For d=1,2, the centered maximal operator satisfies ||Mf||_p ≥ (1+ε(K,p))||f||_p for all p>1, and for generic shapes in d≥3 the fixed-point threshold exceeds that of the ball.
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Lower bounds and fixed points for the centered Hardy--Littlewood maximal operator
For d=1,2, the centered maximal operator satisfies ||Mf||_p ≥ (1+ε(K,p))||f||_p for all p>1, and for generic shapes in d≥3 the fixed-point threshold exceeds that of the ball.