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We prove an a priori estimate for the second derivatives of local minimizersof integral functionals of calculus of variation with convex integrand withrespect to the gradient variable, assuming that the function that measures theoscillation of the integrand with respect to the x variable belongs to asuitable Sobolev space. The novelty here is that we deal with integrandssatisfying subquadratic growth conditions with respect to gradient variable.

Regularity for minimizers of non-autonomous non-quadratic functionals in the case 1 < p < 2: an a priori estimate