Averages over annuli or tubes on the moving planes

Abstract

We consider the L² bounds of the three types of the classical maximal averages over (i) annuli on the plane, (ii) tubes on the plane, and (iii) tubes along the cones. We study those averages over tubes and annuli embedded on the planes of ℝ³ or ℝ⁴ varying on the space variables x. In particular, the planes in ℝ³ containing tubes and annuli have their normal vectors (A(x), −1) where A is 2×2 matrix. The model is the Heisenberg group plane. For this case the average of f given by 1/|R| ∫[R] f(x−y, x₃−〈E(x), y〉) dy where E is the skew symmetric 2×2 matrix. In this paper, we introduce an author's recent classification of L² norm of the annulus or Nikodym maximal functions according to the rank condition of the two different types of matrices AE+(AE)[T] or A+A[T]. Finally, we obtain the bound for the cone-type Nikodym maximal operator associated with the moving planes.