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 the Lebesgue space $L^{p} (ℝ^{d})$ and $g$ is in $L^{q} (ℝ^{d})$ and

$\frac{1}{p} + \frac{1}{q} = \frac{1}{r} + 1$

with $1 \leq p, q, r \leq \infty .$ Then

$‖ f * g ‖_{r} \leq ‖ f ‖_{p} ‖ g ‖_{q} .$

Here the star denotes convolution, $L^{p}$ is Lebesgue space, and

$‖ f ‖_{p} = (\int_{ℝ^{d}} | f (x) |^{p} d x)^{1 / p}$

denotes the usual $L^{p}$ norm.

Equivalently, if $p, q, r \geq 1$ and $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 2$ then

$| \int_{ℝ^{d}} \int_{ℝ^{d}} f (x) g (x - y) h (y) d x d y | \leq {(\int_{ℝ^{d}} | f |^{p})}^{\frac{1}{p}} {(\int_{ℝ^{d}} | g |^{q})}^{\frac{1}{q}} {(\int_{ℝ^{d}} | h |^{r})}^{\frac{1}{r}}$

Generalizations

Young's convolution inequality has a natural generalization in which we replace $ℝ^{d}$ by a $σ$ -compact unimodular group $G .$ If we let $μ$ be a bi-invariant Haar measure on $G$ and we let $f, g : G \to ℝ$ or $ℂ$ be integrable functions, then we define $f * g$ by

$f * g (x) = \int_{G} f (y) g (y^{- 1} x) d μ (y) .$

Then in this case, Young's inequality states that for $f \in L^{p} (G, μ)$ and $g \in L^{q} (G, μ)$ and $p, q, r \in [1, \infty]$ such that

$\frac{1}{p} + \frac{1}{q} = \frac{1}{r} + 1$

we have a bound

$‖ f * g ‖_{r} \leq ‖ f ‖_{p} ‖ g ‖_{q} .$

Equivalently, if $p, q, r \geq 1$ and $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 2$ then

$| \int_{G} \int_{G} f (x) g (y^{- 1} x) h (y) d μ (x) d μ (y) | \leq {(\int_{G} | f |^{p})}^{\frac{1}{p}} {(\int_{G} | g |^{q})}^{\frac{1}{q}} {(\int_{G} | h |^{r})}^{\frac{1}{r}} .$

Since $ℝ^{d}$ 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 $G$ and $μ$ be as before and assume $1 < p, q, r < \infty$ satisfy $\frac{1}{p} + \frac{1}{q} = \frac{1}{r} + 1.$ Then there exists a constant $C$ such that for any $f \in L^{p} (G, μ)$ and any measurable function $g$ on $G$ that belongs to the weak $L^{q}$ space $L^{q, w} (G, μ),$ which by definition means that the following supremum

$‖ g ‖_{q, w}^{q} := sup_{t > 0} t^{q} μ (| g | > t)$

is finite, we have $f * g \in L^{r} (G, μ)$ and

$‖ f * g ‖_{r} \leq C ‖ f ‖_{p} ‖ g ‖_{q, w} .$

Applications

An example application is that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the $L^{2}$ norm (that is, the Weierstrass transform does not enlarge the $L^{2}$ norm).

Proof

Proof by Hölder's inequality

By the triangle inequality for integrals, $| f * g | \leq | f | * | g |$ , and so it suffices to consider the case where $f$ and $g$ are non-negative real-valued functions. If $r = \infty$ , then $\frac{1}{p} + \frac{1}{q} = 1$ , and so by Hölder's inequality we have $(f * g) (x) = \int_{ℝ^{d}} f (y) g (x - y) d y \leq ‖ f ‖_{p} ‖ g ‖_{q}$ for any $x \in ℝ^{d}$ , and so $‖ f * g ‖_{\infty} \leq ‖ f ‖_{p} ‖ g ‖_{q}$ . If $r < \infty$ , then take $p^{'}$ and $q^{'}$ to be the Hölder conjugate exponents of $p$ and $q$ respectively, and note that $\frac{1}{r} + \frac{1}{q^{'}} = \frac{1}{p}$ and $\frac{1}{r} + \frac{1}{p^{'}} = \frac{1}{q}$ .

Thus we can decompose $f (y) g (x - y) = (f (y)^{p} g (x - y)^{q})^{1 / r} f (y)^{p / q^{'}} g (x - y)^{q / p^{'}} .$ Then since $\frac{1}{r} + \frac{1}{q^{'}} + \frac{1}{p^{'}} = 1$ , we can use the generalized Hölder inequality and the translation invariance of the Lebesgue measure to estimate $\begin{matrix} (f * g) (x) & \leq {(\int_{ℝ^{d}} f (y)^{p} g (x - y)^{q} d y)}^{1 / r} {(\int_{ℝ^{d}} f (y)^{p} d y)}^{1 / q^{'}} {(\int_{ℝ^{d}} g (x - y)^{q} d y)}^{1 / p^{'}} \\ = ‖ f ‖_{p}^{p / q^{'}} ‖ g ‖_{q}^{q / p^{'}} {(\int_{ℝ^{d}} f (y)^{p} g (x - y)^{q} d y)}^{1 / r} . \end{matrix}$ Thus by using Tonelli's theorem and translation invariance, $\begin{matrix} ‖ f * g ‖_{r} & \leq ‖ f ‖_{p}^{p / q^{'}} ‖ g ‖_{q}^{q / p^{'}} {(\int_{ℝ^{d}} \int_{ℝ^{d}} f (y)^{p} g (x - y)^{q} d y d x)}^{1 / r} \\ = ‖ f ‖_{p}^{p / q^{'}} ‖ g ‖_{q}^{q / p^{'}} {(\int_{ℝ^{d}} f (y)^{p} \int_{ℝ^{d}} g (x - y)^{q} d x d y)}^{1 / r} \\ = ‖ f ‖_{p}^{p / q^{'}} ‖ g ‖_{q}^{q / p^{'}} ‖ f ‖_{p}^{p / r} ‖ g ‖_{q}^{q / r} \\ = ‖ f ‖_{p} ‖ g ‖_{q} . \end{matrix}$ The same proof works in the case when $ℝ^{d}$ is replaced with any $σ$ -compact unimodular group $G$ along with its Haar measure $μ$ .

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

$‖ f * g ‖_{r} \leq c_{p, q} ‖ f ‖_{p} ‖ g ‖_{q} .$

where the constant $c_{p, q} < 1.$ When this optimal constant is achieved, the function $f$ and $g$ are multidimensional Gaussian functions.