Gradient regularity for double-phase orthotropic functionals

We prove higher integrability for local minimizers of the double-phase orthotropic functional $$\sum _{i=1}^n{\int }_{\mathrm{\Omega }}({\mid u_{x_i}\mid }^p+a(x){\mid u_{x_i}\mid }^q)dx$$ when the weight function $a\ge 0$ is assumed to be $\alpha$-Hölder continuous, while the exponents $p,q$ are such that $2\le p\le q$ and $\frac{q}{p}<1+\frac{\alpha }{n}$. Under natural Sobolev regularity of~$a$, we further obtain explicit Lipschitz regularity estimates for local minimizers.