpolation and extrapolation methods, we are able to extend the Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae, as well as the Bourgain–Brezis–Mironescu convergence theorem, to fractional Sobolev spaces. On the other hand, we disprove a conjecture of Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159) suggesting fractional convergence results given in terms of classical Gagliardo seminorms. We also solve a problem proposed in Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159) concerning sharp forms of the fractional Sobolev embedding.

Bourgain–Brezis–Mironescu convergence via Triebel-Lizorkin spaces

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 $$s\rightarrow 1$$ , 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 $$p \ne 2$$ 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 BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown.