$L^1- L^\infty$ estimate for the semi group of wave equations

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.

• Do you understand in your second source, Proof of Lemma 2.1, it says you can assume that $\phi\in {\mathcal D}({\mathbb R}^m)$ can be taken as a finite sum each supported on a sphere? I understand quite a bit of the other parts, except this one. Feb 20 at 6:47
• I am trying to communicate what I begin to understand. I think the paper of Littman requires the principal curvatures of the surface $S$ to be nonzero. If you make the surface the level surface of your $P$, then the number of nonzero principal curvatures is the same as the rank of the Hessian matrix of $P$. I think that is the main relation. Feb 20 at 7:26
• This is still a mystery for me. I want to understand how the author in the OP used the result of Littman and where the term $(1+|t|)^{-\rho/2 }$ came from. Feb 20 at 10:20