Young's convolution inequality explained

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

Statement

Euclidean space

In real analysis, the following result is called Young's convolution inequality:^[1]

Suppose

is in the Lebesgue space and is in and$$\backslash frac\; +\; \backslash frac\; =\; \backslash frac\; +\; 1$$with Then$$\backslash |f\; *\; g\backslash |\_r\; \backslash leq\; \backslash |f\backslash |\_p\; \backslash |g\backslash |\_q.$$

Here the star denotes convolution,

is Lebesgue space, and$$\backslash |f\backslash |\_p\; =\; \backslash Bigl(\backslash int\_\; |f(x)|^p\backslash ,dx\; \backslash Bigr)^$$denotes the usual norm.

Equivalently, if

and $\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 2$ then $$\backslash left|\backslash int\_\; \backslash int\_\; f(x)\; g(x\; -\; y)\; h(y)\; \backslash ,\backslash mathrmx\; \backslash ,\backslash mathrmy\backslash right|\backslash leq\; \backslash left(\backslash int\_\; \backslash vert\; f\backslash vert^p\backslash right)^\backslash frac\; \backslash left(\backslash int\_\; \backslash vert\; g\backslash vert^q\backslash right)^\backslash frac\; \backslash left(\backslash int\_\; \backslash vert\; h\backslash vert^r\backslash right)^\backslash frac$$

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$$f*g(x)\; =\; \backslash int\_G\; f(y)g(y^x)\backslash ,\backslash mathrm\backslash mu(y).$$Then in this case, Young's inequality states that for and and such that $$\backslash frac\; +\; \backslash frac\; =\; \backslash frac\; +\; 1$$we have a bound $$\backslash lVert\; f*g\; \backslash rVert\_r\; \backslash leq\; \backslash lVert\; f\; \backslash rVert\_p\; \backslash lVert\; g\; \backslash rVert\_q.$$Equivalently, if and $\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 2$ then $$\backslash left|\backslash int\_G\; \backslash int\_G\; f(x)\; g(y^x)\; h\; (y)\; \backslash ,\backslash mathrm\backslash mu(x)\; \backslash ,\backslash mathrm\backslash mu(y)\backslash right|\backslash leq\; \backslash left(\backslash int\_G\; \backslash vert\; f\backslash vert^p\backslash right)^\backslash frac\; \backslash left(\backslash int\_G\; \backslash vert\; g\backslash vert^q\backslash right)^\backslash frac\; \backslash left(\backslash int\_G\; \backslash vert\; h\backslash vert^r\backslash right)^\backslash frac.$$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 $\backslash tfrac\; +\; \backslash tfrac\; =\; \backslash tfrac\; +\; 1.$ 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$$\backslash |g\backslash |\_^q\; ~:=~\; \backslash sup\_\; \backslash ,\; t^q\; \backslash mu(|g|\; >\; t)$$is finite, we have and$$\backslash |f\; *\; g\backslash |\_r\; ~\backslash leq~\; C\; \backslash ,\; \backslash |f\backslash |\_p\; \backslash ,\; \backslash |g\backslash |\_.$$

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.^[2]

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 $p(2\; -\; \backslash tfrac\; -\; \backslash tfrac)\; =\; q(2\; -\; \backslash tfrac\; -\; \backslash tfrac)\; =\; r(2\; -\; \backslash tfrac\; -\; \backslash tfrac)\; =\; 1$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$$\backslash |f*g\backslash |\_r\; \backslash leq\; c\_\; \backslash |f\backslash |\_p\; \backslash |g\backslash |\_q.$$where the constant ^{[3] [4]} When this optimal constant is achieved, the function and are multidimensional Gaussian functions.

External links

• Young's Inequality for Convolutions at ProofWiki

Notes and References

1. , Theorem 3.9.4
2. Book: Lieb, Elliott H.. Analysis. Loss. Michael. 2001. American Mathematical Society. 978-0-8218-2783-3. 2nd. Graduate Studies in Mathematics. Providence, R.I.. 100. 45799429. Elliott H. Lieb.
3. Beckner. William. 1975. Inequalities in Fourier Analysis. 1970980. Annals of Mathematics. 102. 1. 159–182. 10.2307/1970980.
4. 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. 10.1016/0001-8708(76)90184-5.