Limiting fractional and Lorentz spaces estimates of differential forms

极限吸收原理与分数霍尔次方程的辐射条件

Limiting Absorption Principle and Radiation Condition for the Fractional Helmholtz Equation

本文研究了分数霍尔次方程的辐射条件，通过极限吸收原理计算了自由空间格林函数，并证明了其渐进行为与经典霍尔次方程一致，为分数阶霍尔次方程的散射理论提供了理论基础。

Essential self-adjointness of the wave operator and the limiting absorption principle on Lorentzian scattering spaces

We discuss the essential self-adjointness of wave operators, as well as the limiting absorption principle, in generalizations of asymptotically Minkowski settings. This is obtained via using a Fredholm framework for inverting the spectral family first, and then refining its conclusions to show its dense range in L^2 when acting on an appropriate subdomain.

Strengthened Fractional Sobolev Type Inequalities in Besov Spaces

The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of th…

全空间 $\mathbb R^N$ 中的非局部 Schrödinger-Poisson 系统：分数阶 Sobolev 极限情形

Nonlocal Schrödinger-Poisson systems in $\mathbb R^N$: the fractional Sobolev limiting case

本文研究全空间中非局部 Schrödinger-Poisson 系统在分数阶 Sobolev 极限情形下的正解存在性，通过变分逼近方法处理符号变化对数 Riesz 核带来的困难，并建立了解的对称性、正则性与衰减性。

具有分数阶型耗散的拟线性波方程的极限行为

Limiting behavior of quasilinear wave equations with fractional-type dissipation

本文研究具有时间非局部耗散的Westervelt型拟线性波方程，在记忆核弱奇异的最小假设下证明其适定性，并分析小参数趋于零时解的极限行为。

Global existence and limiting behavior of unidirectional flocks for the fractional Euler Alignment system

In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels $ϕ(x):=|x|^{-(n+α)}$ for $α\in(0,2)$. Here, we consider the critical case $α=1$ and establish a couple of global existence results of smooth solutions, together with a full description of their long time dynamics. The first one is obtained via Schauder-type estimates under a null initial entropy condition and the other is a small data result. In fact, using Duhamel's approach we get that any solution is almost Lipschitz-continuous in space. We exte…

Small order asymptotics for nonlinear fractional problems

We study the limiting behavior of solutions to boundary value nonlinear problems involving the fractional Laplacian of order $2s$ when the parameter $s$ tends to zero. In particular, we show that least-energy solutions converge (up to a subsequence) to a nontrivial nonnegative least-energy solution of a limiting problem in terms of the logarithmic Laplacian, i.e., the pseudodifferential operator with Fourier symbol $\ln(|ξ|^2)$. These results are motivated by some applications of nonlocal models where a small value for the parameter $s$ yields the optimal choice. Our approach is based on varia…

Limit of Fractional Power Sobolev Inequalities

We derive the Moser-Trudinger-Onofri inequalities on the 2-sphere and the 4-sphere as the limiting cases of the fractional power Sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by Thomas P. Branson.

A global fractional Caccioppoli-type estimate for solutions to nonlinear elliptic problems with measure data

We prove a global fractional differentiability result via the fractional Caccioppoli-type estimate for solutions to nonlinear elliptic problems with measure data. This work is in fact inspired by the recent paper [B. Avelin, T. Kuusi, G. Mingione, {\em Nonlinear Calderón-Zygmund theory in the limiting case}, Arch. Rational. Mech. Anal. {\bf 227}(2018), 663--714], that was devoted to the local fractional regularity for the solutions to nonlinear elliptic equations with right-hand side measure, of type $-\mathrm{div}\, \mathcal{A}(\nabla u) = μ$ in the limiting case. Being a contribution to rece…

三维Navier–Stokes方程与Euler方程的分数阶Voigt正则化：全局适定性与极限行为

Fractional Voigt-regularization of the 3D Navier--Stokes and Euler equations: Global well-posedness and limiting behavior

本研究提出带分数阶幂次的Voigt正则化推广方法，证明三维周期情形下fNSV、fEV方程的全局适定性，分析多类极限收敛行为并推导爆破准则，为湍流建模与解的爆破研究提供工具。

Lorentz gradient estimates for a class of elliptic p-Laplacian equations with a Schrödinger term

We prove in this paper the global Lorentz estimate in term of fractional-maximal function for gradient of weak solutions to a class of p-Laplace elliptic equations containing a non-negative Schrödinger potential which belongs to reverse Hölder classes. In particular, this class of p-Laplace operator includes both degenerate and non-degenerate cases. The interesting idea is to use an efficient approach based on the level-set inequality related to the distribution function in harmonic analysis.

离散分数阶极大算子在加权Lorentz序列空间之间有界性的刻画

Characterization of the boundedness of the discrete Fractional maximal operator between weighted in Lorentz sequence spaces

本文完整刻画了离散分数阶极大算子在加权Lorentz序列空间间的有界性，通过尖锐重排估计将其归结为上确界算子的有界性，并利用近期结果得到最终判定。

Lorentz improving estimates for the $p$-Laplace equations with mixed data

The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet $p$-Laplace equation of type \begin{align*} \begin{cases} \mathrm{div}(|\nabla u|^{p-2}\nabla u) &= f+ \ \mathrm{div}(|\mathbf{F}|^{p-2}\mathbf{F}) \qquad \text{in} \ \ Ω, \\ \hspace{1.2cm} u &=\ g \hspace{3.1cm} \text{on} \ \ \partial Ω, \end{cases} \end{align*} in Lorentz space, with given data $\mathbf{F} \in L^p(Ω;\mathbb{R}^n)$, $f \in L^{\frac{p}{p-1}}(Ω)$, $g \in W^{1,p}(Ω)$ for $p>1$ and $Ω\subset \ma…

分数抛物理论作为分数椭圆理论的高维极限

Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory

本文延续相关研究计划，将高维极限技术应用于分数算子领域，基于新的椭圆定理和精细极限论证，证明退化抛物方程的三类泛函单调性，其中Alt-Caffarelli-Friedman单调性结果为新成果。

非齐次分数阶 Schrödinger 方程约束极小元的极限行为

Limiting Behavior of Constraint Minimizers for Inhomogeneous Fractional Schrödinger Equations

本文研究非齐次分数阶 Schrödinger 方程的 $L^2$ 约束极小化问题，刻画临界参数 $a^*$ 处极小元的存在性分类及非负极小元的极限行为。

A Limiting absorption principle for high-order Schrödinger operators in critical spaces

In this paper, we prove a limiting absorption principle for high-order Schrödinger operators with a large class of potentials which generalize some results by A. Ionescu and W. Schlag. Our main idea is to handle the boundary operators by the restriction theorem of Fourier transform. Two key tools we use in this paper are the Stein--Tomas theorem in Lorentz spaces and a sharp trace lemma given by S. Agmon and L. Hörmander

$L^2$-临界伪相对论费米系统的极限行为

On the Limiting Behavior of $L^2$-Critical Pseudo-Relativistic Fermi Systems

研究伪相对论费米系统在$L^2$临界情况下的基态存在性及极限行为，证明基态存在当且仅当吸引力强度$a$满足$0 < a < D_{4/3,2}$，并探讨$a$趋近于$D_{4/3,2}$时基态的行为。

Interpolation inequalities in function spaces of Sobolev-Lorentz type

Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters of the function spaces. Several results on necessary conditions are also provided. Next, utilizing the interpolation inequalities together with some embedding results, we prove Gagliardo-Nirenberg inequalities for fractional derivatives in Lorentz spaces, which do hold even for the limiting case when one of the parameters is equal to 1 or $\infty$.

On the limiting behaviour of some nonlocal seminorms: a new phenomenon

In this note we study the behaviour as $s\to 0^+$ of some semigroup based Besov seminorms associated with a non-symmetric and hypoelliptic diffusion with a drift. Our results generalise a previous one of Maz'ya and Shaposhnikova for the classical fractional Sobolev spaces $W^{s,p}$, and they also underscore a new phenomenon caused by the presence of the drift.

关于参数化非局部–分数阶传输问题及相关函数空间

On parameterized nonlocal-fractional transmission problems and associated function spaces

针对跨界面耦合的非局部与分数阶模型，论文由参数化能量变分建立严格数学框架，证明适定性并分析多种极限行为。

关于 Herz-Bochkarev 极限问题

On Herz-Bochkarev limiting problem

本文利用新型 grand Lorentz 空间技术研究 $L_{p,q}$ 中 Hausdorff-Young 型不等式常数随 $p$ 的变化，得出 $p\ ightarrow 2$ 极限情形的最优界，解决 Herz-Bochkarev 问题并改进已有估计。

Strong competition versus fractional diffusion: the case of Lotka-Volterra interaction

We consider a system of differential equations with nonlinear Steklov boundary conditions, related to the fractional problem $$(-Δ)^s u_i = f_i(x,u_i) - βu_i^p \sum_{j\neq i} a_{ij} u_j^p,$$ where $i = i,\dots, k$, $s\in(0,1)$, $p>0$, $a_{ij}>0$ and $β>0$. When $k=2$ we develop a quasi-optimal regularity theory in $C^{0,α}$, uniformly w.r.t. $β$, for every $α< α_{\mathrm opt}=min(1,2s)$; moreover we show that the traces of the limiting profiles as $β\to+\infty$ are Lipschitz continuous and segregated. Such results are extended to the case of $k\geq3$ densities, with some restrictions on $s$, $…

On the two-power nonlinear Schrödinger equation with non-local terms in Sobolev-Lorentz spaces

We are concerned with the two-power nonlinear Schrödinger-type equations with non-local terms. We consider the framework of Sobolev-Lorentz spaces which contain singular functions with infinite-energy. Our results include global existence, scattering and decay properties in this singular setting with fractional regularity index. Solutions can be physically realized because they have finite local $L^2$-mass. Moreover, we analyze the asymptotic stability of solutions and, although the equation has no scaling, show the existence of a class of solutions asymptotically self-similar w.r.t. the scali…

Fractional Integration and Optimal Estimates for Elliptic Systems

In this paper we give an affirmative answer to the Euclidean analogue of a question of Bourgain and Brezis concerning the optimal Lorentz estimate for a Div-Curl system: The function $Z=\operatorname*{curl} (-Δ)^{-1} F$ satisfies \begin{align*} \operatorname*{curl} Z = F \newline \operatorname*{div} Z = 0 \end{align*} and there exists a constant $C>0$ such that \begin{align*} \| Z\|_{L^{3/2,1}(\mathbb{R}^3;\mathbb{R}^3)} \leq C\| F\|_{L^{1}(\mathbb{R}^3;\mathbb{R}^3)}. \end{align*} Our proof relies on a new endpoint Hardy-Littlewood-Sobolev inequality for divergence free measures which we obta…

Endpoint $L^1$ estimates for Hodge systems

In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$ \|I_αF\|_{\dot{B}^{0,1}_{d/(d-α),1}(\mathbb{R}^d;\mathbb{R}^k)} \leq C \|F \|_{L^1(\mathbb{R}^d;\mathbb{R}^k)} $$ for all $F \in L^1(\mathbb{R}^d;\mathbb{R}^k)$ which satisfy a first order cocancelling differential constraint. We show how this implies endpoint Besov-Lorentz estimates for Hodge systems with $L^1$ data via fractional integration for exterior derivatives.

Fractional Adams-Moser-Trudinger type inequalities

Extending several works, we prove a general Adams-Moser-Trudinger type inequality for the embedding of Bessel-potential spaces $\tilde H^{\frac{n}{p},p}(Ω)$ into Orlicz spaces for an arbitrary domain $Ω\subset \mathbb{R}^n$ with finite measure. In particular we prove $$\sup_{u\in \tilde H^{\frac{n}{p},p}(Ω), \;\|(-Δ)^{\frac{n}{2p}}u\|_{L^{p}(Ω)}\leq 1}\int_Ωe^{α_{n,p} |u|^\frac{p}{p-1}}dx \leq c_{n,p}|Ω|, $$ for a positive constant $α_{n,p}$ whose sharpness we also prove. We further extend this result to the case of Lorentz-spaces (i.e. $(-Δ)^\frac{n}{2p}u\in L^{(p,q)})$. The proofs are simple…

具有逆平方渐近性的径向薛定谔算子分数幂的尖锐破幂洛伦兹估计

Sharp Broken-Power Lorentz Estimates for Fractional Powers of Radial Schrödinger Operators with Inverse-Square Asymptotics

研究具有逆平方渐近性的径向薛定谔算子分数幂，在特定假设下确定\(H^{-s/2}\)核双边估计的区间，给出\(1 < p, q < \infty\)且\(1 \leq u, v \leq \infty\)时破幂估计的充要分类，涵盖多种情况，证明结合多种分析和定理。