Averages and maximal averages over Product j-varieties in finite fields

Abstract

We study both averaging and maximal averaging problems for Product j-varieties defined by Πj={x∈Fqd:∏k=1dxk=j} for j∈Fq⁎, where Fqd denotes a d-dimensional vector space over the finite field Fq with q elements. We prove the sharp Lp→Lr boundedness of averaging operators associated to Product j-varieties. We also obtain the optimal Lp estimate for a maximal averaging operator related to a family of Product j-varieties {Πj}j∈Fq⁎.