One-Parameter Families of Operators in $\mathbb{C}$

TL;DR

This paper introduces one-parameter families of operators in the complex plane that characterize solutions to the ar problem, establishing boundedness properties and a correspondence with nonisotropic smoothing operators on polynomial models.

Contribution

It develops a new class of OPF operators for the ar problem, proves their boundedness on L^q spaces, and establishes a Fourier transform-based correspondence with NIS operators.

Findings

Order 0 OPF operators are bounded on L^q for 1<q<

Bound depends on q and degree of p, not on ar operator parameters

One-to-one correspondence between OPF and NIS operators via Fourier transform

Abstract

We develop classes of one-parameter families (OPF) of operators on $C^\infty_c(\mathbb{C})$ which characterize the behavior of operators associated to the $\bar\partial$-problem in $L^2(\mathbb{C},e^{-2p})$ where $p$ is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from $L^q(\mathbb{C})$ to itself, $1<q<\infty$, with a bound that depends on $q$ and the degree of $p$ but not on the parameter $\tau$ or the coefficients of $p$. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in $\tau$ between OPF operators of order $m\leq 2$ and nonisotropic smoothing (NIS) operators of order $m\leq 2$ on polynomial models in $\mathbb{C}^2$.