A geometric bound for maximal functions associated to convex bodies

Detlef Müller

Vol. 142 (1990), No. 2, 297–312

DOI: 10.2140/pjm.1990.142.297

Abstract

For a convex symmetric body B in ℝⁿ let M_B denote the centered maximal operator

for f ∈ L_loc¹(ℝⁿ). We associate with B two linear invariants σ(B) and Q(B), and show that for p > 1 the norm of the operator M_B on L^p(ℝⁿ) is bounded by a constant which may depend on p, σ(B) and Q(B), but not explicitly on the dimension n. In particular, if B_q denotes the unit ball in ℝⁿ with respect to the l^q-norm, we can prove that M_Bq has a bound on L^p(ℝⁿ) which is independent of n, provided that 1 ≤ q < ∞.

Mathematical Subject Classification 2000

Primary: 42B25

Milestones

Received: 29 April 1988

Published: 1 April 1990