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Recall that the convolution of two functions is given by $$f*g(y)=\int f(x)g(y-x)dx.$$ The well known inequality known as Young's inequality, say that $$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$ provided $1/p + 1/q = 1 + 1/r$ and $1\le p,q,r\le\infty$. Obvious implications is that

  • $L^1$ is a Banach algebra
  • $L^\infty$ is an $L^1$-module with respect to convolution

Do you have any deeper applications/examples?

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Could I convince you to write $(f*g)(y)$ rather than just $f*g$ on the definition (but obviously not in the inequality)? –  Michael Hardy Nov 17 '11 at 18:16
    
@MichaelHardy Thanks! –  AD. Nov 17 '11 at 18:17
    
You could make that $L^\infty$ an $L^r$, of course. –  t.b. Nov 17 '11 at 18:30
    
@t.b. Sure, thanks and yes - I just took the first examples that popped up in my head. ;) –  AD. Nov 17 '11 at 19:24
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@wildildildlife Well, it is a matter of taste I agree, I am used to $f*g(x)$. (Could I convince you to write $*$ instead of $\star$?) –  AD. Nov 18 '11 at 8:20

1 Answer 1

The solution of many initial value problems for linear partial differential equations is of the form $K*u_0$, where $u_0$ is the initial value and $K$ is a kernel associated with the equation. Young's inequality gives estimates of the of the solution in $L^p$ spaces in terms of the size of the initial value.

Consider for instance the heat equation in $\mathbb{R}^n\times[0,\infty)$: $$ u_t-\Delta u=0,\quad u(x,0)=u_0(x). $$ The solution is $u(x,t)=K_t*u_0(x)$, where $$ K_t(x)=(4\,\pi\,t)^{-\tfrac{n}{2}}\,e^{-\tfrac{|x|^2}{4t}} $$ is the heat or Gauss kernel. It is in $L^P(\mathbb{R}^n)$ for all $p\ge1$, and $$ \|K_t\|_p\le C_p\,t^{-\tfrac{n}{2}\bigl(1-\tfrac{1}{p}\bigr)}. $$ Then, if $u_0\in L^q(\mathbb{R}^n)$ and $p^{-1}+q^{-1}=1+r^{-1}$, the decay in time of the solution in the $L^r$ norm is $$ \|u(\,\cdot\,,t)\|_r\le C_pt^{-\tfrac{n}{2}\bigl(1-\tfrac{1}{p}\bigr)}\|u_0\|_q. $$

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