Young's convolution inequality


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:
Suppose f is in Lp and g is in Lq and
with 1 ≤ p, q ≤ r ≤ ∞. Then
Here the star denotes convolution, Lp is Lebesgue space, and
denotes the usual Lp 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 with the Lebesgue measure the desired Haar measure, this is in fact a generalization.

Applications

An example application is that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the L2 norm.

Proof

Proof by Hölder's inequality

Young's inequality has an elementary proof with the non-optimal constant 1.
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 p, q > 1 Young's inequality can be strengthened to a sharp form, via
where the constant cp,q < 1. When this optimal constant is achieved, the function and are multidimensional Gaussian functions.