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Let $f$ and $g$ be real functions such that $\int_0^\infty(f(x))^2dx<\infty$ and $\int_0^\infty(g(x)^2dx<\infty$. Prove that:

$$\left(\int_0^\infty\int_0^\infty\frac{f(x)g(y)}{x+y}dxdy \right)^2\leq C\int_0^\infty(f(x))^2dx\int_0^\infty(g(y))^2dy$$

where $C$ is an universal constant (independent of $f$ and $g$)

My attempt: using Cauchy-Schwarz inequality I evaluate:

$$LHS\leq(\int_0^\infty (g(y))^2dy)\times (\int_0^\infty (\int_0^\infty \frac{f(x)}{x+y}dx)^2dy)\leq$$ $$\leq(\int_0^\infty(g(y))^2dy)\times(\int_0^\infty((\int_0^\infty(f(x))^2dx)\times(\int_0^\infty\frac{dx}{(x+y)^2}))dy)\leq$$ $$\leq\left(\int_0^\infty(f(x))^2dx\int_0^\infty(g(y))^2dy\right)\times\int_0^\infty\int_0^\infty\frac{dxdy}{(x+y)^2}$$

which is useless since $\int_0^\infty\int_0^\infty\frac{dxdy}{(x+y)^2}$ is not finite.

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    $\begingroup$ One issue is that in going from your first line to second line, you're applying the Cauchy-Schwarz inequality to the functions $f(x)$ and $\frac{1}{x+y}$ (where $x$ is the variable and $y$ is fixed). Clearly, the latter function does not have a finite integral, so making this step is useless. $\endgroup$
    – Andrew
    Sep 2, 2014 at 17:07

1 Answer 1

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Assume without loss of generality that $f,g$ are nonnegative (in order to use safely Fubini's Theorem), and denote by $I$ the double integral in the left-hand side. We have \begin{eqnarray} I&=&\int_0^\infty g(y)\left(\int_y^\infty f(u-y)\,\frac{du}u \right)dy\\ &=&\int_0^\infty g(y)\left(\int_1^\infty f(y(v-1))\,\frac{dv}v \right)dy\\ &=&\int_0^\infty g(y)\left(\int_0^\infty f(yu)\frac{du}{u+1} \right)dy\\ &=&\int_0^\infty\left(\int_0^\infty g(y)f(uy)\, dy\right)\frac{du}{u+1}\cdot \end{eqnarray} Now, apply Cauchy-Scharz to the inner integral: this gives \begin{eqnarray} \int_0^\infty g(y)f(uy)\, dy&\leq& \Vert g\Vert_2\times \left(\int_0^\infty f(uy)^2 dy\right)^{1/2}\\ &=&\Vert g\Vert_2\times \left(\int_0^\infty f(t)^2 \frac{dt}{u}\right)^{1/2}\\ &=&\Vert g\Vert_2\times \Vert f\Vert_2\times \frac{1}{\sqrt u}\cdot \end{eqnarray} Altogether, we obtain $$I\leq \Vert g\Vert_2\times \Vert f\Vert_2\times\int_0^\infty\frac{du}{(u+1)\sqrt u}\, , $$ which gives the result since the integral in the right-hand side is finite.

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  • $\begingroup$ awesome! very elegant answer $\endgroup$ Sep 3, 2014 at 5:32
  • $\begingroup$ Just mentioning that $\int_0^\infty\frac{du}{(u+1)\sqrt u} = \pi$. $\endgroup$
    – Martin R
    Sep 16, 2020 at 13:26
  • $\begingroup$ @Martin R On can even justify the value of this integral (thanks to Wolfram Alpha) by using the fact that a primitive function of $\dfrac{1}{(u+1)\sqrt{u}}$ is $2 \ \text{arctan} \sqrt{u}$. $\endgroup$
    – Jean Marie
    Sep 16, 2020 at 20:03

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