I'm reading the book 'Classical and multilinear harmonic analysis, Muscalu'. I fail to understand the page 261. Actually, I doubt that the proof is right.
Let $f \in BMO(\mathbb{R^d}$) have compact surpport and mean zero. Let $\phi \in S$ be radial, $supp(\phi) \subset B(0,1) $, $\int \hat{\phi}(t\xi)^2\frac{dt}{t}=1$. The author says that
The Calderon reproduing formula now becomes the following statement: $$f(x)=\int_0^\infty (\phi_t * \phi_t * f)(x)\frac{dt}{t}$$
I know that the above identity holds if $f \in L^2$. But we just assumed that $f \in L^1 \cap BMO$ and $f$ has compact support.