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 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 ≤ p, q, r ≤ ∞. Then

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

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

$\|f\|_{p}={\Bigl (}\int _{\mathbf {R} ^{d}}|f(x)|^{p}\,dx{\Bigr )}^{1/p}$

denotes the usual L^p norm.

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

$\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x)g(x-y)h(y)\,\mathrm {d} x\,\mathrm {d} y\leq \left(\int _{\mathbf {R} ^{d}}\vert f\vert ^{p}\right)^{\frac {1}{p}}\left(\int _{\mathbf {R} ^{d}}\vert g\vert ^{q}\right)^{\frac {1}{q}}\left(\int _{\mathbf {R} ^{d}}\vert h\vert ^{r}\right)^{\frac {1}{r}}$

Generalizations

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

$f*g(x)=\int _{G}f(y)g(y^{-1}x)\,\mathrm {d} \mu (y).$

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

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

we have a bound

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

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

$\int _{G}\int _{G}f(x)g(y^{-1}x)h(y)\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\leq \left(\int _{G}\vert f\vert ^{p}\right)^{\frac {1}{p}}\left(\int _{G}\vert g\vert ^{q}\right)^{\frac {1}{q}}\left(\int _{G}\vert h\vert ^{r}\right)^{\frac {1}{r}}.$

Since $\mathbb {R} ^{d}$ 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.

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 (i.e. the Weierstrass transform does not enlarge 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 $f,g,h:G\to \mathbb {R}$ are nonnegative and integrable, where $G$ is a unimodular group endowed with a bi-invariant Haar measure $\mu$. We use the fact that $\mu (S)=\mu (S^{-1})$ for any measurable $S\subset G$. Since $\textstyle p(2-{\frac {1}{q}}-{\frac {1}{r}})=q(2-{\frac {1}{p}}-{\frac {1}{r}})=r(2-{\frac {1}{p}}-{\frac {1}{r}})=1$

$\int _{G}\int _{G}f(x)g(y^{-1}x)h(y)\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)=\int _{G}\int _{G}\left(f(x)^{p}g(y^{-1}x)^{q}\right)^{1-{\frac {1}{r}}}\left(f(x)^{p}h(y)^{r}\right)^{1-{\frac {1}{q}}}\left(g(y^{-1}x)^{q}h(y)^{r}\right)^{1-{\frac {1}{p}}}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)$

By the Hölder inequality for three functions we deduce that

$\int _{G}\int _{G}f(x)g(y^{-1}x)h(y)\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\leq \left(\int _{G}\int _{G}f(x)^{p}g(y^{-1}x)^{q}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right)^{1-{\frac {1}{r}}}\left(\int _{G}\int _{G}f(x)^{p}h(y)^{r}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right)^{1-{\frac {1}{q}}}\left(\int _{G}\int _{G}g(y^{-1}x)^{q}h(y)^{r}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right)^{1-{\frac {1}{p}}}.$

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.

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.

External links

• Young's Inequality for Convolutions at ProofWiki