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Young's convolution inequality

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In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions,[1] named after William Henry Young.

Statement

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Euclidean space

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In real analysis, the following result is called Young's convolution inequality:[2]

Suppose is in the Lebesgue space and is in and with Then

Here the star denotes convolution, is Lebesgue space, and denotes the usual norm.

Equivalently, if and then

Generalizations

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Young's convolution inequality has a natural generalization in which we replace by a unimodular group If we let be a bi-invariant Haar measure on and we let or be integrable functions, then we define by Then in this case, Young's inequality states that for and and such that we have a bound Equivalently, if and then Since is in fact a locally compact abelian group (and therefore unimodular) with the Lebesgue measure the desired Haar measure, this is in fact a generalization.

This generalization may be refined. Let and be as before and assume satisfy Then there exists a constant such that for any and any measurable function on that belongs to the weak space which by definition means that the following supremum is finite, we have and[3]

Applications

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An example application is that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the norm (that is, the Weierstrass transform does not enlarge the norm).

Proof

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Proof by Hölder's inequality

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Young's inequality has an elementary proof with the non-optimal constant 1.[4]

We assume that the functions are nonnegative and integrable, where is a unimodular group endowed with a bi-invariant Haar measure We use the fact that for any measurable Since By the Hölder inequality for three functions we deduce that The conclusion follows then by left-invariance of the Haar measure, the fact that integrals are preserved by inversion of the domain, and by Fubini's theorem.

Proof by interpolation

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Young's inequality can also be proved by interpolation; see the article on Riesz–Thorin interpolation for a proof.

Sharp constant

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In case Young's inequality can be strengthened to a sharp form, via where the constant [5][6][7] When this optimal constant is achieved, the function and are multidimensional Gaussian functions.

See also

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Notes

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  1. ^ Young, W. H. (1912), "On the multiplication of successions of Fourier constants", Proceedings of the Royal Society A, 87 (596): 331–339, doi:10.1098/rspa.1912.0086, JFM 44.0298.02, JSTOR 93120
  2. ^ Bogachev, Vladimir I. (2007), Measure Theory, vol. I, Berlin, Heidelberg, New York: Springer-Verlag, ISBN 978-3-540-34513-8, MR 2267655, Zbl 1120.28001, Theorem 3.9.4
  3. ^ Bahouri, Chemin & Danchin 2011, pp. 5–6.
  4. ^ Lieb, Elliott H.; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics (2nd ed.). Providence, R.I.: American Mathematical Society. p. 100. ISBN 978-0-8218-2783-3. OCLC 45799429.
  5. ^ Beckner, William (1975). "Inequalities in Fourier Analysis". Annals of Mathematics. 102 (1): 159–182. doi:10.2307/1970980. JSTOR 1970980.
  6. ^ Brascamp, Herm Jan; Lieb, Elliott H (1976-05-01). "Best constants in Young's inequality, its converse, and its generalization to more than three functions". Advances in Mathematics. 20 (2): 151–173. doi:10.1016/0001-8708(76)90184-5.
  7. ^ Fournier, John J. F. (1977), "Sharpness in Young's inequality for convolution", Pacific Journal of Mathematics, 72 (2): 383–397, doi:10.2140/pjm.1977.72.383, MR 0461034, Zbl 0357.43002

References

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