  Planar functions, introduced by Dembowski and Ostrom, have attracted much
attention in the last decade. As shown in this paper, we present a new class of
planar functions of the form $\operatorname{Tr}(ax^{q+1})+\ell(x^2)$ on an
extension of the finite field $\mathbb F_{q^n}/\mathbb F_q$. Specifically, we
investigate those functions on $\mathbb F_{q^2}/\mathbb F_q$ and construct
several typical kinds of planar functions. We also completely characterize them
on $\mathbb F_{q^3}/\mathbb F_q$. When the degree of extension is higher, it
will be proved that such planar functions do not exist given certain
conditions.
