Plancherel theorem on half axis

Question

let $f,g$ be nice. Consider the following integral: $$\int_0^\infty dt\,\int_{-\infty}^\infty dx\, f(x)e^{ixt}\int_{-\infty}^\infty dy\, g(y)e^{iyt}.$$ Fubini's theorem doesn't apply here because $|e^{i(x+y)t}|$ is not integrable on $(0,\infty)$. Assume it does (with embarrassment), then by calculating the Fourier transform of the indicator $\chi_{x+y>0}$ one obtains $$\int_{\mathbb R^2}dxdy\, f(x)g(y)\left(\frac{i}{x+y}+\pi\delta(x+y)\right).$$ I wonder if there is any theorem stating this, or how can one prove this with rigor?

Answer 1

I will elaborate on my comment. The Hilbert transform is defined as the linear operator $H$ given by $H(f)^\wedge(t) = -i \, \mathrm{sgn}(\xi) \, \widehat{f}(t)$. Then, the Fourier multiplier $S$ whose symbol is given by $\chi_{\{\xi \geq 0\}}$ is the operator $$ S = \frac{1}{2} \, \big( \mathrm{id} + i \, H \big). $$ The expression on your first line is the quadratic form $(f,g) \mapsto \langle \widehat{S(f)}, \widehat{g} \rangle = \langle \widehat{f}, \widehat{S(g)} \rangle$. This form is bounded in $L^2$ since $S$ is by Plancherel. Therefore you can apply the usual Plancherel identity to get that this form is equal to $\langle f, S(g) \rangle$. The only thing left to do is to express $S(f)$ as a convolution identity, you already did this (symbolically), $$ \langle \widehat{S(f)}, \widehat{g} \rangle = \int_{-\infty}^\infty f(x) \, \bigg( \int_{-\infty}^\infty g(y) \frac{i}{x - y} d y + \pi \, g(x)\bigg). $$ The only thing left to do is to justify the last identity. That can be done using:

1. Choosing $g$ a Schwartz function.
2. Express the integral of $i /(x - y)$ as a principal value by taking limit $\epsilon \to 0$ of the integral over $|y| > \epsilon$. That will give you that the integrand is a tempered distribution.
3. By density of the Schwartz class in $L^2$ and $L^2$-boundedness of the Hilbert transform you can extend the operator to general $L^2$ functions.

All this is done in many places, like Chapter 3 of

Duoandikoetxea, Javier, Fourier analysis. Transl. from the Spanish and revised by David Cruz-Uribe, Graduate Studies in Mathematics. 29. Providence, RI: American Mathematical Society (AMS). xviii, 222 p. (2001). ZBL0969.42001.