How to obtain the decay of Fourier transform of the Bochner-Riesz multipliers? For $\lambda>0$ define: $$ \hat{m_{\lambda}}(x)=\int\limits_{\mathbb{R}^d} (1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \xi}d\xi,\label{1}\tag{1} $$ then $|\hat{m_{\lambda}}(x)|\sim |x|^{-\frac{d+1}{2}-\lambda}$, for $|x|\rightarrow \infty$.
The classic way to do this is to compute them exactly using Bessel functions, while Tao mentions in his notes (1999) that there is another more robust way called "fuzzier", but I don't get his point. Can someone explain this in details or provide other methods without using special functions?
P.S.: The "fuzzier" method he mentions consists in the following three steps. Assume $x=\lambda e_d$: then
we can "restrict" the Fourier transform \eqref{1} to the unit sphere and write $$ \int\limits_{|\xi|\leq 1}(1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \xi}d\xi=\int\limits_{|\xi_{d}-1|\ll1}+\int\limits_{|\xi_{d}+1|\ll1}+\int\limits_\text{other},$$ where the "other" integral term is easily evaluable.
Then we can express the $$ (1-|\xi|^2)_{+}^{\lambda}=(fd\sigma)*\mu+\text{error}, $$ where
- $f$ is an appropriate function on $\mathbb{S}^{d-1}$ and
- $\mu(\xi)=\delta_0(\xi^{'})\eta(\xi_d)(-\xi_d)_{+}^{\lambda}$ such that the "error" term vanishes to order $\lambda+1$.
we can reiterate the step above and get an error term with sufficiently higher order in the to get the decay, and the front term could also be tackled.
I don't understand what happens in the second and third steps:
- How to select $f$?
- What is the "vanishing order" which can lead to the desired decay?
