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

Abstract

We develop classes of one-parameter families (OPF) of operators on $C_c^{\infty }(\mathbb{C})$ 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 $m\le 2$ and nonisotropic smoothing (NIS) operators of order $m\le 2$ on polynomial models in ${\mathbb{C}}^2$.

v2: 18 pages. AMS-LaTeX. Updated references, numerous errors/typos fixed