On Some Modifications of the Fueter Operator

Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation

In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as \begin{align*} \partial_t ω+ {\mathcal L}_u ω&= 0\\ u &= δ\tilde η^{-1} Δ^{-1} ω\end{align*} where $ω$ is the vorticity $2$-form, ${\mathcal L}_u$ denotes the Lie derivative with respect to the velocity field $u$, $Δ$ is the Hodge Laplacian, $δ$ is the codifferential (the negative of the divergence operator), and $\tilde η^{-1}$ is the canonical map from $2$-forms to $2$-vector fields induced by the Euclidean metric $η$. In this paper we consider a generali…

Regularity and solvability of linear differential operators in Gevrey spaces: omitted proofs

This is an addendum to a previous article, which aims to provide the proofs of some results in that paper (Theorem 7.5 and Proposition 9.15) which were removed from its final version. The reason for such omission is that these proofs follow quite closely others already present in the literature, with minor modifications. I make them publicly available for the sake of completeness.

A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates

Let $L$ be a nonnegative, self-adjoint operator satisfying Gaussian estimates on $L^2(\RR^n)$. In this article we give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(\R)$ in terms of the nontangential maximal functions associated with the heat semigroup of $L$, and this leads eventually to characterizations of Hardy spaces associated to $L$, via atomic decomposition or the nontangential maximal functions. The proofs are based on a modification of technique due to A. Calderón \cite{C}.

Boltzmann方程的一类各向同性修正的全局存在性

Global existence for an isotropic modification of the Boltzmann equation

针对非截断Boltzmann方程大数据全局存在性公开问题，提出各向同性Boltzmann模型，证明其在空间齐次极软势部分区域的全局存在性，证明借鉴各向同性Landau方程研究思路并运用加权分数阶Hardy不等式新进展。

Mittag-Leffler's function, Vekua transform and an inverse obstacle scattering problem

This paper studies a prototype of inverse obstacle scattering problems whose governing equation is the Helmholtz equation in two dimensions. An explicit method to extract information about the location and shape of unknown obstacles from the far field operator with a fixed wave number is given. The method is based on: an explicit construction of a modification of Mittag-Leffler's function via the Vekua transform and the study of the asymptotic behaviour; an explicit density in the Herglotz wave function that approximates the modification of Mittag-Leffler's function in the bounded domain surro…