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Kakeya Maximal Conjecture

The Kakeya maximal conjecture asserts that in R^n with n>=2, for every epsilon>0, the Kakeya maximal function satisfies

||K_deltaf||_(L^n(S^(n-1)))<=C_epsilondelta^(-epsilon)||f||_(L^n(R^n))

uniformly for 0<delta<1, using the usual surface measure on S^(n-1).

This is a quantitative estimate for averages over tubes. A full Hausdorff dimension statement for a Kakeya set does not by itself establish this maximal estimate. The continuous needle-rotation question is the separate Kakeya needle problem.

See also

Fourier Restriction Conjecture, Hausdorff Dimension, Kakeya Maximal Function, Kakeya Needle Problem, Kakeya Set

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References

Wolff, T. H. Lectures on Harmonic Analysis. (Ed. I. Łaba and C. Shubin). Providence, RI: Amer. Math. Soc., 2003. https://doi.org/10.1090/ulect/029.

Cite this as:

Weisstein, Eric W. "Kakeya Maximal Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KakeyaMaximalConjecture.html

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