In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young.
In real analysis, the following result is called Young's convolution inequality:
Suppose f is in Lp(Rd) and g is in Lq(Rd) 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
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.
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 (i.e. the Weierstrass transform does not enlarge the L2 norm).
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 .
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.
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.
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