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 MB be the geometric maximal operator defined by (Equation presented). We show that either MB is bounded on Lp(ℝ2) for every 1 < p ≤ ∞ or that MB is unbounded on Lp(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 Lp(ℝ2) for every 1 < p ≤ ∞.