L^p(ℝ²) Bounds for Geometric Maximal Operators Associated with Homothecy Invariant Convex Bases

Let B be a nonempty homothecy invariant collection of convex sets of positive finite measure in ℝ². Let M_B be the geometric maximal operator defined by (Equation presented). We show that either M_B is bounded on L^p(ℝ²) for every 1 < p ≤ ∞ or that M_B is unbounded on L^p(ℝ²) for every 1 ≤ p < ∞. As a corollary, we have that any density basis that is a homothecy invariant collection of convex sets in ℝ² must differentiate L^p(ℝ²) for every 1 < p ≤ ∞.