An optimal trace estimate for microlocal square functions on quadratic surfaces

We study a local trace estimate for the microlocal angular square function $$G_R f := \left(\sum_\Theta |f_\Theta|^2\right)^{1/2}$$ associated with a parabolic decomposition of the frequency annulus of radius $R$ in $\mathbb{R}^3$. The measure under consideration is $$\mu_Q=\chi\, H^2\lfloor S_Q,$$ where $\chi\in L^\infty(S_Q)$ is a measurable nonnegative density compactly supported in the patch, and $$S_Q=\{(u_1,u_2,Q(u_1,u_2)):u\in U\}, \qquad Q(u_1,u_2)=\frac12(\lambda_1u_1^2+\lambda_2u_2^2), \qquad \lambda_1\lambda_2 >0.$$ Writing $\rho=R^{-1/2}$, we prove $$\| G_R f\|_{L^2(\mathrm d\mu_Q)} \lesssim R^{1/8}\|f\|_{L^2(\mathbb R^3)}.$$ Under local positivity of the density near the tangency point, the factor $R^{1/8}$ is attained by a tangent wave packet test and hence cannot be improved within this elliptic quadratic model, at this parabolic scale and for this angular square function. In particular, it measures the failure of a trace bound uniform in $R$ within this class. Its source is the extreme tangential interaction between a tube of radius $\rho$ and $S_Q$: the relevant surface measure is $\sim\rho^{3/2}$, whereas an $L^2$-normalized wave packet has quadratic size $\sim\rho^{-2}$. Thus the optimal quadratic cost is $\rho^{-1/2}$, producing the norm factor $\rho^{-1/4}=R^{1/8}$.