A fractional version of Rivière's GL(N)-gauge

Note on the lifespan estimate of solutions for non-gauge invariant semilinear massless semirelativistic equations with some scaling critical nonlinearity

In this manuscript, in the $L^1$ scaling critical case, a lifespan estimate of solutions to the Cauchy problem for non-gauge invariant semilinear semirelativistic equations is considered. The lifespan estimate is given by the modified test function method with a fractional Laplace operator. The main obstacle to obtaining the lifespan estimate is the non-locality of the fractional Laplace operator. To treat the non-locality, special test functions are introduced.

分数阶方程的弱解序列

Sequences of weak solutions for fractional equations

本文研究了一类含分数阶积分微分算子的非局部方程无穷多个弱解的存在性。这些方程具有变分结构，我们发现了一个序列的非平凡的弱解，他们利用${\mathbb{Z}}_2$对称版本的山路定理。为了使非线性方法工作，对所涉及的分数空间进行一些仔细的分析是必要的。作为一个特殊的情况下，我们推导出一个存在性定理的分数拉普拉斯方程，发现非平凡的解决方案的方程$$ \left\{\begin{array}{ll} (-Δ)^s u=f(x,u) & {\mbox{in}} Ω\\ u=0 & {\mbox{in}} \erre^n\setminus Ω. \end{array} \right. $$据我们所知，所有这些结果是新的，并表示一个分数版本的经典定理与拉普拉斯方程。

A self-similar solution to time-fractional Stefan problem

We derive the fractional version of one-phase one-dimensional Stefan model. We assume that the diffusive flux is given by the time-fractional Riemann-Liouville derivative, i.e. we impose the memory effect in the examined model. Furthermore, we find a special solution to this problem.

分数阶Kadomtsev-Petviashvili方程的Cauchy问题

The Cauchy problem for the fractional Kadomtsev-Petviashvili equations

本文的目的是证明一类分数阶Kadomtsev-Petviashvili（KP）方程的Cauchy问题的各种不适定性和适定性结果，包括Benjamin-Ono方程的KP形式和中间长波方程.

On a Fractional Version of a Murat Compactness Result and Applications

The paper provides an extension, to fractional order Sobolev spaces, of the classical result of Murat and Brezis which states that the positive cone of elements in $H^{-1}(Ω)$ compactly embeds in $W^{-1,q}(Ω)$, for every $q < 2$ and for any open and bounded set $Ω$ with Lipschitz boundary. In particular, our proof contains the classical result. Several new analysis tools are developed during the course of the proof to our main result which are of wider interest. Subsequently, we apply our results to the convergence of convex sets and establish a fractional version of the Mosco convergence resu…

分数阶Sobolev不等式与Hardy-Littlewood-Sobolev不等式

Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities

本文研究了一个改进的分数阶Sobolev不等式，它的余项涉及最近被证明的Hardy-Littlewood-Sobolev不等式.通过将最近关于标准拉普拉斯算子的一个结果推广到分数阶情形，我们提供了一个新的、更简单的证明，并对所涉及的最佳常数提供了新的估计。利用端点微分，我们还获得了球面上Moser-Trudinger-Onofri型不等式的改进版本。作为直接结果，我们利用球极平面投影得到了欧几里得空间上Onofri不等式的一个改进形式。

Extension of Mikhlin Multiplier Theorem to Fractional Derivatives and Stable Processes

In this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between fractional derivatives and stable processes and prove a version of Mikhlin theorem under a condition given in terms of the infinitesimal generator of symmetric stable process. The classical Mikhlin theorem is shown to be a corollary of this new generalized version in this paper.

分数次极大算子的加权范数不等式--Bellman函数方法

Weighted norm inequalities for fractional maximal operators--a Bellman function approach

本文研究了$\R^d$上的分数阶极大算子的加权$L^p\to L^q$不等式，该不等式最初是由Muckenhoupt和Wheeden在70年代证明的.我们建立了一个稍强的版本，这个不等式的使用一个新的扩展Bellman函数方法。更准确地说，估计推导出存在的某一特殊函数享有适当的优化和优化。从这个结果和“$A_{p-\varepsilon}$定理”的一个显式版本，也用Bellman函数导出，我们得到了Lacey，Moen，Pérez和Torres的尖锐不等式。

Heisenberg群上次拉普拉斯算子分数幂的Hardy不等式

Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group

证明了Heisenberg群$\mathbb{H}^n$上次拉普拉斯算子的共形不变分数幂的Hardy不等式.我们证明了两个版本的不等式取决于所涉及的权重是否是非齐次或齐次。在第一种情况下，哈迪不等式中的常数是最优的。为了得到我们的结果，我们将使用基态表示。得到后者的关键是次拉普拉斯算子的分数幂的显式整表示和M. Cowling和U.哈格鲁普证明整表示的方法是通过半群的语言。作为Hardy不等式的结果，我们还得到了分数阶次拉普拉斯算子的海森堡不确定性不等式。