Young's Convolution Inequality
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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 Lp(Rd) and g is in Lq(Rd) and

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

with 1 p, q r

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

Here the star denotes convolution, Lp is Lebesgue space, and

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

denotes the usual Lp norm.

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

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

By the Hölder inequality for three functions we deduce that

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

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.

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

## Notes

1. ^ Young, W. H. (1912), "On the multiplication of successions of Fourier constants", Proceedings of the Royal Society A, 87 (596): 331-339, doi:10.1098/rspa.1912.0086, JFM 44.0298.02, JSTOR 93120
2. ^ Bogachev, Vladimir I. (2007), Measure Theory, I, Berlin, Heidelberg, New York: Springer-Verlag, ISBN 978-3-540-34513-8, MR 2267655, Zbl 1120.28001, Theorem 3.9.4
3. ^ Lieb, Elliott H.; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics (2nd ed.). Providence, R.I.: American Mathematical Society. p. 100. ISBN 978-0-8218-2783-3. OCLC 45799429.
4. ^ Beckner, William (1975). "Inequalities in Fourier Analysis". Annals of Mathematics. 102 (1): 159-182. doi:10.2307/1970980. JSTOR 1970980.
5. ^ Brascamp, Herm Jan; Lieb, Elliott H (1976-05-01). "Best constants in Young's inequality, its converse, and its generalization to more than three functions". Advances in Mathematics. 20 (2): 151-173. doi:10.1016/0001-8708(76)90184-5.
6. ^ Fournier, John J. F. (1977), "Sharpness in Young's inequality for convolution", Pacific J. Math., 72 (2): 383-397, doi:10.2140/pjm.1977.72.383, MR 0461034, Zbl 0357.43002