The best constant for the centered Hardy-Littlewood maximal inequality

By A. Melas. 2003

94 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e157e518a31e956a918df0616947ed8f3dc52a69

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In Applied Mathematics

1. Zunwei Fu
2. Terence Tao
3. Assaf Naor
4. Shan-zhen Lu
5. Loukas Grafakos
6. J. M. Aldaz
7. Fayou Zhao
8. Sanghyuk Lee

In Mathematical Physics

1. Loukas Grafakos
2. J. M. Aldaz
3. Fayou Zhao
4. Zunwei Fu
5. Assaf Naor
6. Shan-zhen Lu
7. Terence Tao
8. Van H. Vu

Citations in the corpus

• Sharp bounds for m-linear Hardy and Hilbert Operators (2012, background): “[7], [11]) but still requires the use of certain beautiful and elegant ideas.”
• THE WEAK TYPE (1,1) BOUNDS FOR THE MAXIMAL FUNCTION ASSOCIATED TO CUBES GROW TO INFINITY WITH THE DIMENSION (2008, methodology): “8 , thereby refuting the conjecture that c 1 = 3/2 (cf. [BH, Problem 7.74 c]) and showing that c 1 < 2, which is the best constant in the uncentered case. Discretization was also used in [Me1] and [Me2], where the exact value of c 1 = 11+ √ 61 12 was found. No best constants are known for dimensi”
• Random Martingales and localization of maximal inequalities (2009, background): “In most cases of interest it is probably impossible to compute ‖M‖L1(X)→L1,∞(X) exactly; notable exceptions to this statement are ultrametric spaces, where the weak (1, 1) norm of M equals 1 (we will return to the class of ultrametric spaces presently), and the real line R, equipped with the usual m”