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
with 1 ≤ p, q, r ≤ ∞. Then
Here the star denotes convolution, Lp is Lebesgue space, and
denotes the usual Lp 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.
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 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.
Sharp constant
In case p, q > 1 Young's inequality can be strengthened to a sharp form, via
where the constant cp,q < 1.[4][5][6] When this optimal constant is achieved, the function and are multidimensional Gaussian functions.
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, 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 J. Math., 72 (2): 383&ndash, 397, doi:10.2140/pjm.1977.72.383, MR 0461034, Zbl 0357.43002
External links
- Young's Inequality for Convolutions at ProofWiki