IUMJ

Title: Scalar and vector Muckenhoupt weights

Authors: Michael Lauzon and Sergei Treil

Issue: Volume 56 (2007), Issue 4, 1989-2015

Abstract:

We inspect the relationship between the $\mathcal{A}_{p,q}$ condition for families of norms on vector valued functions and the $A_p$ condition for scalar weights. In particular, we will show if we are considering a norm-valued function $\rho_{(\cdotp)}$ such that, uniformly in all nonzero vectors $x$, $\rho_{(\cdotp)}(x)^p \in A_p$ and $\rho_{(\cdotp)}^*(x)^q \in A_q$, then the following hold: If $p=q=2$, and functions take values in $\mathbb{R}^2$, then $\rho \in \mathcal{A}_{2,2}$. If $p=q=2$ and functions take values in $\mathbb{R}^n$, $n \geq 6$, $\rho$ need not be an $\mathcal{A}_{2,2}$ weight. If $\rho$ satisfies the relatively weak $A_{0,0}$ condition in addition to the scalar conditions mentioned above, then $\rho \in \mathcal{A}_{p,q}$.