Triebel-Lizorkin Spaces Research Articles

Besov and Triebel–Lizorkin spaces on metric spaces: Embeddings and pointwise multipliers

In this paper, we obtain the Franke–Jawerth embedding property of Hajłasz–Besov and Hajłasz–Triebel–Lizorkin spaces on a measure metric space (X,d,μ) which is Ahlfors regular with dimension “Q”. As applications, we show that, when (X,d,μ) is doubling and satisfies an Ahlfors lower bound condition with Q, then the Hajłasz–Besov space Np,qs(X) with p∈(Q,∞], s∈(Qp,1] and q∈(0,∞] and the Hajłasz–Triebel–Lizorkin space Mp,qs(X) with p∈(Q,∞), s∈(Qp,1] and q∈(QQ+s,∞] are algebras under pointwise multiplication and, moreover, when X is Ahlfors Q-regular, we characterize the class of all pointwise multipliers on the Hajłasz–Triebel–Lizorkin space Mp,qs(X) for p∈(Q,∞), s∈(Qp,1] and q∈(QQ+s,∞] by its related uniform space.

Pointwise characterizations of Besov and Triebel-Lizorkin spaces in terms of averages on balls

Pointwise characterizations of Besov and Triebel-Lizorkin spaces in terms of averages on balls

On the regularity of maximal operators supported by submanifolds

We prove that the maximal operators supported by submanifolds are bounded and continuous on the Triebel–Lizorkin spaces Fsp,q(Rd) and Besov spaces Bsp,q(Rd) for 0<s<1 and 1<p,q<∞. As a corollary, we also show that these operators are bounded and continuous on the fractional Sobolev spaces Ws,p(Rd) for 0≤s<1 and 1<p<∞. Our main results represent significant improvements as well as natural extensions of what was known previously.

A T(1) Theorem for Fractional Sobolev Spaces on Domains

Given any uniform domain $\Omega$, the Triebel-Lizorkin space $F^s_{p,q}(\Omega)$ with $0<s<1$ and $1<p,q<\infty$ can be equipped with a norm in terms of first order differences restricted to pairs of points whose distance is comparable to their distance to the boundary. Using this characterization, originally due to Seeger and reproven here, we prove a T(1)-theorem for fractional Sobolev spaces with $0<s<1$ for any uniform domain and for a large family of Calder\'on-Zygmund operators in any ambient space $\mathbb{R}^d$ as long as $sp>d$.

Variable exponent Herz type Besov and Triebel–Lizorkin spaces

Abstract We establish the boundedness of the vector-valued Hardy–Littlewood maximal operator in variable exponent Herz spaces, which were introduced by Samko in [33]. We also introduce variable exponent Herz type Besov and Triebel–Lizorkin spaces and give characterizations of these new spaces by maximal functions.

Variable integral and smooth exponent Triebel-Lizorkin spaces associated with a non-negative self-adjoint operator

Variable integral and smooth exponent Triebel-Lizorkin spaces associated with a non-negative self-adjoint operator

On the Triebel-Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces

On the Triebel-Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces

Bilinear Pseudo-differential Operators with Symbols in $$BS^{m}_{1,1}$$ B S 1 , 1 m on Triebel–Lizorkin Spaces

In this paper, bilinear pseudo-differential operators with symbols in the bilinear Hormander symbol class $$BS^{m}_{1,1}$$ on Triebel–Lizorkin spaces are discussed. As a result, we can obtain the Kato–Ponce inequality in local Hardy spaces.

Homogeneous Besov and Triebel–Lizorkin spaces associated to non-negative self-adjoint operators

Homogeneous Besov and Triebel–Lizorkin spaces with complete set of indices are introduced in the general setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The main step in this theory is the development of distributions modulo generalized polynomials. Some basic properties of the general homogeneous Besov and Triebel–Lizorkin spaces are established, in particular, a discrete (frame) decomposition of these spaces is obtained.

Traces of Muckenhoupt weighted function spaces in case of distant singularities

Abstract We study Besov and Triebel–Lizorkin spaces with Muckenhoupt weights which have their singularities at some distance from the set where the trace is taken. We exemplify our results for special weights of type w ⁢ ( x ) ∼ | x | α {w(x)\sim|x|^{\alpha}} . The approach is based on atomic decomposition results.

A note on transport equation in quasiconformally invariant spaces

Abstract In this note, we study the well-posedness of the Cauchy problem for the transport equation in the BMO space and certain Triebel–Lizorkin spaces.

Nikolskii inequality and Besov, Triebel-Lizorkin, Wiener and Beurling spaces on compact homogeneous manifolds

In this paper we prove Nikolskii's inequality (also known as the reverse Holder inequality) on general compact Lie groups and on compact homogeneous spaces with the constant interpreted in terms of the eigenvalue counting function of the Laplacian on the space, giving the best constant for certain indices, attained on the Dirichlet kernel. Consequently, we establish embedding theorems between Besov spaces on compact homogeneous spaces, as well as embeddings between Besov spaces and Wiener and Beurling spaces. We also analyse Triebel-Lizorkin spaces and beta-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces.

Harmonic Besov and Triebel–Lizorkin Spaces on the Ball

Harmonic Besov and Triebel–Lizorkin spaces on the unit ball in \({\mathbb R}^d\) with full range of parameters are introduced and studied. It is shown that these spaces can be identified with respective Besov and Triebel–Lizorkin spaces of distributions on the sphere. Frames consisting of harmonic functions are also developed and frame characterization of the harmonic Besov and Triebel–Lizorkin spaces is established.

Well-posedness of degenerate differential equations with fractional derivative in vector-valued functional spaces

In this paper, we study the well-posedness of the degenerate differential equations with fractional derivative Dα(Mu)(t)=Au(t)+f(t),(0≤t≤2π) in Lebesgue–Bochner spaces Lp(T;X), periodic Besov spaces Bp,qs(T;X) and periodic Triebel–Lizorkin spaces Fp,qs(T;X), where A and M are closed linear operators in a complex Banach space X satisfying D(A)⊂D(M), α>0 and Dα is the fractional derivative in the sense of Weyl. Using known operator-valued Fourier multiplier results, we completely characterize the well-posedness of this problem in the above three function spaces by the R-bounedness (or the norm boundedness) of the M-resolvent of A.

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let \(p\in (1,\infty )\) and \(q\in [1,\infty )\). In this article, the authors characterize the Triebel-Lizorkin space \({F}^{\alpha }_{p,q}(\mathbb {R}^{n})\) with smoothness order α ∈ (0, 2) via the Lusin-area function and the \(g_{\lambda }^{*}\)-function in terms of difference between f(x) and its ball average \(B_{t}f(x):=\frac 1{|B(x,t)|}{\int }_{B(x,t)}f(y)\,dy\) over the ball B(x, t) centered at \(x\in \mathbb {R}^{n}\) with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) via pointwise inequalities, involving ball averages, in spirit close to Hajlasz gradients, here some interesting phenomena naturally appear that, in the end-point case when α = 2, some of these pointwise inequalities characterize the Triebel-Lizorkin spaces \(F^{2}_{p,2}(\mathbb {R}^{n})\), while not \(F^{2}_{p,\infty }(\mathbb {R}^{n})\), and that some of other obtained pointwise characterizations are only known to hold true for \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) with \(p\in (1,\infty )\), α ∈ (0, 2) or α ∈ (n/p, 2). In particular, some new pointwise characterizations of Hajlasz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than 1 on spaces of homogeneous type.

Besov and Triebel–Lizorkin regularity for the Hodge decomposition and applications to magnetic potentials

The Hodge decomposition is a fundamental result in the theory of differential forms. It has been extensively studied from different perspectives and it is the source of several applications in physics and mathematics. For example, it is important in the analysis of Navier–Stokes' and Maxwell's equations, it can be used to prove results on inverse scattering problems in quantum mechanics and it is useful for the study of magnetic potentials associated to a magnetic field. The regularity properties of the Hodge decomposition have been studied in several works in the last decades, most prominently it has been proved that Sobolev regularity holds true. In this paper we prove regularity for the Hodge decomposition in a much larger scale of function spaces: Besov and Triebel–Lizorkin spaces on compact Riemannian manifolds with boundary. We prove regularity in such spaces for the Hodge–Morrey decomposition and the Hodge–Morrey–Friedrichs decomposition. We, furthermore, state and prove a refinement for the Hodge–Morrey–Friedrichs decomposition (with regularity) that allows expressing every differential form as the sum of the differential of a form plus the co-differential of another form plus a harmonic field belonging to a finite dimensional space. This holds true even if the original form is barely differentiable (with sth order of differentiability, for every real number s>0). We finally provide an application to the study of magnetic potentials associated to magnetic fields. More precisely, we prove the existence of a magnetic potential associated to a magnetic field. The former being one time more differentiable than the latter, in terms of Besov and Triebel–Lizorkin regularity. This is an important motivation for us, since we have the intention to apply our results to further investigations in relativistic inverse (time dependent) scattering problems in quantum mechanics, where Besov and Triebel–Lizorkin regularity for magnetic potentials is relevant.

Lower Bounds for Haar Projections: Deterministic Examples

In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev–Triebel–Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel–Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.

Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces of revolution

We consider the boundedness of the rough singular integral operator TΩ,ψ,h along a surface of revolution on the Triebel-Lizorkin space \(\dot F_{p,q}^\alpha (\mathbb{R}^n )\) for Ω ∈ H1(Sn−1) and Ω ∈ L log+L(Sn−1) ∪ (U1<q<∞Bq(0,0) (Sn−1)), respectively.

Global well-posedness of the 3D incompressible porous media equation with critical dissipation in Triebel-Lizorkin spaces

In this paper, we study the global well-posedness of the 3D incompressible critical dissipative porous media equation with small initial data in the Triebel-Lizorkin space $F^{s}_{p,q}(\mathbb{R}^{3})$ . By a pointwise exponential decay estimate on the Poisson semigroup $e^{-t\nu\sqrt{-\Delta}}$ and the Fourier localization technique, we generalize the global well-posedness in the Sobolev spaces $H_{p}^{s}(\mathbb{R}^{3})=F^{s}_{p,2}(\mathbb{R}^{3})$ and $H^{s}(\mathbb {R}^{3})=F^{s}_{2,2}(\mathbb{R}^{3})$ into the general Triebel-Lizorkin spaces $F^{s}_{p,q}(\mathbb{R}^{3})$ with $s>\frac{3}{p}$ , $p, q\in (1,\infty)$ .

Haar projection numbers and failure of unconditional convergence in Sobolev spaces

For $$1<p<\infty $$ we determine the precise range of $$L_p$$ Sobolev spaces for which the Haar system is an unconditional basis. We also consider the natural extensions to Triebel–Lizorkin spaces and prove upper and lower bounds for norms of projection operators depending on properties of the Haar frequency set.