I am looking for a proof of the following lemma
for the case where:
- $y= (y_1,\cdots, y_n)\mapsto P(y) = \|y\|_2= \sqrt{y_1^2+ \cdots + y_n^2}.$ In this case the rank of the mentioned matrix is $n-1$ for $y\neq 0$
- ${\rm supp}(v)= \{y \in \mathbb{R}^n , 1<\|y\|_2 <2 \}$
The author referred to the following paper for the proof in the general case as stated above where It was done in the frame of Fourier transform of surface carried measures and its behaviour at the infinity. I wonder if there is another (more direct) proof which uses the usual techniques of functional analysis ($L^p$ estimates, interpolation estimates ...etc.)
Thank you for any hint. EDIT:
Here is what I found in the literature for the general proof:
In the following paper (see picture below): the Lemma 2.1 seems to have a result of the same nature
where the author referred again the this paper from which I took a screenshot of the main result
I wonder what the role of the assumption on the Hessian and the parameter $t$ is.