View source for Young's convolution inequality

{{Short description|Mathematical inequality about the convolution of two functions}}
In [[mathematics]], '''Young's convolution inequality''' is a [[inequality (mathematics)|mathematical inequality]] about the [[convolution]] of two functions,<ref>{{citation | first1 = W. H. | last1 = Young | author-link = William Henry Young | year = 1912 | title = On the multiplication of successions of Fourier constants | journal = [[Proceedings of the Royal Society A]] | volume = 87 | issue = 596 | pages = 331–339 | doi = 10.1098/rspa.1912.0086 | jfm = 44.0298.02 | jstor = 93120| doi-access = free }}</ref> named after [[William Henry Young]].

==Statement==

===Euclidean space===

In [[real analysis]], the following result is called Young's convolution inequality:<ref>{{citation
 | last       = Bogachev
 | first      = Vladimir I.
 | title      = Measure Theory
 | volume     = I
 | publisher  = Springer-Verlag
 | location   = Berlin, Heidelberg, New York
 | year       = 2007
 | isbn       = 978-3-540-34513-8
 | mr       = 2267655
 | zbl       = 1120.28001 
}}, Theorem&nbsp;3.9.4</ref>

Suppose <math>f</math> is in the [[Lp space|Lebesgue space]] <math>L^p(\Reals^d)</math> and <math>g</math> is in <math>L^q(\Reals^d)</math> and
<math display=block>\frac{1}{p} + \frac{1}{q} = \frac{1}{r} + 1</math>
with <math>1 \leq p, q, r \leq \infty.</math> Then
<math display=block>\|f * g\|_r \leq \|f\|_p \|g\|_q.</math>

Here the star denotes [[convolution]], <math>L^p</math> is [[Lp space|Lebesgue space]], and
<math display=block>\|f\|_p = \Bigl(\int_{\Reals^d} |f(x)|^p\,dx \Bigr)^{1/p}</math>
denotes the usual <math>L^p</math> norm.

Equivalently, if <math>p, q, r \geq 1</math> and <math display="inline"> \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 2</math> then 
<math display=block>\left|\int_{\Reals^d} \int_{\Reals^d} f(x) g(x - y) h(y) \,\mathrm{d}x \,\mathrm{d}y\right|
\leq \left(\int_{\Reals^d} \vert f\vert^p\right)^\frac{1}{p} \left(\int_{\Reals^d} \vert g\vert^q\right)^\frac{1}{q} \left(\int_{\Reals^d} \vert h\vert^r\right)^\frac{1}{r}</math>

===Generalizations===

Young's convolution inequality has a natural generalization in which we replace <math>\Reals^d</math> by a [[unimodular group]] <math>G.</math> If we let <math>\mu</math> be a bi-invariant [[Haar measure]] on <math>G</math> and we let <math>f, g : G \to\Reals</math> or <math>\Complex</math> be integrable functions, then we define <math>f * g</math> by
<math display=block>f*g(x) = \int_G f(y)g(y^{-1}x)\,\mathrm{d}\mu(y).</math>
Then in this case, Young's inequality states that for <math>f\in L^p(G,\mu)</math> and <math>g\in L^q(G,\mu)</math> and <math>p, q, r \in [1,\infty]</math> such that 
<math display=block>\frac{1}{p} + \frac{1}{q} = \frac{1}{r} + 1</math>
we have a bound 
<math display=block>\lVert f*g \rVert_r \leq \lVert f \rVert_p \lVert g \rVert_q.</math>
Equivalently, if <math>p, q, r \ge 1</math> and <math display="inline"> \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 2</math> then 
<math display=block>\left|\int_G \int_G f(x) g(y^{-1}x) h (y) \,\mathrm{d}\mu(x) \,\mathrm{d}\mu(y)\right|
\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}.</math>
Since <math>\Reals^d</math> 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 <math>G</math> and <math>\mu</math> be as before and assume <math>1 < p, q, r < \infty</math> satisfy <math display=inline>\tfrac{1}{p} + \tfrac{1}{q} = \tfrac{1}{r} + 1.</math> Then there exists a constant <math>C</math> such that for any <math>f \in L^p(G,\mu)</math> and any measurable function <math>g</math> on <math>G</math> that belongs to the [[Lp space#Weak Lp|weak <math>L^q</math> space]] <math>L^{q,w}(G, \mu),</math> which by definition means that the following [[supremum]]
<math display=block>\|g\|_{q,w}^q ~:=~ \sup_{t > 0} \, t^q \mu(|g| > t)</math>
is finite, we have <math>f * g \in L^r(G, \mu)</math> and{{sfn|Bahouri|Chemin|Danchin|2011|pp=5-6}}
<math display=block>\|f * g\|_r ~\leq~ C \, \|f\|_p \, \|g\|_{q,w}.</math>

==Applications==

An example application is that Young's inequality can be used to show that the [[Weierstrass transform|heat semigroup]] is a contracting semigroup using the <math>L^2</math> norm (that is, the [[Weierstrass transform]] does not enlarge the <math>L^2</math> norm).

==Proof==

===Proof by Hölder's inequality===

Young's inequality has an elementary proof with the non-optimal constant 1.<ref>{{Cite book|title=Analysis|last=Lieb|first=Elliott H.|last2=Loss|first2=Michael|date=2001|publisher=American Mathematical Society|isbn=978-0-8218-2783-3|edition=2nd|series=Graduate Studies in Mathematics|location=Providence, R.I.|pages=100|oclc=45799429|author-link=Elliott H. Lieb}}</ref>

We assume that the functions <math>f, g, h : G \to \Reals</math> are nonnegative and integrable, where <math>G</math> is a unimodular group endowed with a bi-invariant Haar measure <math>\mu.</math> We use the fact that <math>\mu(S)=\mu(S^{-1})</math> for any measurable <math>S \subseteq G.</math>
Since <math display="inline">p(2 - \tfrac{1}{q} - \tfrac{1}{r}) = q(2 - \tfrac{1}{p} - \tfrac{1}{r}) = r(2 - \tfrac{1}{p} - \tfrac{1}{q}) = 1</math>
<math display=block>\begin{align}
&\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{align}</math>
By the [[Hölder inequality]] for three functions we deduce that 
<math display=block>\begin{align}
&\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{align}</math>
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 theorem#Convolution_operators|Riesz–Thorin interpolation]] for a proof.

==Sharp constant==

In case <math>p, q > 1,</math> Young's inequality can be strengthened to a sharp form, via
<math display=block>\|f*g\|_r \leq c_{p,q} \|f\|_p \|g\|_q.</math>
where the constant <math>c_{p,q} < 1.</math><ref>{{Cite journal|last=Beckner|first=William|date=1975|title=Inequalities in Fourier Analysis|jstor=1970980|journal=Annals of Mathematics|volume=102|issue=1|pages=159–182|doi=10.2307/1970980}}</ref><ref>{{Cite journal|last=Brascamp|first=Herm Jan|last2=Lieb|first2=Elliott H|date=1976-05-01|title=Best constants in Young's inequality, its converse, and its generalization to more than three functions|journal=[[Advances in Mathematics]]|volume=20|issue=2|pages=151–173|doi=10.1016/0001-8708(76)90184-5|doi-access=}}</ref><ref>{{Citation
| last = Fournier
| first = John J. F.
| title = Sharpness in Young's inequality for convolution
| journal = [[Pacific Journal of Mathematics]]
| volume = 72
| issue = 2
| pages = 383&ndash;397
| year = 1977
| url = http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1102811121&page=record
| doi = 10.2140/pjm.1977.72.383
| zbl = 0357.43002
| mr = 0461034|doi-access=free
}}
</ref> 
When this optimal constant is achieved, the function <math>f</math> and <math>g</math> are [[Gaussian function|multidimensional Gaussian functions]].

==See also==

* {{annotated link|Minkowski inequality}}

==Notes==

{{reflist}}

==References==
{{sfn whitelist|CITEREFBahouriCheminDanchin2011}}
* {{Bahouri Chemin Danchin Fourier Analysis and Nonlinear Partial Differential Equations 2011}}

==External links==

* [https://proofwiki.org/wiki/Young's_Inequality_for_Convolutions ''Young's Inequality for Convolutions''] at ProofWiki

{{Lp spaces}}
{{DEFAULTSORT:Young's Convolution Inequality}}

[[Category:Inequalities]]
[[Category:Lp spaces]]