I am having an annoying conceptual problem trying to solve problem 46 in Chapter 8 of Folland's "Real Analysis". I'll try to explain my problem as briefly as possible.
Consider the wave equation $\partial_{t}^{2}u-\Delta u=0$ in $\mathbb{R}^{3}$ with initial data $u(x,0)=f(x)$ and $\partial_{t}u(x,0)=g(x).$
Taking Fourier transform, we swiftly get: $\widehat{u}(\xi,t)=\widehat{f}(\xi)\cos (2 \pi \vert \xi \vert)+\widehat{g}(\xi) \dfrac{\sin (2 \pi \vert \xi \vert)}{2 \pi \vert \xi \vert}.$
Following Folland's notation, we write $u(x,t)=f \ast \partial_{t}W_{t}(x)+g \ast W_{t}(x),$ where $W_{t}$ is the inverse Fourier transform of $ \dfrac{\sin (2 \pi \vert \xi \vert)}{2 \pi \vert \xi \vert}.$
In the three-dimensional space, we have that the inverse transform of $\displaystyle \left(\dfrac{\sin (2 \pi \vert \xi \vert)}{2 \pi \vert \xi \vert} \right) $ is $\dfrac{\sigma_{t}(x)}{4 \pi t},$ with $\sigma_{t}(x)$ the surface measure in the sphere $\vert x \vert =t.$
My problem now is that I do not understand at all how to compute $f \ast \dfrac{\sigma_{t}(x)}{4 \pi t},$ for I do not know even where to integrate! What does it mean to take the convolution of $f$ with a surface measure?? Folland defines previously in his book the convolution of $f \in L^{1}\left( \mathbb{R}^{n} \right)$ with Borel measures in $\mathbb{R}^{n}.$
$f \ast W_{t}$ should be a integral over the whole space, but then I do not know if $\displaystyle \int_{\mathbb{R}^{3}} f(x-y)\dfrac{\sigma_{t}(y)}{4 \pi t}dy $ makes any sense....
I have also tried supposing that $f \ast W_{t}$ should be understood as $\displaystyle \dfrac{1}{4 \pi} \int_{\partial B(0,t} f(x-y) \dfrac{\sigma_{t}(y)}{t} $... but what do I do then? use spherical coordinates and write the measure as $t^{2}\sin \theta$....
I do not know how to interpret this convolution with a surface measure nor how to compute it, but even proceeding as I just indicated, I do not see how are we going to recover Kirchhoff's solution, which one tipically gets with the method of spherical means.
Thank you so much for any help!
And surface measure is an example of a Borel measure, so the aforementioned definition applies. Without having a book with me, I'll venture a guess: it is
$$(f*\mu) (x) = \int_{\mathbb R^n} f(x-y)\,d\mu(y)$$
Let's look at the one-dimensional case first. Here the surface of sphere of radius $t$ is the two-point set $\{-t,t\}$. The normalized measure on this "sphere" is $\mu=\frac12 (\delta_{-t}+\delta_t)$. The convolution of $f$ with this measure is $$(f*\mu)(x) = \frac12 (f(x+t)+f(x-t))$$ which you can recognize as a part of d'Alembert's formula.
Same in three dimensions. The convolution of $f$ with the normalized surface measure $\mu=\frac{1}{4\pi t}d\sigma_t(y)$ is the spherical mean of $f$: $$(f*\mu) (x) = \frac{1}{4\pi t}\int_{|y|=t} f(x-y)\,d\sigma_t(y) $$