Zero modes on product Riemannian manifolds

This paper is concerned with the zero mode equation $D_g\varphi=iA\cdot\varphi$ on product of closed spin manifolds $(M_1^{n_1}\times M_2^{n_2},g_1+g_2,\sigma)$ of dimensions $n_1\leq n_2$ respectively. Here $A$ is a real vector field on $M^n=M_1^{n_1}\times M_2^{n_2}$. Under non-increasing condition on $|\varphi|$ we prove that $$\parallel A\parallel_n^2\geq\frac{n_2}{4(n_2-1)}Y(M^n,[g]),$$ where $Y(M^n,[g])$ is the Yamabe constant of $(M^n,g)$. This estimate is sharp in even dimensions. We also obtain a similar estimate for non trivial solutions of the zero mode type equation $D_g\varphi=f\varphi$, where $f$ is a scalar function.