Triebel-Lizorkin Spaces Research Articles

Estimates for Certain Rough Multiple Singular Integrals on Triebel–Lizorkin Space

This paper focuses on studying the mapping properties of singular integral operators over product symmetric spaces. The boundedness of such operators is established on Triebel–Lizorkin spaces whenever their rough kernel functions belong to the Grafakos and Stefanov class. Our findings generalize, extend and improve some previously known results on singular integral operators.

Persistence of the solution to the Euler equations in an end‐point critical Triebel–Lizorkin space

Persistence of the solution to the Euler equations in an end‐point critical Triebel–Lizorkin space

Solution of Linear Damped Wave Equation on Triebel–Lizorkin Spaces

Solution of Linear Damped Wave Equation on Triebel–Lizorkin Spaces

Asymptotic estimates of solution to damped fractional wave equation

It is known that the damped fractional wave equation has the diffusive structure as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$t\\rightarrow \\infty $\\end{document}. Let u(t,x)=e−tcosh(tL)f(x)+e−tsinh(tL)L(f(x)+g(x))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(t,x)=e^{-t}\\cosh (t\\sqrt{L})f(x)+e^{-t} \\frac{\\sinh (t\\sqrt{L})}{\\sqrt{L}}(f(x)+g(x))$\\end{document} be the solution of the Cauchy problem for the damped fractional wave equation, where L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\sqrt{L}$\\end{document} involves the fractional Laplacian (−△)α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(-\ riangle )^{\\alpha}$\\end{document} on the space variable. We can study the decay estimate of the solution u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(t,x)$\\end{document} over the time t by means of the Cauchy problem for the parabolic equation. In this paper, we consider, for 0<α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0<\\alpha <1$\\end{document}, the Cauchy problem in the two- and three-dimensional spaces for the damped fractional wave equation and the corresponding parabolic equation and obtain the Triebel–Lizorkin space estimate of the difference of solutions. At the same time, we also consider, for α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha =1$\\end{document}, the case of the Cauchy problem in the four-dimensional space and obtain a Triebel–Lizorkin space estimate.

Classification of anisotropic local Hardy spaces and inhomogeneous Triebel–Lizorkin spaces

This paper provides a characterization of when two expansive matrices yield the same anisotropic local Hardy and inhomogeneous Triebel–Lizorkin spaces. The characterization is in terms of the coarse equivalence of certain quasi-norms associated to the matrices. For nondiagonal matrices, these conditions are strictly weaker than those classifying the coincidence of the corresponding homogeneous function spaces. The obtained results complete the classification of anisotropic Besov and Triebel–Lizorkin spaces associated to general expansive matrices.

Matrix-weighted Besov–Triebel–Lizorkin spaces with logarithmic smoothness

In this article, the authors study the matrix-weighted Besov–Triebel–Lizorkin spaces with logarithmic smoothness. Via first obtaining the Lp(Rn)-boundedness and the Fefferman–Stein type vector-valued inequality of matrix-weighted Peetre-type maximal functions with the exquisite ranges of indices in terms of the Ap dimension of matrix weights under consideration, the authors establish an equivalent characterization of these spaces in terms of the matrix-weighted Peetre-type maximal functions, which further implies that these spaces are well defined. As an application, the authors obtain the boundedness of some pointwise multipliers on these spaces and, even back to classical Besov–Triebel–Lizorkin spaces, some of them are also new.

Nemytzkij operators on Besov and Triebel-Lizorkin spaces with power weights: necessary conditions

Nemytzkij operators on Besov and Triebel-Lizorkin spaces with power weights: necessary conditions

Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss–Weierstrass semi–groups

AbstractWe present a new proof of the caloric smoothing related to the fractional Gauss–Weierstrass semi–group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.

Capacity in Besov and Triebel–Lizorkin spaces with generalized smoothness

Abstract We prove a lower bound estimate for capacities in Hajłasz–Besov, Hajłasz–Triebel–Lizorkin and Hajłasz–Sobolev spaces with generalized smoothness defined on metric spaces in terms of Netrusov–Hausdorff content or Hausdorff content. These results are improvements of the results obtained in [Z. Li, D. Yang and W. Yuan, Lebesgue points of Besov and Triebel–Lizorkin spaces with generalized smoothness, Mathematics 9 2021, 10.3390/math9212724].

Weighted estimates for product singular integral operators in Journé’s class on RD-spaces

Abstract An RD-space 𝑀 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝑀. In this paper, firstly, the authors give the Plancherel–Pôlya characterization of product weighted Triebel–Lizorkin spaces and product weighted Besov spaces on RD-spaces and make some estimates for the product singular integral operators in Journé’s class on these function spaces. As a result of these conclusions, they present some sufficient conditions for the boundedness of product singular integral operators on the product Lipschitz spaces and product weighted Hardy spaces. Secondly, by the boundedness of lifting and projection operators, they also obtain that the dual spaces of the product weighted Hardy spaces are product weighted Carleson measure spaces. Using the idea of dual, the authors obtain the weighted boundedness of singular integral operators on the product weighted Carleson measure spaces.

Relations between product and flag Triebel-Lizorkin spaces

Nagel, Ricci and Stein proved that product kernels are finite sums of flag kernels in the Euclidean space. We show that the product Triebel-Lizorkin space is the intersection of two flag Triebel-Lizorkin spaces. This extends a main result in [Chang D-C, Han Y, Wu X. Relations between product and flag Hardy spaces. J Geom Anal. 2021;31(7):6601–6623]. As an application, we provide a new proof of the boundedness of product singular integral operators on product Triebel-Lizorkin spaces.

Bourgain-Brezis-Mironescu formula for W^{s,p}_q-spaces in arbitrary domains

Under certain restrictions on s, p, q, the Triebel-Lizorkin spaces can be viewed as generalised fractional Sobolev spaces W^{s,p}_q. In this article, we show that the Bourgain-Brezis-Mironescu formula holds for W^{s,p}_q-seminorms in arbitrary domain. This addresses an open question raised by Brazke-Schikorra-Yung (Calc Var Partial Differ Equ 62(2):41–33, (2023).

Inhomogeneous Besov and Triebel-Lizorkin spaces associated with a para-accretive function and their applications

Inhomogeneous Besov and Triebel-Lizorkin spaces associated with a para-accretive function and their applications

Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel–Lizorkin space $$F^{d+1}_{1, \infty }(\mathbb {R}^d)$$

Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel–Lizorkin space $$F^{d+1}_{1, \infty }(\mathbb {R}^d)$$

Long-time dynamics of classical Keller–Segel equation

In this article, we obtain global dynamics of 2D Keller–Segel equation (KS for short) in subcritical regime with no additional assumptions. With initial data small in some Triebel-Lizorkin space, we also characterize the long-time asymptotics of global mild solutions to the KS equation in higher dimensions where the spatial dimension n≥3.

The Tb theorem for some inhomogeneous Besov and Triebel-Lizorkin spaces over space of homogeneous type

In this paper, the authors give some new characterizations of the inhomogeneous Besov spaces and inhomogeneous Triebel-Lizorkin spaces respectively over space of homogeneous type, and establish the Tb theorems for the boundedness of the Calderón-Zygmund singular integral operator with non-convolution kernel on these new inhomogeneous Besov and Triebel-Lizorkin spaces.

Triebel–Lizorkin regularity and bi-Lipschitz maps: Composition operator and inverse function regularity

We study the stability of Triebel–Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel–Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain the results we provide an equivalent norm for the Triebel–Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.

Dahlberg degeneracy for homogeneous Besov and Triebel–Lizorkin spaces

AbstractWe consider the composition operators acting on the real‐valued homogeneous Besov or Triebel–Lizorkin spaces, realized as dilation invariant subspaces of , denoted as . If and , then any function acting by composition on is necessarily linear. The above conditions are optimal: (i) in case , (Besov space), (Triebel–Lizorkin space), is a quasi‐Banach algebra for the pointwise product, (ii) in case , , , any function such that is a finite measure, and , acts by composition on .

Local solvability for real-analytic involutive structures of tube type of corank one in Besov and Triebel-Lizorkin spaces

This study shows the local solvability, in degree one, for the differential complex associated to a real-analytic, involutive structure of tube type of corank one in mixed spaces involving Besov and Triebel-Lizorkin spaces.

A new regularity criterion for the 3D MHD equations in Triebel-Lizorkin spaces

We consider the regularity criterion for the 3D magnetohydrodynamics (MHD) equations in Triebel-Lizorkin spaces. It is proved that if∇π∈L8p11p−12(0,T;F˙p,8p12−3p0(R3)), with1211<p<4,then the weak solution (u,b) becomes a regular solution on (0,T].