Fourier-restriction on Hyperplane

Question

In Muscalu, Schlag "Classical and multilinear harmonic analysis, Volume 1" (2013), Excercise 11.1 is to prove, basically, that there exists a function $f\in L^p \quad \forall\ p>1$ such, that the Fourier-transform $\mathcal{F}f$ of $f$ is infinite on a (previously chosen) hyperplane $$H(\theta, s) := \{ x\in \mathbb{R}^n | x\cdot \theta = s\} \qquad (\theta \in S^{n-1}, s\in \mathbb{R})$$ I have arrived at the function $$f_{\theta, s}(x) =\frac{1}{1+|x\cdot \theta -s|}$$ (the $\cdot$ denotes the standard inner product) I could show, that $f\in L^p(\mathbb{R}^n) \quad \forall\ p>1$, but I have failed to show, that it's Fourier-transform is identical to $\infty$ on $H(\theta,s)$. How can I show this?

The hint which lead me to this choice of $f_{\theta,s}$ is from this book: Fourier Analysis And Convexity, p. 218

EDIT:
Thanks to @paulgarret I was able to prove the following
Let $U$ be a unitary matrix with $U\theta = e_n$ and let $$f(x) = \frac{e^{-\sum_{j=1}^{n-1} x_j^2}}{1+|x_n|}$$ Then by $f_{\theta, s} := f(Ux-se_n)$ we have $$\widehat{f_{\theta,s}}(\xi) =\infty \quad \forall \ \xi \in H(\theta,s)$$ and $$\Vert f_{\theta,s} \Vert_p^p = \Vert f \Vert_p^p = \frac{2}{p-1} \left ( \frac{\pi}{p} \right ) ^{\frac{n-1}{2}} < \infty \qquad \forall \ p>1$$ The only remaining Question is, how can we construct such a $U$ to give a concrete example?

Answer 1

To write a preliminary answer as expanded comment, to explain why the tentative answer has a certain problem, although it does make progress: first, a simpler analogue where a function is constant in one direction: take $f(x,y)={1\over 1+x^2}$. Then $$ \int_{\mathbb R^2} f(x,y)\;dx\,dy\;=\; \int_{\mathbb R^2}{dx\,dy\over 1+x^2} \;=\;\int_{\mathbb R} {dx\over 1+x^2}\cdot \int_{\mathbb R}1\;dy $$ The integral over $x$ is fine, but the integral in $y$ is the integral of $1$ over the whole real line, giving $+\infty$. Thus, as in your attempt, as noted by @GiuseppeNegro, this glitch has to be addressed.

By rotation, we might as well suppose the hyperplane is the $x_n=0$ hyperplane. Thus, you want a function to blow up as $x_n\to 0$, but/and be integrable on the whole $\mathbb R^n$. Ok, as a function of $x_n$, something like ${1\over \sqrt{|x_n|}(1+x_n^2)}$. In $x_1,\ldots,x_{n-1}$, you want reasonable behavior, so a Gaussian (handy for FT, also) like $e^{-\pi(x_1^2+\ldots+x_{n-1}^2)}$ would do. The product has the right sort of behavior. Take FT, ...

(Although one should generally tend to believe "authorities", there is the auxiliary issue of being alert to disconnects between one's understanding of what is being asserted/asked, what is really asked/asserted, and, still, what one has actually done. Partial but incomplete progress should not be confused with "failure", nor with "success", for example.)

Answer 2

Here is an interesting idea, for two dimensions, and I'm sure it can be generalized. Let $f(x,y) = \frac{1}{\log |x| (1 + |yx|)}$. This function is in $L^p$: $\int |f(x,y)|^p dy dx = \int \frac{1}{(\log |x|)^p} \int \frac{du}{|x|(1+|u|)^p} dx < \infty$, and just barely (needs $p>1$ for the logarithm)!

Now take the Fourier transform. Integrating in $y$ first is already divergent (though I'm not sure about being $\infty$. The F.T. seems to exist as a principal value). It's not very clear here. If $\xi = 0$ then we definitely have $\infty$, but in other places the function is not Lebesgue integrable, so I don't know what to make of it.

To generalize this specific example, maybe $f(x_1,\ldots,x_{n-1},x_n) = \frac{1}{\log r (1+|x_n| r_{}^{n-1})}$ where $r= \sqrt{ |x_1|^2 + \ldots + |x_{n-1}|^2}$