The autoconvolution inequality, also called the Martin-O'Bryant autoconvolution inequality, concerns the least constant for which

for every nonzero nonnegative function , where is the convolution of with itself and is the L^p norm. The bound with follows by integrating . Martin and O'Bryant (2009) proposed , which Matolcsi and Vinuesa (2010) refuted.

Crmarić et al. (2026) showed that the ratio of the left-hand side to the product of the two norms on the right can approach 1 for nonnegative compactly supported step functions. Thus is sharp. The authors credit AI with developing the proof.

See also

Convolution, Step Function

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References

Crmarić, T.; Kovač, V.; and Rupčić, J. "Sharpness of the Martin-O'Bryant Autoconvolution Inequality." 9 Nov 2026. https://arxiv.org/abs/2611.12685v1.Martin, G. and O'Bryant, K. "The Supremum of Autoconvolutions, with Applications to Additive Number Theory." Illinois J. Math. 53, 219-235, 2009. https://doi.org/10.1215/ijm/1264170847.Matolcsi, M. and Vinuesa, C. "Improved Bounds on the Supremum of Autoconvolutions." J. Math. Anal. Appl. 372, 439-447.e2, 2010. https://doi.org/10.1016/j.jmaa.2010.07.030.

Cite this as:

Weisstein, Eric W. "Autoconvolution Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AutoconvolutionInequality.html

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