Nikodym Maximal Functions Associated with Variable Planes in R-3

作者：Kim Joonil^*

来源：Integral Equations and Operator Theory, 2012, 73(4): 455-480.

DOI：10.1007/s00020-012-1985-5

Given a vector field a on R-3, we consider a mapping x -> a(x) that assigns to each x. R3, a plane Pi(a)(x) containing x, whose normal vector is a(x). Associated with this mapping, we define a maximal operator M-N(a) on L-loc(1)(R-3) for each N >> 1 by
M-N(a) f(X) = sup(x is an element of tau) 1/vertical bar tau vertical bar integral(tau) vertical bar f(y)vertical bar dy
where the supremum is taken over all 1/N x 1/N x 1 tubes t whose axis is embedded in the plane Pi(a)(x). We study the behavior of M-N(a) according to various vector fields a. In particular, we classify the operator norms of M-N(a) on L-2(R-3) when a( x) is the linear function of the form (a(11)x(1) + a(21)x(2), a(12)x(1) + a(22)x(2), 1). The operator norm of M-N(a) on L 2 ( R3) is related with the number given by D = (a(12) + a(21))(2) - 4a(11)a(22).

• 出版日期2012-8