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

In real analysis, the following result is called Young's convolution inequality:^[2]

Suppose f is in L^p(R^d) and g is in L^q(R^d) and

{\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1

with 1 ≤ p, q, r ≤ ∞. Then

\|f*g\|_{r}\leq \|f\|_{p}\|g\|_{q}.

Here the star denotes convolution, L^p is Lebesgue space, and

\|f\|_{p}={\Bigl (}\int _{\mathbf {R} ^{d}}|f(x)|^{p}\,dx{\Bigr )}^{1/p}

denotes the usual L^p norm.

Equivalently, if $p,q,r\geq 1$ and $\textstyle {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=2$ then

\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x)g(x-y)h(y)\,\mathrm {d} x\,\mathrm {d} y\leq \left(\int _{\mathbf {R} ^{d}}\vert f\vert ^{p}\right)^{\frac {1}{p}}\left(\int _{\mathbf {R} ^{d}}\vert g\vert ^{q}\right)^{\frac {1}{q}}\left(\int _{\mathbf {R} ^{d}}\vert h\vert ^{r}\right)^{\frac {1}{r}}

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 $f$ , $g$ and $h$ are nonnegative. Since $\textstyle p(2-{\frac {1}{q}}-{\frac {1}{r}})=q(2-{\frac {1}{p}}-{\frac {1}{r}})=r(2-{\frac {1}{p}}-{\frac {1}{r}})=1$

\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x)g(x-y)h(y)\,\mathrm {d} x\,\mathrm {d} y=\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}\left(f(x)^{p}g(x-y)^{q}\right)^{1-{\frac {1}{r}}}\left(f(x)^{p}h(y)^{r}\right)^{1-{\frac {1}{q}}}\left(g(x-y)^{q}(x)h(y)^{r}\right)^{1-{\frac {1}{p}}}\,\mathrm {d} x\,\mathrm {d} y

By the Hölder inequality for three functions we deduce that

\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x)g(x-y)h(y)\,\mathrm {d} x\,\mathrm {d} y\leq \left(\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x)^{p}(x)g(x-y)^{q}\,\mathrm {d} x\,\mathrm {d} y\right)^{1-{\frac {1}{r}}}\left(\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x)^{p}h(y)^{r}\,\mathrm {d} x\,\mathrm {d} y\right)^{1-{\frac {1}{q}}}\left(\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}g(x-y)^{q}(x)h(y)^{r}\,\mathrm {d} x\,\mathrm {d} y\right)^{1-{\frac {1}{p}}}.

The conclusion follows then by a change of variable and by Fubini's theorem.

Sharp constant

In case p, q > 1 Young's inequality can be strengthened to a sharp form, via

\|f*g\|_{r}\leq c_{p,q}\|f\|_{p}\|g\|_{q}.

where the constant c_p,q < 1.^[4]^[5]^[6] When this optimal constant is achieved, the function $f$ and $g$ 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, JFM 44.0298.02, JSTOR 93120, doi:10.1098/rspa.1912.0086
↑ 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 ed.). Providence, R.I.: American Mathematical Society. p. 100. ISBN 978-0-8218-2783-3. OCLC 45799429. CS1 maint: Extra text (link)
↑ Beckner, William (1975). "Inequalities in Fourier Analysis". Annals of Mathematics. 102 (1): 159–182. doi:10.2307/1970980.
↑ 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): 383–397, MR 0461034, Zbl 0357.43002, doi:10.2140/pjm.1977.72.383

External links

Young's Inequality for Convolutions at ProofWiki

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