In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young.
Contents
Statement
Euclidean space
In real analysis, the following result is called Young's convolution inequality:
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}}}
Generalizations
Young's convolution inequality has a natural generalization in which we replace
R
d
{\displaystyle \mathbb {R} ^{d}}
by a
σ
{\displaystyle \sigma }
-compact 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
Applications
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
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
→
R
{\displaystyle f,g,h:G\to \mathbb {R} }
are nonnegative and integrable, where
G
{\displaystyle G}
is a
σ
{\displaystyle \sigma }
-compact unimodular group endowed with a bi-invariant
σ
{\displaystyle \sigma }
-finite Haar measure
μ
.
{\displaystyle \mu .}
We use the fact that
μ
(
S
)
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
,
{\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