Young's convolution inequality
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In mathematics, Young's convolution inequality is a mathematical inequality about the
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of two functions, named after
William Henry Young William Henry Young FRS (London, 20 October 1863 – Lausanne, 7 July 1942) was an English mathematician. Young was educated at City of London School and Peterhouse, Cambridge. He worked on measure theory, Fourier series, differential ca ...
.


Statement


Euclidean Space

In
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, the following result is called Young's convolution inequality: Suppose f is in the Lebesgue spaceL^p(\Reals^d) and g is in L^q(\Reals^d) and \frac + \frac = \frac + 1 with 1 \leq p, q, r \leq \infty. Then \, f * g\, _r \leq \, f\, _p \, g\, _q. Here the star denotes
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
, L^p is Lebesgue space, and \, f\, _p = \Bigl(\int_ , f(x), ^p\,dx \Bigr)^ denotes the usual L^p norm. Equivalently, if p, q, r \geq 1 and \frac + \frac + \frac = 2 then \left, \int_ \int_ f(x) g(x - y) h(y) \,\mathrmx \,\mathrmy \ \leq \left(\int_ \vert f\vert^p\right)^\frac \left(\int_ \vert g\vert^q\right)^\frac \left(\int_ \vert h\vert^r\right)^\frac


Generalizations

Young's convolution inequality has a natural generalization in which we replace \Reals^d by a unimodular group G. If we let \mu be a bi-invariant Haar measure on G and we let f, g : G \to\Reals or \Complex be integrable functions, then we define f * g by f*g(x) = \int_G f(y)g(y^x)\,\mathrm\mu(y). Then in this case, Young's inequality states that for f\in L^p(G,\mu) and g\in L^q(G,\mu) and p, q, r \in ,\infty/math> such that \frac + \frac = \frac + 1 we have a bound \lVert f*g \rVert_r \leq \lVert f \rVert_p \lVert g \rVert_q. Equivalently, if p, q, r \ge 1 and \frac + \frac + \frac = 2 then \left, \int_G \int_G f(x) g(y^x) h (y) \,\mathrm\mu(x) \,\mathrm\mu(y)\ \leq \left(\int_G \vert f\vert^p\right)^\frac \left(\int_G \vert g\vert^q\right)^\frac \left(\int_G \vert h\vert^r\right)^\frac. Since \Reals^d 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 G and \mu be as before and assume 1 < p, q, r < \infty satisfy \tfrac + \tfrac = \tfrac + 1. Then there exists a constant C such that for any f \in L^p(G,\mu) and any measurable function g on G that belongs to the weak L^q space L^(G, \mu), which by definition means that the following supremum \, g\, _^q ~:=~ \sup_ \, t^q \mu(, g, > t) is finite, we have f * g \in L^r(G, \mu) and \, f * g\, _r ~\leq~ C \, \, f\, _p \, \, g\, _.


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 norm (that is, the
Weierstrass transform In mathematics, the Weierstrass transform of a function , named after Karl Weierstrass, is a "smoothed" version of obtained by averaging the values of , weighted with a Gaussian centered at ''x''. Specifically, it is the function define ...
does not enlarge the 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 \to \Reals are nonnegative and integrable, where G is a unimodular group endowed with a bi-invariant Haar measure \mu. We use the fact that \mu(S)=\mu(S^) for any measurable S \subseteq G. Since p(2 - \tfrac - \tfrac) = q(2 - \tfrac - \tfrac) = r(2 - \tfrac - \tfrac) = 1 \begin &\int_G \int_G f(x) g(y^x) h(y) \,\mathrm\mu(x) \,\mathrm\mu(y) \\ =& \int_G \int_G \left(f(x)^p g(y^x)^q\right)^ \left(f(x)^p h(y)^r\right)^ \left(g(y^x)^q h(y)^r\right)^\,\mathrm\mu(x) \,\mathrm\mu(y) \end By the
Hölder inequality Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modul ...
for three functions we deduce that \begin &\int_G \int_G f (x) g (y^x) h(y) \,\mathrm\mu(x) \,\mathrm\mu(y) \\ &\leq \left(\int_G \int_G f(x)^p g(y^x)^q \,\mathrm\mu(x) \,\mathrm\mu(y)\right)^ \left(\int_G \int_G f(x)^p h(y)^r \,\mathrm\mu(x) \,\mathrm\mu(y)\right)^ \left(\int_G \int_G g(y^x)^q h(y)^r \,\mathrm\mu(x) \,\mathrm\mu(y)\right)^. \end 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 In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
.


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, Young's inequality can be strengthened to a sharp form, via \, f*g\, _r \leq c_ \, f\, _p \, g\, _q. where the constant c_ < 1. When this optimal constant is achieved, the function f and g are multidimensional Gaussian functions.


See also

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Notes


References

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External links


''Young's Inequality for Convolutions''
at ProofWiki {{DEFAULTSORT:Young's Convolution Inequality Inequalities