Nonlinear generalised functions on manifolds

A nonlinear theory of distributional geometry

This paper builds on the theory of generalised functions begun in [1]. The Colombeau theory of generalised scalar fields on manifolds is extended to a nonlinear theory of generalised tensor fields which is diffeomorphism invariant and has the sheaf property. The generalised Lie derivative for generalised tensor fields is introduced and it is shown that this commutes with the embedding of distributional tensor fields. It is also shown that the covariant derivative of generalised tensor fields commutes with the embedding at the level of association. The concept of generalised metric is introduce…

$L^p$-$L^q$ boundedness of $(k, a)$-Fourier multipliers with applications to Nonlinear equations

The $(k,a)$-generalised Fourier transform is the unitary operator defined using the $a$-deformed Dunkl harmonic oscillator.The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k, a)$-generalised Fourier multipliers. To show the boundedness we first establish Paley inequality and Hausdorff-Young-Paley inequality for $(k, a)$-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations.

广义柯西测度与凸测度的庞加莱不等式与积分曲率-维数准则

Poincar{é} inequalities and integrated curvature-dimension criterion for generalised Cauchy and convex measures

本文在黎曼流形上建立一般测度的尖锐加权庞加莱不等式，并由此统一证明广义柯西测度在全部参数范围内的加权庞加莱不等式，同时给出最优谱间隙、误差项与极值函数。