Classification of a class of planar quadrinomials

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