Classification of a class of planar quadrinomials

Let $p$ be an odd prime, $k,\ell$ be positive integers, $q=p^k, Q=p^{\ell}$. In this paper we characterise planar functions of the form $f_{\underline{c}}(X)=c_0X^{qQ+q}+c_1X^{qQ+1}+c_2X^{Q+q}+c_3X^{Q+1}$ over $\mathbb{F}_{q^2}$ for any $\underline{c}=(c_0,c_1,c_2,c_3) \in \mathbb{F}_{q^2}^4$ in terms of linear equivalence.