In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions,[1] named after William Henry Young.
In real analysis, the following result is called Young's convolution inequality:[2] Suppose
is in the Lebesgue space
Here the star denotes convolution,
is Lebesgue space, and
denotes the usual
Young's convolution inequality has a natural generalization in which we replace
by a unimodular group
be a bi-invariant Haar measure on
be integrable functions, then we define
Then in this case, Young's inequality states that for
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.
Then there exists a constant
and any measurable function
that belongs to the weak
which by definition means that the following supremum
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
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
for any measurable
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.
Young's inequality can also be proved by interpolation; see the article on Riesz–Thorin interpolation for a proof.
Young's inequality can be strengthened to a sharp form, via
[5][6][7] When this optimal constant is achieved, the function
are multidimensional Gaussian functions.