Lizorkin Morrey Spaces Research Articles

Oscillations and differences in Triebel–Lizorkin–Morrey spaces

Oscillations and differences in Triebel–Lizorkin–Morrey spaces

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).

Triebel–Lizorkin–Morrey spaces and differences

AbstractWe study the Triebel–Lizorkin–Morrey spaces and prove that under certain sufficient conditions on the parameters these spaces can be characterised in terms of higher‐order differences. Moreover we show that some of the mentioned conditions are also necessary.

Extrapolation to weighted Morrey spaces with variable exponents and applications

AbstractThis paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

Truncation in Besov–Morrey and Triebel–Lizorkin–Morrey spaces

We will prove that under certain conditions on the parameters the operators T+f=max(f,0) and Tf=|f| are bounded mappings on the Triebel–Lizorkin–Morrey and Besov–Morrey spaces. Moreover we will show that some of the conditions we mentioned before are also necessary. Furthermore we prove that in many cases the Triebel–Lizorkin–Morrey spaces do not have the Fubini property.

Decompositions with Atoms and Molecules for Variable Exponent Triebel–Lizorkin–Morrey Spaces

We continue the study of the variable exponent Morreyfied Triebel-Lizorkin spaces introduced in a previous paper. Here we give characterizations by means of atoms and molecules. We also show that in some cases the number of zero moments needed for molecules, in order that an infinite linear combination of them (with coefficients in a natural sequence space) converges in the space of tempered distributions, is much smaller than what is usually required. We also establish a Sobolev type theorem for related sequence spaces, which might have independent interest.

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

AbstractIn this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.

Triebel–Lizorkin–Morrey spaces associated to Hermite operators

The aim of this article is to establish molecular decomposition of homogeneous and inhomogeneous Triebel–Lizorkin–Morrey spaces associated to the Hermite operator \(\mathbb {H} \equiv -\Delta +|x|^2\) on the Euclidean space \(\mathbb {R}^n\). As applications of the molecular decomposition theory, we show the Triebel–Lizorkin–Morrey boundedness of Riesz potential, Bessel potential and spectral multipliers associated to the operator \({\mathbb {H}}\). These results generalize the corresponding results in Bui and Duong (J Fourier Anal Appl 21:405–448, 2015).

Traces and extensions of generalized smoothness Morrey spaces on domains

We study traces on the boundary of generalized smoothness Morrey spaces on Ck domains Ω. These spaces are equipped with three parameters s,p,q and a function parameter φ. Our results remain valid for the usual Besov–Morrey spaces Nu,p,qs(Ω), Triebel–Lizorkin–Morrey spaces Eu,p,qs(Ω), and Triebel–Lizorkin type spaces Fp,qs,τ(Ω), which are all included in our scales as special cases. Moreover, to complete our investigation, we also study traces of Besov-type spaces Bp,qs,τ(Ω), which are not covered by our approach and give some applications in terms of a priori estimates generalizing results from El Baraka (2005).

Well-posedness of the Euler equation in Triebel–Lizorkin–Morrey spaces

ABSTRACTThis paper establishes the local existence, uniqueness and a blow-up criterion for the solutions to the inviscid incompressible Euler equation in Triebel–Lizorkin–Morrey space . As an application, we also derive the global persistence of the initial regularity in Triebel–Lizorkin–Morrey space for the solutions of 2-D Euler equation. These results are established using the logarithmic inequality of Beal–Kato–Majda type, the Moser type of inequality and commutator estimates in Triebel–Lizorkin–Morrey space.

Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions, local means and non‐smooth atomic decompositions

AbstractIn this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents . These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents as well as the classical Triebel–Lizorkin spaces , even the case when . Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.

Y spaces and global smooth solution of fractional Navier–Stokes equations with initial value in the critical oscillation spaces

For fractional Navier–Stokes equations and critical initial spaces X, one used to establish the well-posedness in the solution space which is contained in C(R+,X). In this paper, for heat flow, we apply parameter Meyer wavelets to introduce Y spaces Ym,β where Ym,β is not contained in C(R+,B˙∞1−2β,∞). Consequently, for 12<β<1, we establish the global well-posedness of fractional Navier–Stokes equations with small initial data in all the critical oscillation spaces. The critical oscillation spaces may be any Besov–Morrey spaces (B˙p,qγ1,γ2(Rn))n or any Triebel–Lizorkin–Morrey spaces (F˙p,qγ1,γ2(Rn))n where 1≤p,q≤∞,0≤γ2≤np,γ1−γ2=1−2β. These critical spaces include many known spaces. For example, Besov spaces, Sobolev spaces, Bloch spaces, Q-spaces, Morrey spaces and Triebel–Lizorkin spaces etc.

Equivalent Quasi-Norms of Besov–Triebel–Lizorkin-Type Spaces via Derivatives

In this paper, applying the Peetre maximal function characterizations and the boundedness of Fourier multipliers on Besov-type and Triebel–Lizorkin-type spaces, as well as Besov–Morrey and Triebel–Lizorkin–Morrey spaces, the authors present some equivalent quasi-norms of these spaces in terms of derivatives.

Littlewood–Paley characterizations of Triebel–Lizorkin–Morrey spaces via ball averages

Let 1<p≤u<∞ and q∈(1,∞]. In this article, the authors establish equivalent characterizations of the Triebel–Lizorkin–Morrey space Eu,q,pα(Rn) with smoothness order α∈(0,2) in terms of the Lusin-area function and the gλ∗-function generated by the difference between f(x) and its ball average B2−jf(x):=1|B(x,2−j)|∫B(x,2−j)f(y)dy,∀j∈{1,2,…},∀x∈Rn, where, for any j∈{1,2,…} and x∈Rn, B(x,2−j):={y∈Rn:|y−x|<2−j}. As applications, the authors obtain several characterizations of Eu,∞,pα(Rn) via pointwise inequalities involving ball averages.

On Sobolev and Franke–Jawerth embeddings of smoothness Morrey spaces

We study embeddings of Triebel–Lizorkin–Morrey spaces $${\mathcal {E}}^{s}_{p,u,q}({\mathbb {R}}^{d})$$ within that scale as well as to classical spaces like $$C({\mathbb {R}}^{d})$$ or $$L_r({\mathbb {R}}^{d})$$ . Here we obtain necessary and sufficient conditions for the continuity of it. Similarly we can deal with the situation when $${\mathbb {R}}^{d}$$ is replaced by a bounded smooth domain $$\Omega \subset {\mathbb {R}}^{d}$$ , now focussing, in addition, on compactness criteria. The second goal are embeddings of so-called Franke–Jawerth type, that is, $$\mathcal{N}^{s_1}_{p_1,u_1,q_1}({\mathbb {R}}^{d}) \hookrightarrow {\mathcal {E}}^{s}_{p,u,q}({\mathbb {R}}^{d}) \hookrightarrow \mathcal{N}^{s_2}_{p_2,u_2,q_2}({\mathbb {R}}^{d})$$ , where the differential dimension is fixed, $$s_1 - \frac{d}{p_1} = s_2 - \frac{d}{p_2}=s-\frac{d}{p}$$ , and $$s_1 > s>s_2$$ .

Hausdorff Besov-type and Triebel–Lizorkin-type spaces and their applications

Let s∈R,p∈(1,∞) and τ∈(0,1/p′], where p′ denotes the conjugate index of p. In this article, the authors first introduce the Hausdorff Besov-type space BH˙p,qs,τ(Rn) with q∈[1,∞) and the Hausdorff Triebel–Lizorkin-type space FH˙p,qs,τ(Rn) with q∈(1,∞) via a class of weights defined on the upper plane, and then establish some equivalent characterizations of BH˙p,qs,τ(Rn) and FH˙p,qs,τ(Rn) via some classes of weights defined on Rn. The authors then prove that their dual spaces are Besov–Morrey and Triebel–Lizorkin–Morrey spaces, respectively. The relations between these spaces and the known Besov–Triebel–Lizorkin–Hausdorff spaces are also studied. As an application, for p∈(1,∞) and λ∈(0,n), the authors obtain the coincidence between the space FH˙p,20,(n−λ)/(np′)(Rn) and the predual space, Hp,λ(Rn), of the Morrey space Lp,λ(Rn). Moreover, characterizations of BH˙p,qs,τ(Rn) and FH˙p,qs,τ(Rn) via local means and Peetre maximal functions, as well as the boundedness of Riesz potential operators and some singular integrals on BH˙p,qs,τ(Rn) and FH˙p,qs,τ(Rn) are also obtained.

Vector-valued singular integral operators on Morrey type spaces and variable Triebel-Lizorkin-Morrey spaces

A criteria on the vector-valued Banach function spaces X (B) is obtained so that whenever a vector-valued singular integral operator is bounded on X (B), it can be extended to be a bounded linear operator on the corresponding Morrey type spaces. Using this result, we define the generalized Triebel–Lizorkin–Morrey spaces and obtain the atomic and molecular decompositions. As a particular example of the generalized Triebel–Lizorkin–Morrey spaces, we introduce and study the variable Triebel–Lizorkin–Morrey spaces.

New applications of Besov-type and Triebel–Lizorkin-type spaces

In this paper, the authors prove that Besov–Morrey spaces are proper subspaces of Besov-type spaces B ˙ p , q s , τ ( R n ) and that Triebel–Lizorkin–Morrey spaces are special cases of Triebel–Lizorkin-type spaces F ˙ p , q s , τ ( R n ) . The authors also establish an equivalent characterization of B ˙ p , q s , τ ( R n ) when τ ∈ [ 0 , 1 / p ) . These Besov-type spaces B ˙ p , q s , τ ( R n ) and Triebel–Lizorkin-type spaces F ˙ p , q s , τ ( R n ) were recently introduced to connect Besov spaces and Triebel–Lizorkin spaces with Q spaces. Moreover, for the spaces B ˙ p , q s , τ ( R n ) and F ˙ p , q s , τ ( R n ) , the authors investigate their trace properties and the boundedness of the pseudo-differential operators with homogeneous symbols in these spaces, which generalize the corresponding classical results of Jawerth and Grafakos–Torres by taking τ = 0 .

Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces

The aim of this paper is to define the Besov–Morrey spaces and the Triebel– Lizorkin–Morrey spaces and to present a decomposition of functions belonging to these spaces. Our results contain an answer to the conjecture proposed by Mazzucato.