The aim of this short review is to attract the attention of the pseudo-differential community to possibilities in the development of operator calculus for symbols (depending on p-adic conjugate variables) taking values in fields of p-adic numbers. Essentials of this calculus were presented in works of the authors of this paper in order to perform p-adic valued quantization. Unfortunately, this ...

We consider here pseudo-differential operators whose symbol σ(x, ξ) is not infinitely smooth with respect to x. Decomposing such symbols into four -sometimes fivecomponents and using tools of paradifferential calculus, we derive sharp estimates on the action of such pseudo-differential operators on Sobolev spaces and give explicit expressions for their operator norm in terms of the symbol σ(x, ...

Pseudo-Hermiticity as a generalization of usual Hermiticity is a rather common feature of (differential) operators emerging in various physical setups. Examples are Hamiltonians of PTand CPT-symmetric quantum mechanical systems [1] as well as the operator of the spherically symmetric α-dynamo [2] in magnetohydrodynamics (MHD). In order to solve the inverse spectral problem for these operators, ...

This article proves the regularity for Boltzmann equation without angular cutoff with hard potential. By sharpening coercivity and upper bound estimate on collision operator, analyzing Poisson bracket between transport operator some weighted pseudo-differential we prove regularizing effect in space velocity variables when initial data has mild regularity.

We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the farfield operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the orde...

In this paper, we derive a nonlinear Picone identity to the pseudo p-Laplace operator, which contains some known Picone identities and removes a condition used in many previous papers. Some applications are given including a Liouville type theorem to the singular pseudo p-Laplace system, a Sturmian comparison principle to the pseudo p-Laplace equation, a new Hardy type inequality with weight an...

In this paper, applying two theorems of Ricceri and Bonanno, we will establish the existence of three weak solutions for a quasilinear elliptic system. Indeed, we will assign a differentiable nonlinear operator to a differential equation system such that the critical points of this operator are weak solutions of the system. In this paper, applying two theorems of R...