Homogeneous Sobolev Spaces Research Articles

Integro-differential diffusion equations on graded Lie groups

We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on L p ( G ) ( 1 < p < + ∞, G is a graded Lie group). We show the explicit solution of the considered equation. The equation involves a nonlocal in time operator (with a general kernel) and a positive Rockland operator acting on G. Also, we provide L p ( G ) − L q ( G ) ( 1 < p ⩽ 2 ⩽ q < + ∞) norm estimates and time decay rate for the solutions. In fact, by using some contemporary results, one can translate the latter regularity problem to the study of boundedness of its propagator which strongly depends on the traces of the spectral projections of the Rockland operator. Moreover, in many cases, we can obtain time asymptotic decay for the solutions which depends intrinsically on the considered kernel. As a complement, we give some norm estimates for the solutions in terms of a homogeneous Sobolev space in L 2 ( G ) that involves the Rockland operator. We also give a counterpart of our results in the setting of compact Lie groups. Illustrative examples are also given.

On homogeneous Sobolev and Besov spaces on the whole and the half space

On homogeneous Sobolev and Besov spaces on the whole and the half space

Existence and regularity of steady-state solutions of the Navier-Stokes equations arising from irregular data

We analyze the forced incompressible stationary Navier-Stokes flow in R+n, n>2. Existence of a unique solution satisfying a global integrability property measured in a scale of tent spaces is established for small data in homogeneous Sobolev space with s=−12 degree of smoothness. The velocity field is shown to be locally Hölder continuous while the pressure belongs to Llocp for any p∈(1,∞). Our approach is based on the analysis of the inhomogeneous Stokes system for which we derive a new solvability result involving Dirichlet data in Triebel-Lizorkin classes with negative amount of smoothness and is of independent interest.

A note on the inhomogeneous fractional nonlinear Schrödinger equation

This paper investigates some well-posedness issues of the fractional inhomogeneous Schrödinger equation iu˙-(-Δ)γu=±|x|ρ|u|p-1u,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} i\\dot{u}-(-\\Delta )^\\gamma u=\\pm |x|^\\rho |u|^{p-1}u, \\end{aligned}$$\\end{document}where 0<γ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\gamma <1$$\\end{document} and ρ<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho <0$$\\end{document}. Here, one considers the inter-critical regime 0<sc:=N2-2γ+ρp-1<γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<s_c:=\\frac{N}{2}-\\frac{2\\gamma +\\rho }{p-1}<\\gamma $$\\end{document}, where sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_c$$\\end{document} is the energy critical exponent, which is the only one real number satisfying ‖κ2γ+ρp-1u0(κ·)‖H˙sc=‖u0‖H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert \\kappa ^\\frac{2\\gamma +\\rho }{p-1}u_0(\\kappa \\cdot )\\Vert _{\\dot{H}^{s_c}}=\\Vert u_0\\Vert _{\\dot{H}^{s_c}}$$\\end{document}. In order to avoid a loss of regularity in Strichartz estimates, one assumes that the datum is spherically symmetric. First, using a sharp Gagliardo–Nirenberg-type estimate, one develops a local theory in the space H˙γ∩H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^\\gamma \\cap \\dot{H}^{s_c}$$\\end{document}. Then, one investigates the LN(p-1)ρ+2γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\frac{N(p-1)}{\\rho +2\\gamma }}$$\\end{document} concentration of finite-time blow-up solutions bounded in H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^{s_c}$$\\end{document}. Finally, one proves the existence of non-global solutions with negative energy. Since one considers the homogeneous Sobolev space H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^{s_c}$$\\end{document}, the main difficulty here is to avoid the mass conservation law.

Brezis–Seeger–Van Schaftingen–Yung-type characterization of homogeneous ball Banach Sobolev spaces and its applications

Let [Formula: see text] and [Formula: see text] be a ball Banach function space satisfying some extra mild assumptions. Assume that [Formula: see text] or [Formula: see text] is an [Formula: see text]-domain for some [Formula: see text]. In this paper, the authors prove that a function [Formula: see text] belongs to the homogeneous ball Banach Sobolev space [Formula: see text] if and only if [Formula: see text] and [Formula: see text] where [Formula: see text] is related to [Formula: see text]. This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey [Bourgain–Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which is new even in all these special cases; in particular, it coincides with the well-known result of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung when [Formula: see text] with [Formula: see text], while it is still new even when [Formula: see text] with [Formula: see text]. The novelty of this paper exists in that, to establish the characterization of [Formula: see text], the authors provide a machinery via using the generalized Brezis–Seeger–Van Schaftingen–Yung formula on [Formula: see text], the extension theorem on [Formula: see text], the Bourgain–Brezis–Mironescu-type characterization of the inhomogeneous ball Banach Sobolev space [Formula: see text], and the method of extrapolation to overcome those difficulties caused by that [Formula: see text] might be neither the rotation invariance nor the translation invariance and that the norm of [Formula: see text] has no explicit expression.

Global existence and decay estimate of solution to rate type viscoelastic fluids

We are concerned with the global well-posedness of rate type viscoelastic fluids system (1.1)–(1.2). First, we prove the global existence and decay estimate of solution to (1.1)–(1.2) with stress diffusion near a constant state in HN(R3) (N≥4). We only assume that the H3-norm of initial data is small, but the L2-norm of the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev or Besov spaces, we obtain the optimal decay rates of the solution and its higher order derivatives. Next, we establish the local existence of smooth solution to (1.1)–(1.2) without stress diffusion and present a blow-up criterion in R2. From which and the Bony decomposition, we prove the global existence of smooth solution.

Existence and uniqueness of limits at infinity for homogeneous Sobolev functions

We establish the existence and uniqueness of limits at infinity along infinite curves outside a zero modulus family for functions in a homogeneous Sobolev space under the assumption that the underlying space is equipped with a doubling measure which supports a Poincaré inequality. We also characterize the settings where this conclusion is nontrivial. Secondly, we introduce notions of weak polar coordinate systems and radial curves on metric measure spaces. Then sufficient and necessary conditions for existence of radial limits are given. As a consequence, we characterize the existence of radial limits in certain concrete settings.

Existence of solutions and their behavior for the anisotropic quasi-geostrophic equation in Sobolev and Sobolev-Gevrey spaces

This paper establishes that the anisotropic quasi-geostrophic equation, with orders of fractional dissipation α and β, admits a unique mild solution θ in homogeneous Sobolev and Sobolev-Gevrey spaces H˙a,σs(R2) (with a≥0, σ≥1, α∈(12,34], β∈(12,34] and max⁡{2α−1,2β−1}≤s<1−|α−β|) provided that the initial data θ0 is small enough in these spaces (the critical cases α=12 and β=12 are studied as well). Moreover, this work also studies the behavior of this same solution θ through the following decay rate:limsupt→∞tκ2max⁡{α,β}‖θ(t)‖H˙a,σκ=0, for all κ≥0, where a≥0, σ≥1, α∈(12,34], β∈(12,34] and max⁡{2α−1,2β−1}≤s≤min⁡{2−2α,2−2β}). It is important to emphasize that the limit superior above is a consequence of the Gevrey regularity of θ and of the fact thatlimt→∞⁡‖θ(t)‖L2=0, if it is assumed that θ0∈L2(R2).

On Sobolev norms involving Hardy operators in a half-space

We consider Hardy operators on the half-space, that is, ordinary and fractional Schrödinger operators with potentials given by the appropriate power of the distance to the boundary. We show that the scales of homogeneous Sobolev spaces generated by the Hardy operators and by the fractional Laplacian are comparable with each other when the coupling constant is not too large in a quantitative sense. Our results extend those in the whole Euclidean space and rely on recent heat kernel bounds.

Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces

Abstract Our main purpose is to establish Gagliardo-Nirenberg-type inequalities using fractional homogeneous Sobolev spaces and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in previous studies.

Asymptotics of singular values for quantised derivatives on noncommutative tori

We obtain Weyl type asymptotics for pseudodifferential operators on quantum torus Tθd of the form TIθ−1. Here T stands for classical pseudodifferential operators on quantum torus of order 0, and Iθ−1 is the Riesz potential of order −1. As an application, we obtain Weyl type asymptotics for the quantised derivative ▪ of an operator x from the homogeneous Sobolev space W˙d1(Tθd). The asymptotic coefficient is equivalent to the norm of ▪ in the principal ideal Ld,∞, as well as the norm of x in W˙d1(Tθd). We precise and rectify earlier results in our previous paper (Comm. Math. Phys. 371 (2019), no. 3, 1231–1260) on quantum integration formula for ▪.

Approximate extension in Sobolev space

Let Lm,p(Rn) be the homogeneous Sobolev space for p∈(n,∞), μ be a Borel regular measure on Rn, and Lm,p(Rn)+Lp(dμ) be the space of Borel measurable functions with finite seminorm ‖f‖Lm,p(Rn)+Lp(dμ):=inff1+f2=f⁡{‖f1‖Lm,p(Rn)p+∫Rn|f2|pdμ}1/p. We construct a linear operator T:Lm,p(Rn)+Lp(dμ)→Lm,p(Rn), that nearly optimally decomposes every function in the sum space: ‖Tf‖Lm,p(Rn)p+∫Rn|Tf−f|pdμ≤C‖f‖Lm,p(Rn)+Lp(dμ)p with C dependent on m, n, and p only. For E⊂Rn, let Lm,p(E) denote the space of all restrictions to E of functions F∈Lm,p(Rn), equipped with the standard trace seminorm. For p∈(n,∞), we construct a linear extension operator T:Lm,p(E)→Lm,p(Rn) satisfying Tf|E=f|E and ‖Tf‖Lm,p(Rn)≤C‖f‖Lm,p(E), where C depends only on n, m, and p. We show these operators can be expressed through a collection of linear functionals whose supports have bounded overlap.

Global solutions to the dissipative quasi-geostrophic equation with dispersive forcing

We consider the initial value problem for the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing and prove the global existence of a unique solution in the scaling subcritical Sobolev spaces $H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) and the scaling critical space $H^{2-\alpha}(\mathbb{R}^2)$. More precisely, for the scaling subcritical case, we establish a unique global solution for a given initial data $\theta_{0} \in H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) if the size of dispersion parameter is sufficiently large and also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. For the scaling critical case, we find that the size of dispersion parameter to ensure the global existence is determined by each subset $K \subset H^{2-\alpha}(\mathbb{R}^{2})$, which is precompact in some homogeneous Sobolev spaces.

Function spaces via fractional Poisson kernel on Carnot groups and applications

We provide a new characterization of homogeneous Besov and Sobolev spaces in Carnot groups using the fractional heat kernel and Poisson kernel. We apply our results to study commutators involving fractional powers of the sub-Laplacian.

On the Oseen-Type Resolvent Problem Associated with Time-Periodic Flow past a Rotating Body

Consider the time-periodic flow of an incompressible viscous fluid past a body performing a rigid motion with non-zero translational and rotational velocity. We introduce a framework of homogeneous Sobolev spaces that renders the resolvent problem of the associated linear problem well posed on the whole imaginary axis. In contrast to the cases without translation or rotation, the resolvent estimates are merely uniform under additional restrictions, and the existence of time-periodic solutions depends on the ratio of the rotational velocity of the body motion to the angular velocity associated with the time period. Provided that this ratio is a rational number, time-periodic solutions to both the linear and, under suitable smallness conditions, the nonlinear problem can be established. If this ratio is irrational, a counterexample shows that in a special case there is no uniform resolvent estimate and solutions to the time-periodic linear problem do not exist.

On the Stokes-Type Resolvent Problem Associated with Time-Periodic Flow Around a Rotating Obstacle

Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.

Embeddings of function spaces via the Caffarelli–Silvestre extension, capacities and Wolff potentials

Let Pαf(x,t) be the Caffarelli–Silvestre extension of a smooth function f(x):Rn→R+n+1≔Rn×(0,∞). The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the Lebesgue spaces Lp(Rn) to the Lq(R+n+1,μ). On one hand, these embeddings will be characterized by using a newly introduced Lp−capacity associated with the Caffarelli–Silvestre extension. In doing so, the mixed norm estimates of Pαf(x,t), the dual form of the Lp−capacity, the Lp−capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when p>q>1, these embeddings will also be characterized in terms of the Hedberg–Wolff potential of μ. Secondly, we characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the homogeneous Sobolev spaces Ẇβ,p(Rn) to the Lq(R+n+1,μ) in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.

OSEEN SYSTEM WITH CORIOLIS TERM

This paper is devoted to solutions of the Dirichlet problem for the Oseen system with Coriolis term –�u(z) + �� 1 u(z) – (� × z) � �u(z) + �� × u(z) + �p(z) = f (z), � . � in �, u = g on �� in the homogeneous Sobolev space 1, 3 ( ; ) ( )q q W L� � �� � with 2 � q &lt; 3. Here �� � � 3 is an exterior domain. Kracmar, Necasová and Penel proved that if � has boundary of class � 2 , g � 0 and f � D –1,q (�; � 3 ), then there exists a unique solution of the problem. This paper shows that this result holds true even for domains with Lipschitz boundary. Moreover, we prove unique solvability of the problem for general g � W 1–1/q,q (��;� � 3 ) and f � D –1,q (�; � 3 ).

Sampling in spaces of entire functions of exponential type in [formula omitted

In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose on the entire functions, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary bU of the Siegel upper half-space U and it is fundamental that bU can be identified with the Heisenberg group Hn. We consider entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restriction to bU are square integrable with respect to the Haar measure on Hn. For these functions we prove a version of the Whittaker–Kotelnikov–Shannon Theorem. Instrumental in our work are spaces of entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restrictions to bU belong to some homogeneous Sobolev space on Hn. For these spaces, using the group Fourier transform on Hn, we prove a Paley–Wiener type theorem and a Plancherel–Pólya type inequality.

A density result for homogeneous Sobolev spaces

We show that in a bounded Gromov hyperbolic domain Ω smooth functions with bounded derivatives C∞(Ω)∩Wk,∞(Ω) are dense in the homogeneous Sobolev spaces Lk,p(Ω).