For every 1<p<∞ there exists ε_p>0 such that ||Mf||_p ≥ (1+ε_p)||f||_p for all f in L^p(R), resolving the Ivanisvili-Zbarsky conjecture.
Grafakos, Classical and modern Fourier analysis, Prentice Hall, 2004
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Lower bounds for the centered Hardy-Littlewood maximal operator on the real line
For every 1<p<∞ there exists ε_p>0 such that ||Mf||_p ≥ (1+ε_p)||f||_p for all f in L^p(R), resolving the Ivanisvili-Zbarsky conjecture.