I am trying to prove that the conditions of the Strichartz estimates of the homogeneous wave equation, \begin{align} \begin{cases} (\partial_{tt}-\Delta)u=0,\\ u(0,x)=f(x),\\ \partial_tu(0,x)=g(x), \end{cases} \end{align} is necessary. That is to say,
Let $u:\mathbb{R}^{1+n} \rightarrow \mathbb{C}$ be a solution to the homogeneous wave equation and $s\le0$, $2\le q \le \infty$ and $2\le r < \infty$. If $u$ satisfies the following the Strichartz estimates, \begin{equation} \|u\|_{L_t^q L_x^r} \lesssim \|f\|_{\dot{H}^s}+\|g\|_{\dot{H}^{s-1}} \end{equation} where $\dot{H}^{s}$ denotes homogeneous Sobolev space. Then, the conditions are \begin{equation} \frac{1}{q}+\frac{n}{2}=\frac{n}{2}-s \quad \text{and} \quad \frac{2}{q}+\frac{n-1}{r}\leq \frac{n-1}{2}. \end{equation}
The first condition can be shown by using scaling argument, but my purpose is second one. To show this, I found an example, called Knapp counterexample, in Exercise 2.43 of the book, Nonlinear Dispersive Equations Local and Global Analysis written by Terence Tao, \begin{equation} u(t,x)=\int_{1\le\xi_1\le2; |\xi_2|,...,|\xi_n|\le\varepsilon}e^{ix\cdot \xi}e^{it|\xi|}d\xi \end{equation} where $0<\epsilon<1$. According to this book, $u(t,x)$ is comparable to $\varepsilon^{n-1}$ (i.e. $c\varepsilon^{n-1} \le |u(t,x)|\le C\varepsilon^{n-1} $ for some $c$ and $C$, we denote $|u(t,x)|\sim \varepsilon$) whenever $|t| \ll 1/\varepsilon, |x_1-t|\ll \varepsilon$ and $|x_2|,...,|x_n|\ll 1$. And it would be used to show my question.
I proved already that $u(t,x)$ is comparable to $\varepsilon^{n-1}$ using direct calculation. In order to prove that the second condtion holds, I tried to show that $\varepsilon^{\alpha} \leq \|u\|_{L_t^q L_x^r} \lesssim \|u(0,\cdot)\|_{\dot{H}^s}+\|\partial_t u(0,\cdot)\|_{\dot{H}^{s-1}} \leq \varepsilon^{\beta}$ where $\alpha, \beta$ are some numbers depending on $q,r,n$ satisfying $\varepsilon^{\alpha} \le \varepsilon^{\beta}$ which yield the condition. However, I am failing it to find that $\alpha, \beta$. Can anyone advise me?