On Lipschitz regularity for bounded minimizers of functionals with (p,q) growth

We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q<p+2$ with $p \geq 2$. This relation improves existing restrictions when $p \leq N-1$, moreover our result is sharp in the range $N > \frac{p(2+p)}{2} + 1$. The standard Lipschitz regularity takes the form $W^{1,\infty}_{\text{loc}} - W^{1,p}_{\text{loc}}$, whereas we obtain $W^{1,\infty}_{\text{loc}} - L^{\infty}_{\text{loc}}$ regularity estimate and then make use of existing sharp $L^{\infty}_{\text{loc}}$ bounds to obtain the required conclusion.