On growth conditions for quasiconvex integrands

We prove that, for $1\le p<2$, if a $W^{1,p}$-quasiconvex integrand $f\text{\hspace{0.17em}⁣}:{\mathbb{R}}^{N\times n}\to \mathbb{R}$ has linear growth from above on the rank-one cone, then it must satisfy this growth for all matrices in ${\mathbb{R}}^{N\times n}$. An immediate corollary of this is, for example, that there can be no quasiconvex integrand that has genuinely superlinear $p$ growth from above for $1<p<2$, but only linear growth in rank-one directions. The key element of this proof involves constructing a Sobolev function which maps points in a cube to some one-dimensional frame, and moreover preserves boundary values. This construction is an inductive process on the dimension $n$, and involves using a Whitney decomposition. This technique also allows us to generalise this result for $W^{1,p}$-quasiconvex integrands where $1\leq p < k\leq \min{n,N}$.