A Harmonic Analysis Theorem and Applications to Homogenization C RISTIAN E. GUTI ´ ERREZ & I RENEO P ERAL In memory of Filippo Chiarenza ABSTRACT . We consider a one-parameter family of functions {f ε } ε>0 and establish uniform estimates of the L p -norms. The result is applied to show that the gradients of higher order cor- rectors in linear periodic homogenization converge in L p -spaces. 1. I NTRODUCTION The purpose of this paper is to establish uniform estimates in L p for a one- parameter family of functions {f ε } ε>0 satisfying certain compactness-interpolation conditions, and to obtain as a consequence convergence in L p as ε → 0. The mo- tivation for this convergence comes from the theory of homogenization, where the functions f ε are the gradients of the so–called microscopic solutions. The main result, Theorem 2.2, has independent interest and extends some ideas from [CP]. Roughly speaking, to obtain the desired L p convergence, we assume two conditions on the f ε : a uniform weak estimate in some L p , p> 1, and an invari- ance condition under re–scalings of the functions in the family; see (A) and (B) in Section 2. We prove that this implies uniform geometric growth of the measure of the level sets for all the f ε which yields uniform interior integrability in L q for 1 <q<p. This coupled with the convergence in L 1 gives convergence in all L q for 1 <q<p. We apply the main result to prove that higher order correctors in linear periodic homogenization converge in L p -spaces, Theorem 4.3. The paper is organized as follows. In Section 2 we introduce notation, state the main result and give the preliminary set up. The proofs of the main lemmas as well as the main result are the contents of Section 3. Finally, Section 4 contains the applications to homogenization. 1651 Indiana University Mathematics Journal c , Vol. 50, No. 4 (2001)