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We establish an a priori estimate for the second derivatives of local minimizers of integral functionals of the form $$ℱ(\nu ,\mathrm{\Omega })={\int }_{\mathrm{\Omega }}f(x,D\nu (x))𝑑x$$ with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the $x$ variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.