Young's convolution inequality

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In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions,[1] named after William Henry Young.

Statement[edit]

Euclidean Space[edit]

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[edit]

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[edit]

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[edit]

Proof by Hölder's inequality[edit]

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[edit]

In case pq > 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[edit]

  1. ^ 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 
  2. ^ 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
  3. ^ 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. 
  4. ^ Beckner, William (1975). "Inequalities in Fourier Analysis". Annals of Mathematics. 102 (1): 159–182. doi:10.2307/1970980. JSTOR 1970980. 
  5. ^ 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. 
  6. ^ Fournier, John J. F. (1977), "Sharpness in Young's inequality for convolution", Pacific J. Math., 72 (2): 383–397, doi:10.2140/pjm.1977.72.383, MR 0461034, Zbl 0357.43002 

External links[edit]