Mathematical inequality about the convolution of two functions
In mathematics , Young's convolution inequality is a mathematical inequality about the convolution of two functions,[ 1] named after William Henry Young .
In real analysis , the following result is called Young's convolution inequality:[ 2]
Suppose
f
{\displaystyle f}
is in the Lebesgue space
L
p
(
R
d
)
{\displaystyle L^{p}(\mathbb {R} ^{d})}
and
g
{\displaystyle g}
is in
L
q
(
R
d
)
{\displaystyle L^{q}(\mathbb {R} ^{d})}
and
1
p
+
1
q
=
1
r
+
1
{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1}
with
1
≤
p
,
q
,
r
≤
∞
.
{\displaystyle 1\leq p,q,r\leq \infty .}
Then
‖
f
∗
g
‖
r
≤
‖
f
‖
p
‖
g
‖
q
.
{\displaystyle \|f*g\|_{r}\leq \|f\|_{p}\|g\|_{q}.}
Here the star denotes convolution ,
L
p
{\displaystyle L^{p}}
is Lebesgue space , and
‖
f
‖
p
=
(
∫
R
d
|
f
(
x
)
|
p
d
x
)
1
/
p
{\displaystyle \|f\|_{p}={\Bigl (}\int _{\mathbb {R} ^{d}}|f(x)|^{p}\,dx{\Bigr )}^{1/p}}
denotes the usual
L
p
{\displaystyle L^{p}}
norm.
Equivalently, if
p
,
q
,
r
≥
1
{\displaystyle p,q,r\geq 1}
and
1
p
+
1
q
+
1
r
=
2
{\textstyle {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=2}
then
|
∫
R
d
∫
R
d
f
(
x
)
g
(
x
−
y
)
h
(
y
)
d
x
d
y
|
≤
(
∫
R
d
|
f
|
p
)
1
p
(
∫
R
d
|
g
|
q
)
1
q
(
∫
R
d
|
h
|
r
)
1
r
{\displaystyle \left|\int _{\mathbb {R} ^{d}}\int _{\mathbb {R} ^{d}}f(x)g(x-y)h(y)\,\mathrm {d} x\,\mathrm {d} y\right|\leq \left(\int _{\mathbb {R} ^{d}}\vert f\vert ^{p}\right)^{\frac {1}{p}}\left(\int _{\mathbb {R} ^{d}}\vert g\vert ^{q}\right)^{\frac {1}{q}}\left(\int _{\mathbb {R} ^{d}}\vert h\vert ^{r}\right)^{\frac {1}{r}}}
Young's convolution inequality has a natural generalization in which we replace
R
d
{\displaystyle \mathbb {R} ^{d}}
by a unimodular group
G
.
{\displaystyle G.}
If we let
μ
{\displaystyle \mu }
be a bi-invariant Haar measure on
G
{\displaystyle G}
and we let
f
,
g
:
G
→
R
{\displaystyle f,g:G\to \mathbb {R} }
or
C
{\displaystyle \mathbb {C} }
be integrable functions, then we define
f
∗
g
{\displaystyle f*g}
by
f
∗
g
(
x
)
=
∫
G
f
(
y
)
g
(
y
−
1
x
)
d
μ
(
y
)
.
{\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
f
∈
L
p
(
G
,
μ
)
{\displaystyle f\in L^{p}(G,\mu )}
and
g
∈
L
q
(
G
,
μ
)
{\displaystyle g\in L^{q}(G,\mu )}
and
p
,
q
,
r
∈
[
1
,
∞
]
{\displaystyle p,q,r\in [1,\infty ]}
such that
1
p
+
1
q
=
1
r
+
1
{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1}
we have a bound
‖
f
∗
g
‖
r
≤
‖
f
‖
p
‖
g
‖
q
.
{\displaystyle \lVert f*g\rVert _{r}\leq \lVert f\rVert _{p}\lVert g\rVert _{q}.}
Equivalently, if
p
,
q
,
r
≥
1
{\displaystyle p,q,r\geq 1}
and
1
p
+
1
q
+
1
r
=
2
{\textstyle {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=2}
then
|
∫
G
∫
G
f
(
x
)
g
(
y
−
1
x
)
h
(
y
)
d
μ
(
x
)
d
μ
(
y
)
|
≤
(
∫
G
|
f
|
p
)
1
p
(
∫
G
|
g
|
q
)
1
q
(
∫
G
|
h
|
r
)
1
r
.
{\displaystyle \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}}.}
Since
R
d
{\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.
This generalization may be refined. Let
G
{\displaystyle G}
and
μ
{\displaystyle \mu }
be as before and assume
1
<
p
,
q
,
r
<
∞
{\displaystyle 1<p,q,r<\infty }
satisfy
1
p
+
1
q
=
1
r
+
1.
{\textstyle {\tfrac {1}{p}}+{\tfrac {1}{q}}={\tfrac {1}{r}}+1.}
Then there exists a constant
C
{\displaystyle C}
such that for any
f
∈
L
p
(
G
,
μ
)
{\displaystyle f\in L^{p}(G,\mu )}
and any measurable function
g
{\displaystyle g}
on
G
{\displaystyle G}
that belongs to the weak
L
q
{\displaystyle L^{q}}
space
L
q
,
w
(
G
,
μ
)
,
{\displaystyle L^{q,w}(G,\mu ),}
which by definition means that the following supremum
‖
g
‖
q
,
w
q
:=
sup
t
>
0
t
q
μ
(
|
g
|
>
t
)
{\displaystyle \|g\|_{q,w}^{q}~:=~\sup _{t>0}\,t^{q}\mu (|g|>t)}
is finite, we have
f
∗
g
∈
L
r
(
G
,
μ
)
{\displaystyle f*g\in L^{r}(G,\mu )}
and
‖
f
∗
g
‖
r
≤
C
‖
f
‖
p
‖
g
‖
q
,
w
.
{\displaystyle \|f*g\|_{r}~\leq ~C\,\|f\|_{p}\,\|g\|_{q,w}.}
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
{\displaystyle L^{2}}
norm (that is, the Weierstrass transform does not enlarge the
L
2
{\displaystyle L^{2}}
norm).
Proof by Hölder's inequality[ edit ]
Young's inequality has an elementary proof with the non-optimal constant 1.[ 4]
We assume that the functions
f
,
g
,
h
:
G
→
R
{\displaystyle f,g,h:G\to \mathbb {R} }
are nonnegative and integrable, where
G
{\displaystyle G}
is a unimodular group endowed with a bi-invariant Haar measure
μ
.
{\displaystyle \mu .}
We use the fact that
μ
(
S
)
=
μ
(
S
−
1
)
{\displaystyle \mu (S)=\mu (S^{-1})}
for any measurable
S
⊆
G
.
{\displaystyle S\subseteq G.}
Since
p
(
2
−
1
q
−
1
r
)
=
q
(
2
−
1
p
−
1
r
)
=
r
(
2
−
1
p
−
1
q
)
=
1
{\textstyle 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}
∫
G
∫
G
f
(
x
)
g
(
y
−
1
x
)
h
(
y
)
d
μ
(
x
)
d
μ
(
y
)
=
∫
G
∫
G
(
f
(
x
)
p
g
(
y
−
1
x
)
q
)
1
−
1
r
(
f
(
x
)
p
h
(
y
)
r
)
1
−
1
q
(
g
(
y
−
1
x
)
q
h
(
y
)
r
)
1
−
1
p
d
μ
(
x
)
d
μ
(
y
)
{\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
∫
G
∫
G
f
(
x
)
g
(
y
−
1
x
)
h
(
y
)
d
μ
(
x
)
d
μ
(
y
)
≤
(
∫
G
∫
G
f
(
x
)
p
g
(
y
−
1
x
)
q
d
μ
(
x
)
d
μ
(
y
)
)
1
−
1
r
(
∫
G
∫
G
f
(
x
)
p
h
(
y
)
r
d
μ
(
x
)
d
μ
(
y
)
)
1
−
1
q
(
∫
G
∫
G
g
(
y
−
1
x
)
q
h
(
y
)
r
d
μ
(
x
)
d
μ
(
y
)
)
1
−
1
p
.
{\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 [ edit ]
Young's inequality can also be proved by interpolation; see the article on Riesz–Thorin interpolation for a proof.
In case
p
,
q
>
1
,
{\displaystyle p,q>1,}
Young's inequality can be strengthened to a sharp form, via
‖
f
∗
g
‖
r
≤
c
p
,
q
‖
f
‖
p
‖
g
‖
q
.
{\displaystyle \|f*g\|_{r}\leq c_{p,q}\|f\|_{p}\|g\|_{q}.}
where the constant
c
p
,
q
<
1.
{\displaystyle c_{p,q}<1.}
[ 5] [ 6] [ 7]
When this optimal constant is achieved, the function
f
{\displaystyle f}
and
g
{\displaystyle g}
are multidimensional Gaussian functions .
^ 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
^ Bogachev, Vladimir I. (2007), Measure Theory , vol. I, Berlin, Heidelberg, New York: Springer-Verlag, ISBN 978-3-540-34513-8 , MR 2267655 , Zbl 1120.28001 , Theorem 3.9.4
^ 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 .
^ Beckner, William (1975). "Inequalities in Fourier Analysis". Annals of Mathematics . 102 (1): 159–182. doi :10.2307/1970980 . JSTOR 1970980 .
^ 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 .
^ Fournier, John J. F. (1977), "Sharpness in Young's inequality for convolution" , Pacific Journal of Mathematics , 72 (2): 383–397, doi :10.2140/pjm.1977.72.383 , MR 0461034 , Zbl 0357.43002