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 compactabelian 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
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).
Proofedit
Proof by Hölder's inequalityedit
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
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 interpolationedit
Young's inequality can also be proved by interpolation; see the article on Riesz–Thorin interpolation for a proof.
Sharp constantedit
In case Young's inequality can be strengthened to a sharp form, via
^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
Referencesedit
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.