Triebel-Lizorkin Spaces Research Articles

Convolution inequalities for Besov and Triebel–Lizorkin spaces, and applications to convolution semigroups

We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel–Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{

On the composition operators on Besov and Triebel–Lizorkin spaces with power weights

Let $G:\mathbb R\rightarrow \mathbb R$ be a continuous function. Under some assumptions on $G$, $s,\alpha ,p$ and $q$ we prove that $$\{G(f):f\in A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha })\}\subset A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha })$$ implie

On the relationship and the norm discretization for Lizorkin-Triebel spaces with generalized smoothness

Homogeneous and non-homogeneous Lizorkin-Triebel spaces with generalized smoothness ?F ?(.)pq (Rn) and F?(.)pq (Rn) have been considered. In particular, under some assumptions of the function ?(t): R+ ? R+,?(1) = 1 determining the generalized smoothness properties, the discretization procedure is realized and the relationship is established between these spaces on Rn and their discrete analogues.

Continuous wavelet transform on Triebel-Lizorkin spaces

The continuous wavelet transform in higher dimensions is used to prove the regularity of weak solutions $u \in L^p(\mathbb R^n)$ under $Qu = f$ where $f$ belongs to the Triebel-Lizorkin space $F^{r,q}_p(\mathbb R^n)$ where $1 < p,q < \infty$, $0< r 0$ with positive constant coefficients $c_{\beta}$.

Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators

Abstract In this paper, we establish some sharp maximal function estimates for certain Toeplitz-type operators associated with some fractional integral operators with general kernel. As an application, we obtain the boundedness of the Toeplitz-type operators on the Lebesgue, Morrey and Triebel-Lizorkin spaces. The operators include the Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Elliptic problems in Besov and Sobolev—Triebel—Lizorkin spaces of low regularity

Elliptic problems with additional unknown distributions in boundary conditions are investigated in Besov and Sobolev–Triebel–Lizorkin spaces of low regularity, specifically of an arbitrary negative order. We find that the problems induce Fredholm bounded operators on appropriate pairs of these spaces.

L p Smoothness on Weighted Besov–Triebel–Lizorkin Spaces in terms of Sharp Maximal Functions

It is known, in harmonic analysis theory, that maximal operators measure local smoothness of L p functions. These operators are used to study many important problems of function theory such as the embedding theorems of Sobolev type and description of Sobolev space in terms of the metric and measure. We study the Sobolev-type embedding results on weighted Besov–Triebel–Lizorkin spaces via the sharp maximal functions. The purpose of this paper is to study the extent of smoothness on weighted function spaces under the condition M α # f ∈ L p , μ , where μ is a lower doubling measure, M α # f stands for the sharp maximal function of f , and 0 ≤ α ≤ 1 is the degree of smoothness.

Lebesgue Points of Besov and Triebel–Lizorkin Spaces with Generalized Smoothness

In this article, the authors study the Lebesgue point of functions from Hajłasz–Sobolev, Besov, and Triebel–Lizorkin spaces with generalized smoothness on doubling metric measure spaces and prove that the exceptional sets of their Lebesgue points have zero capacity via the capacities related to these spaces. In case these functions are not locally integrable, the authors also consider their generalized Lebesgue points defined via the γ-medians instead of the classical ball integral averages and establish the corresponding zero-capacity property of the exceptional sets.

Szasz’s theorem and its generalizations

We establish the most general Szasz type estimates for homogeneous Besov and Lizorkin-Triebel spaces, and their realizations.

Fourier multipliers for Triebel–Lizorkin spaces on compact Lie groups

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.

Erratum to: The boundedness of commutators of sublinear operators on Herz Triebel-Lizorkin spaces

Erratum to: The boundedness of commutators of sublinear operators on Herz Triebel-Lizorkin spaces

Difference Characterization of Besov and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type

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 \({{\mathcal {X}}}\) expressed by its dyadic reference points, dyadic cubes, and the (local) lower bound. Moreover, some results when \(p\le 1\) but near to 1 are new even when \({{\mathcal {X}}}\) is an RD-space.

Approximation by quasi-interpolation operators and Smolyak's algorithm

We study approximation of multivariate periodic functions from Besov and Triebel–Lizorkin spaces of dominating mixed smoothness by the Smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding Smolyak algorithm in the Lq-norm for functions from the Besov spaces Bp,θs(Td) and the Triebel–Lizorkin spaces Fp,θs(Td) for all s>0 and admissible 1≤p,θ≤∞ as well as provide analogues of the Littlewood–Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators.

Continuous Bessel wavelet transform of distributions

The continuous Bessel wavelet transform is extended to distributions in Hμ′(0,∞) and obtained continuity results. Boundedness of continuous Bessel wavelet transform is investigated in a generalized Sobolev space, Besov space and Triebel–Lizorkin space.

Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A_{infty }-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the Ap-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

Pseudo-multipliers and Smooth Molecules on Hermite Besov and Hermite Triebel–Lizorkin Spaces

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 Hormander-type adapted to the Hermite setting on spaces for which the smoothness allowed includes non-positive values; 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, we show that Hermite Besov spaces and Hermite Triebel–Lizorkin spaces are closed under non-linearities.

Hyperbolic Wavelet Analysis of Classical Isotropic and Anisotropic Besov\u2013Sobolev Spaces

In this paper we introduce new function spaces which we call anisotropic hyperbolic Besov and Triebel-Lizorkin spaces. Their definition is based on a hyperbolic Littlewood-Paley analysis involving an anisotropy vector only occurring in the smoothness weights. Such spaces provide a general and natural setting in order to understand what kind of anisotropic smoothness can be described using hyperbolic wavelets (in the literature also sometimes called tensor-product wavelets), a wavelet class which hitherto has been mainly used to characterize spaces of dominating mixed smoothness. A centerpiece of our present work are characterizations of these new spaces based on the hyperbolic wavelet transform. Hereby we treat both, the standard approach using wavelet systems equipped with sufficient smoothness, decay, and vanishing moments, but also the very simple and basic hyperbolic Haar system. The second major question we pursue is the relationship between the novel hyperbolic spaces and the classical anisotropic Besov–Lizorkin-Triebel scales. As our results show, in general, both approaches to resolve an anisotropy do not coincide. However, in the Sobolev range this is the case, providing a link to apply the newly obtained hyperbolic wavelet characterizations to the classical setting. In particular, this allows for detecting classical anisotropies via the coefficients of a universal hyperbolic wavelet basis, without the need of adaption of the basis or a-priori knowledge on the anisotropy.

A new formula for the Lp norm

Recently, Brezis, Van Schaftingen and the second author [4] established a new formula for the W˙1,p norm of a function in Cc∞(RN). The formula was obtained by replacing the Lp(R2N) norm in the Gagliardo semi-norm for W˙s,p(RN) with a weak-Lp(R2N) quasi-norm and setting s=1. This provides a characterization of such W˙1,p norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula [1]. In this paper, we obtain an analog for the case s=0. In particular, we present a new formula for the Lp norm of any function in Lp(RN), which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on Lp(RN), which complements a formula by Maz′ya and Shaposhnikova [12]. As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space F2s,p(RN) (i.e. the Bessel potential space (I−Δ)−s/2Lp(RN)), as well as its homogeneous counterpart F˙2s,p(RN), for s∈(0,1), p∈(1,∞).

Density results for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets

We investigate two density questions for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set Ω⊂Rn, D(Ω) is dense in {u∈Hs(Rn):suppu⊂Ω‾} whenever ∂Ω has zero Lebesgue measure and Ω is “thick” (in the sense of Triebel); and (ii) for a d-set Γ⊂Rn (0<d<n), {u∈Hs1(Rn):suppu⊂Γ} is dense in {u∈Hs2(Rn):suppu⊂Γ} whenever −n−d2−m−1<s2≤s1<−n−d2−m for some m∈N0. For (ii), we provide concrete examples, for any m∈N0, where density fails when s1 and s2 are on opposite sides of −n−d2−m. The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether {u∈Hs(Rn):suppu⊂Γ}={0} for a given closed set Γ⊂Rn and s∈R. They also both arise naturally in the study of boundary integral equation formulations of acoustic wave scattering by fractal screens. We additionally provide analogous results in the more general setting of Besov and Triebel–Lizorkin spaces.

Remarks on the Well-Posedness of the Euler Equations in the Triebel–Lizorkin Spaces

We prove the continuous dependence of the solution maps for the Euler equations in the (critical) Triebel–Lizorkin spaces, which was not shown in the previous works [6, 7, 9]. The proof relies on the classical Bona–Smith method as [12], where similar result was obtained in critical Besov spaces $$B^1_{\infty ,1}$$ .