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We find the exact value of the best possible constant C for the weak-type (1, 1) inequality for the one-dimensional centered Hardy-Littlewood maximal operator. We prove that C is the largest root of the quadratic equation 12C 2 − 22C + 5 = 0 thus obtaining C = 1.5675208 . . . . This is the first time the best constant for one of the fundamental inequalities satisfied by a centered maximal operator is precisely evaluated. | Annals of Mathematics The best constant for the centered Hardy-Littlewood maximal inequality By Antonios D. Melas Annals of Mathematics 157 2003 647 688 The best constant for the centered Hardy-Littlewood maximal inequality By Antonios D. Melas Abstract We find the exact value of the best possible constant C for the weak-type 1 1 inequality for the one-dimensional centered Hardy-Littlewood maximal operator. We prove that C is the largest root of the quadratic equation 12C2 22C 5 0 thus obtaining C 1.5675208. . This is the first time the best constant for one of the fundamental inequalities satisfied by a centered maximal operator is precisely evaluated. 1. Introduction Maximal operators play a central role in the theory of differentiation of functions and also in Complex and Harmonic Analysis. In general one considers a certain collection of sets C in Rn and then given any locally integrable function f at each x one measures the maximal average value of f with respect to the collection C translated by x. Then it is of fundamental importance to obtain certain regularity properties of this operators such as weak-type inequalities or Lp-boundedness. These properties are well known if C for example consists of all aD where a 0 is arbritrary and D c Rra is a fixed bounded convex set containing 0 in its interior. Such maximal operators are usually called centered. However little is known about the deeper properties of centered maximal operators even in the simplest cases. And one way to acquire such a deeper understanding is to start asking for the best constants in the corresponding inequalities satisfied by them. In this direction let us mention the result of E. M. Stein and J.-O. Stromberg 13 where certain upper bounds are given for such constants in the case of centered maximal operators as described above and the corresponding still open question raised there see also 3 Problem 7.74b on whether the best constant in the weak-type 1 1 inequality for certain centered .
