Compact embeddings for fractional super and sub harmonic functions with radial symmetry

积分几何中的分数阶积分与切触问题

Fractional Integrals and Tangency Problems in Integral Geometry

本文在欧氏、球面和双曲空间中构造了沿与给定曲面相切的低维测地球面或截面进行积分的Radon型变换新例子，给出反演公式并分析切点奇异性，同时探讨其在数学层析成像中的应用。

分数阶双侧四元数Dunkl变换与Heisenberg型不等式

The Fractional Two-Sided Quaternionic Dunkl Transform and Heisenberg-Type Inequalities

本文研究了二维分数阶双侧四元数Dunkl变换的定义与性质，建立了经典Heisenberg型不确定性原理的广义形式，并推广证明了适用于任意矩的高阶Heisenberg型不等式，最后推导出双侧分数阶四元数Fourier变换的相应不确定性原理。

径向权诱导的分数阶Volterra型算子在Hardy空间上的作用

Fractional Volterra-type operator induced by radial weight acting on Hardy space

本文研究径向加倍权诱导的分数阶Volterra型算子在Hardy空间上的有界性、紧性及Schatten类性质，给出其成立的充要条件，并得到相关函数空间的刻画。

Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$\|φ\|_{L^p(\mathbb{R}^d)}\le C\|φ\|_{\dot H^{s}(\mathbb{R}^d)}^β \left(\iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{|φ(x)|^q\,|φ(y)|^q}{|x - y|^{d-α}} dx dy\right)^γ,$$ involving fractional Sobolev norms with $s>0$ and Coulomb type energies with $0<α<d$ and $q\ge 1$. We establish optimal ranges of parameters for the validity of such inequalities and discuss the existence of the optimisers. In the special case $p=\frac{2d}{d-2s}$ our results include a new refinement of the fractional Sobolev inequality by a Coulomb…

仿射分数阶 Sobolev 不等式与等周不等式

Affine Fractional Sobolev and Isoperimetric Inequalities

本文建立 $\mathbb R^n$ 上的尖锐仿射分数阶 Sobolev 不等式，强化并推出 Almgren–Lieb 不等式，极限下蕴含 Zhang 的仿射 Sobolev 不等式，同时获得分数阶 Petty 投影不等式并证明径向平均体猜想。

分数阶非局部方程的Harnack不等式

Harnack's inequality for fractional nonlocal equations

证明了分数阶非局部方程解的内部Harnack不等式。我们的例子包括分数幂的散度型椭圆算子的潜力，运营商产生的经典正交展开和径向拉普拉斯算子。为了得到这些结果，我们使用了一种基于Caffarelli-Silvestre延拓问题的推广的分析方法，C. E.古铁雷斯和一种转移方法以这种方式，我们应用本地PDE技术的非本地运营商。在此基础上，得到了一类分数阶非局部方程的最大值原理和Liouville定理。

Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials

This paper is concerned with the following fractional Schrödinger equations involving critical exponents: \begin{eqnarray*} (-Δ)^αu+V(x)u=k(x)f(u)+λ|u|^{2_α^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N}, \end{eqnarray*} where $(-Δ)^α$ is the fractional Laplacian operator with $α\in(0,1)$, $N\geq2$, $λ$ is a positive real parameter and $2_α^{*}=2N/(N-2α)$ is the critical Sobolev exponent, $V(x)$ and $k(x)$ are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the…

Bloch 空间的分数阶导数描述

Fractional derivative description of the Bloch space

本文通过径向权重诱导的分数阶导数建立了 Bloch 空间的新刻画，证明了新空间连续嵌入 Bloch 空间，并给出了两者等价的径向权重条件。

Non-Archimedean Radial Calculus: Volterra Operator and Laplace Transform

In an earlier paper (A. N. Kochubei, {\it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirov's fractional differentiation operator $D^α$, $α>0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^α$ that the appropriate change of variables reduces equations with $D^α$ (for radial functions) to integral equations whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary dif…

Sharp bounds for Hardy-type operators on mixed radial-angular spaces

In this paper, by using the rotation method, we calculate that the sharp bound for $n$-dimensional Hardy operator $\mathcal{H}$ on mixed radial-angular spaces. Furthermore, we also obtain the sharp bound for $n$-dimensional fractional Hardy operator $\mathcal{H}_β$ from $L^p_{|x|}L_θ^{\bar{p}}({\Bbb R}^n)$ to $L^q_{|x|}L_θ^{\bar{q}}({\Bbb R}^n)$, where $0<β<n$, $1<p,q,\bar{p},\bar{q}<\infty$ and $1/p-1/q=β/n$. By using duality, the corresponding results for the dual operators $\mathcal{H}^*$ and $\mathcal{H}^*_β$ are also established. In addition, the sharp weak-type estimate for $\mathcal{H}$…