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.
Statement
Euclidean space
In real analysis, the following result is called Young's convolution inequality:[2]
Suppose
is in the Lebesgue space
and
is in
and
with
Then
Here the star denotes convolution,
is Lebesgue space, and
denotes the usual
norm.
Equivalently, if
and
then
Generalizations
Young's convolution inequality has a natural generalization in which we replace
by a unimodular group
If we let
be a bi-invariant Haar measure on
and we let
or
be integrable functions, then we define
by
Then in this case, Young's inequality states that for
and
and
such that
we have a bound
Equivalently, if
and
then
Since
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
and
be as before and assume
satisfy
Then there exists a constant
such that for any
and any measurable function
on
that belongs to the weak
space
which by definition means that the following supremum
is finite, we have
and
Applications
An example application is that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the
norm (that is, the Weierstrass transform does not enlarge the
norm).
Proof
Proof by Hölder's inequality
Young's inequality has an elementary proof with the non-optimal constant 1.[4]
We assume that the functions
are nonnegative and integrable, where
is a unimodular group endowed with a bi-invariant Haar measure
We use the fact that
for any measurable
Since
By the Hölder inequality for three functions we deduce that
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
Young's inequality can be strengthened to a sharp form, via
where the constant
[5][6][7] When this optimal constant is achieved, the function
and
are multidimensional Gaussian functions.
See also
- Minkowski inequality – Inequality that established Lp spaces are normed vector spaces
Notes
- ^ 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
References
- Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128.
External links
- Young's Inequality for Convolutions at ProofWiki