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Hardy-Littlewood Maximal Function

The centered Hardy-Littlewood maximal function averages the absolute value of a locally integrable function over balls centered at the output point and takes the supremum over their radii. For f on R^n, it is

Mf(x)=sup_(r>0)1/(|B(x,r)|)int_(B(x,r))|f(y)|dy,

where B(x,r) is the open Euclidean ball and |B(x,r)| is its Lebesgue measure. The uncentered version instead takes the supremum over all balls containing x.

Lerner and Ombrosi (2026) proved the Muckenhoupt Cp conjecture, characterizing the weights for which the weighted p-norm of the Hilbert transform is bounded by that of the Hardy-Littlewood maximal function for 1<p<infty.

The sharp maximal function instead averages the deviation of f from its interval average, measuring local oscillation (Lerner 2020).

See also

Gradient, Kakeya Maximal Function, Lebesgue Integral, Sobolev Space

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References

Lerner, A. K. "A Characterization of the Weighted Weak Type Coifman-Fefferman and Fefferman-Stein Inequalities." Math. Ann. 378, 425-446, 2020. https://doi.org/10.1007/s00208-020-01980-z.Lerner, A. K. and Ombrosi, S. "The C_p Conjecture of Muckenhoupt." 25 Oct 2026. https://arxiv.org/abs/2610.34099.Tao, T. "Lecture Notes 3 for 247A." UCLA, 2006. https://www.math.ucla.edu/~tao/247a.1.06f/notes3.pdf.

Cite this as:

Weisstein, Eric W. "Hardy-Littlewood Maximal Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hardy-LittlewoodMaximalFunction.html

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