How to prove a tempered distribution is in $L^p(\mathbb{R}^n)$

Question

Given $g \in L^p(\mathbb{R}^n)$, how can I to prove that the tempered distribution $$f=\mathcal{F}^{-1}[(z-4\pi^2|x|^2)^{-1}\mathcal{F}g]$$ is in $L^{p}(\mathbb{R}^n)$ where $z \in \{u \in \mathbb{C}-\mathbb{R}_{+};Re(u) \geq 0\}, x \in \mathbb{R}^n$ and $\mathcal{F}$ and $\mathcal{F}^{-1}$ denotes the Fourier Transforms and Inverse Fouries Tranformes respectively?

This problems appears in a proof line of theorem 2.3.3, pg 40, from book "The Theory of Fractional Powers of Operators", by C. Martínez Carracedo and M. Sanz Alix.

Answer 1

Try to write $k \mapsto (z- 4 \pi^2 |k|^2)^{-1}$ as the Fourier transform of a function $h\in L^1$ (try a multiple of $x \mapsto e^{-\alpha |x|}$, with $\alpha$ a square root of $-z$ - notice the condition on $z$ is what ensures you can pick a square root with strictly positive real part which is needed to get it to be in $L^1$), then recall $$\mathcal F h \cdot \mathcal F g = \mathcal F (h * g),$$so you are looking at $h * g$ which is in $L^p$ by Young's inequality.