  For nonautonomous, nonuniformly elliptic integrals with so-called
$(p,q)$-growth conditions, we show a general interpolation property allowing to
get basic higher integrability results for H\"older continuous minimizers under
improved bounds for the gap $q/p$. For this we introduce a new method, based on
approximating the original, local functional, with mixed local/nonlocal ones,
and allowing for suitable estimates in fractional Sobolev spaces.
