Young's convolution inequality

In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions,^[1] named after William Henry Young.

Statement[]

Euclidean Space[]

In real analysis, the following result is called Young's convolution inequality:^[2]

Suppose f is in L^p(R^d) and g is in L^q(R^d) and

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

with 1 ≤ p, q ≤ r ≤ ∞ . Then

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

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

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

denotes the usual L^p norm.

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

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

Generalizations[]

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

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

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

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

we have a bound

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

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

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

Since ${\displaystyle \mathbb {R} ^{d}}$ 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.^[3]

We assume that the functions ${\displaystyle f,g,h:G\to \mathbb {R} }$ are nonnegative and integrable, where ${\displaystyle G}$ is a unimodular group endowed with a bi-invariant Haar measure ${\displaystyle \mu }$. We use the fact that ${\displaystyle \mu (S)=\mu (S^{-1})}$ for any measurable ${\displaystyle S\subset G}$. Since ${\textstyle p(2-{\frac {1}{q}}-{\frac {1}{r}})=q(2-{\frac {1}{p}}-{\frac {1}{r}})=r(2-{\frac {1}{p}}-{\frac {1}{q}})=1}$

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

By the Hölder inequality for three functions we deduce that

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

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

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

where the constant c_p,q < 1.^[4][5][6] When this optimal constant is achieved, the function ${\displaystyle f}$ and ${\displaystyle g}$ are multidimensional Gaussian functions.

Notes[]

External links[]

• Young's Inequality for Convolutions at ProofWiki