Young's convolution inequality

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

Statement

Euclidean space

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

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

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

Proof by Hölder's inequality

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

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

Sharp constant

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

Notes

  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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