# Pseudo-differential operators in vector-valued spaces and applications

## TL;DR

This paper investigates pseudo-differential operators with parameters in vector-valued spaces, establishing resolvent estimates, regularity properties, and applications to anisotropic and system equations, with implications for analytic semigroup generation.

## Contribution

It introduces new resolvent estimates and regularity results for pseudo-differential operators with parameters, extending their analysis in vector-valued function spaces.

## Key findings

- Operators are positive and generate analytic semigroups.
- Maximal regularity properties are established for pseudo-differential parabolic equations.
- Applications include anisotropic parameter-dependent and system pseudo-differential equations.

## Abstract

Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential abstract parabolic equation are established. Particularly, it is proven that the operators generated by these pseudo-differential equations are positive and aso are generators of analytic semigroups. As an application, the anisotropic parameter dependent pseudo-differential equations and the system of pseudo-differential equations are studied