In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized incompressible Navier-Stokes equations (gNS) in Triebel-Lizorkin space $\dot{F}^{-α,r}_{q_α}(\mathbb{R}^3)$ with $(α,r)\in(1,5/4)\times[2,\infty]$ and $q_α=\frac{3}{α-1}$. Our work establishes a {\it dichotomy} of well-posedness and ill-posedness depending on $r=2$ or $r>2$. Specifically, by combining the new endpoint bilinear estimates in $L^{q_α}_x L^2_T$ with the characterization of Triebel-Lizorkin space via fractional semigroup, we prove the well-posedness of the gNS in $\dot{F}^{-α,r}_{q_α}(\mathbb{R}^3)$ for $r=2$. On the other hand, for any $r>2$, we show that the solution to the gNS can develop {\it norm inflation} in the sense that arbitrarily small initial data in the spaces $\dot{F}^{-α,r}_{q_α}(\mathbb{R}^3)$ can lead the corresponding solution to become arbitrarily large after an arbitrarily short time. In particular, such dichotomy of Triebel-Lizorkin spaces is also true for the classical N-S equations, i.e.\,\,$α=1$. Thus the Triebel-Lizorkin space framework naturally provides better connection between the well-known Koch-Tataru's $BMO^{-1}$ well-posed work and Bourgain-Pavlović's $\dot{B}_\infty^{-1,\infty}$ ill-posed work.
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.
Local stay of the solutions to the Euler equations for an ideal incompressible fluid in the end-point Triebel-Lizorkin spaces $F^s_{1, \infty}(\mathbb{R}^d)$ with $s \geq d + 1$ is clarified.
In this paper we establish optimal local and global Besov-Lipschitz and Triebel-Lizorkin estimates for the solutions to linear hyperbolic partial differential equations. These estimates are based on local and global estimates for Fourier integral operators that span all possible scales (and in particular both Banach and quasi-Banach scales) of Besov-Lipschitz spaces $B^s_{p,q}(\R^n)$, and certain Banach and quasi-Banach scales of Triebel-Lizorkin spaces $F^s_{p,q}(\R^n)$
In this paper we obtain new estimates for bilinear pseudodifferential operators with symbol in the class $BS_{1,1}^m$, when both arguments belong to Triebel-Lizorkin spaces of the type $F_{p,q}^{n/p}(\mathbb{R}^n)$. The inequalities are obtained as a consequence of a refinement of the classical Sobolev embedding $F^{n/p}_{p,q}(\mathbb{R}^n)\hookrightarrow\mathrm{bmo}(\mathbb{R}^n)$, where we replace $\mathrm{bmo}(\mathbb{R}^n)$ by an appropriate subspace which contains $L^\infty(\mathbb{R}^n)$. As an application, we study the product of functions on $F_{p,q}^{n/p}(\mathbb{R}^n)$ when $1<p<\inf…
Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel-Lizorkin space $F^{d+1}_{1, \infty}(\mathbb{R}^d)$
Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel--Lizorkin spaces
伪微分边界算子的Boutet de Monvel演算被推广到Besov和Triebel-Lizorkin空间的全尺度上（尽管后者具有有限的整指数）。本文证明的连续性和Fredholm性质推广了Franke和Grubb以前的结果，关于满射椭圆Green算子的值域补的结果改进了以前的结果，甚至对于具有$1<p<\infty$的经典空间也是如此.处理的符号类是一致估计的。给出了0类迹和奇异Green算子的一般定义的一些精确化。
Sharp regularity for the periodic Camassa--Holm equation in critical Triebel--Lizorkin spaces
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（华中科技大学工程建模与科学计算湖北省重点实验室）
In this work we investigate the boundedness of Fourier multipliers on Triebel-Lizorkin spaces associated to positive Rockland operators on a graded Lie group. The found criterion is expressed in terms of the Hörmander-Mihlin condition on the global symbol of a Fourier multiplier.
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.
Let $G:\mathbb{R\rightarrow R}$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on $G$ such that \begin{equation*} \{G(f):f\in \dot{K}_{p,q}^{α}F_{β}^{s}\}\subset \dot{K}_{p,q}^{α}F_{β}^{s} \end{equation*} holds. Here $\dot{K}_{p,q}^{α}F_{β}^{s}$ are Herz-type Triebel-Lizorkin spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights, Sobolev and Triebel-Lizorkin spaces of power weights. In the second part of this paper we will study local and global Cauchy problems for the semilinear parabo…
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 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.
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…
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 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 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…
Matrix-Weighted Besov-Triebel-Lizorkin Spaces of Optimal Scale: Real-Variable Characterizations, Invariance on Integrable Index, and Sobolev-Type Embedding
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
The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting à la Hedberg$\&$Netrusov, which includes weighted anisotropic mixed-norm Besov and Lizorkin-Triebel spaces. In the special case of the classical Lizorkin-Triebel spaces, the intersection representation gives an improvement of the well-known Fubini property. The main result has applications in the weighted $L_{q}$-$L_{p}$-maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Lizorkin-Triebel spaces occu…
