We consider a class of degenerate Ornstein-Uhlenbeck operators in R N , of the kindTo get this estimate we use in a crucial way the left invariance of L with respect to a Lie group structure in R N +1 and some results on singular integrals on nonhomogeneous spaces recently proved in Bramanti (Revista Matematica Iberoamericana, 2009, in press).

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Abstract. The energy-integral of the calculus of variations (1.1), (1.2) below has a limit behavior when q = np/(n − p), where p is the harmonic average of the exponents pi, i = 1, . . . , n. In fact, if q is larger than in the stated equality, counterexamples to the local boundedness and regularity of minimizers are known. In this paper we prove the local boundedness of minimizers (and also of quasi-minimizers) under this stated limit condition. Some other general and limit growth conditions are also considered.

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