Gabor Functional Multiplier in the Higher Dimensions

For two given full-rank lattices $\mathcal{L}=A\mathbb{Z}^d$ and $\mathcal{K}=B\mathbb{Z}^d$ in $\mathbf{R}^d$, where $A$ and $B$ are nonsingular real $d\times d$ matrices, a function $g(\bf{t})\in L^2(\mathbf{R}^d)$ is called a Parseval Gabor frame generator if $\sum_{\bf{l},\bf{k}\in\mathbb{Z}^d}|\langle f, {e^{2\pi i\langle B\bf{k},\bf{t}\rangle}}g(\bf{t}-A\bf{l})\rangle|^2=\|f\|^2$ holds for any $f(\bf{t})\in L^2(\mathbf{R}^d)$. It is known that Parseval Gabor frame generators exist if and only if $|\det(AB)|\le 1$. A function $h\in L^{\infty}(\mathbf{R}^d)$ is called a functional Gabor frame multiplier if it has the property that $hg$ is a Parseval Gabor frame generator for $L^2(\mathbf{R}^d)$ whenever $g$ is. It is conjectured that an if and only if condition for a function $h\in L^{\infty}(\mathbf{R}^d)$ to be a functional Gabor frame multiplier is that $h$ must be unimodular and $h(\bf{x})\overline{h(\bf{x}-(B^T)^{-1}\bf{k})}=h(\bf{x}-A\bf{l})\overline{h(\bf{x}-A\bf{l}-(B^T)^{-1}\bf{k})},\ \forall\ \bf{x}\in \mathbf{R}^d$ {\em a.e.} for any $\bf{l},\bf{k}\in \mathbb{Z}^d$, $\bf{k}\not=\bf{0}$. The if part of this conjecture is true and can be proven easily, however the only if part of the conjecture has only been proven in the one dimensional case to this date. In this paper we prove that the only if part of the conjecture holds in the two dimensional case.