Gradient regularity for double-phase orthotropic functionals

We prove higher integrability for local minimizers of the double-phase orthotropic functional $$\sum_{i=1}^{n}\int_\Omega\left(\left|u_{x_i}\right|^p+a(x)\left| u_{x_i}\right|^q\right)dx$$ when the weight function $a \geq0$ is assumed to be $\alpha$-H\"older continuous, while the exponents $p, q$ are such that $2 \leq p \leq q$ and $\frac{q}{p} < 1 + \frac{\alpha}{n}$. Under natural Sobolev regularity of~$a$, we further obtain explicit Lipschitz regularity estimates for local minimizers.