# What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate?

I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the reductions except for the one using a rotation.

In this paper, $$Q$$ is a nonsingular quadratic form on $$\mathbb R^n$$, $$n\ge 3$$, given by$$Q(\xi) = -\xi_{1}^2-\dots-\xi_{\mathstrut j}^2 + \xi_{j+1}^2 + \dots \xi_{\mathstrut n}^2.$$ They apparently say: when $$a\neq 0$$ is such that $$Q(a)\neq 0$$, it can be rotated while preserving the form of $$Q$$ so that $$a\in \operatorname{span}((0,\dots,0,1))$$ or $$a\in \operatorname{span}((1,0,\dots,0))$$. Why is this true?

With a rotation in the first $$j$$ components and then another in the remaining ones, I can make $$a\in \operatorname{span}((1,0,\dots,0),(0,\dots,0,1))$$. So 'clearly' I guess I'm missing some sort of rotation that intermingles these two parts of $$Q$$. But what rotations of this type preserve $$Q$$?

Later in the paper near the end, the author specifies in the case where $$Q(D)$$ is the wave operator $$\partial_t^2 -\Delta$$ that one needs hyperbolic rotations. Do I need to figure out an appropriate group of "rotations" for $$Q$$ in the more general case?

[KRS] Kenig, C. E.; Ruiz, A.; Sogge, C. D., Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55, 329-347 (1987). ZBL0644.35012.

It turns out that (1) Yes, I will need to use 'nonstandard' rotations, but (2) the hyperbolic ones suffice. This is because (as already in question body) the usual Euclidean rotations in $$\mathbb R^j$$ and $$\mathbb R^{n-j}$$ turns the vector $$a\in \mathbb R^n$$ essentially into a 2D vector which we can consider as living in a hyperbolic plane. (in the event $$j=0,n$$ there is nothing to do.) So my problem stems from not understanding any hyperbolic geometry at all :) and it could well be that a passer-by could have easily pointed this to me had I better phrased my problem. Alas!
In any case: the question body reduces the issue into the study of $$b=(|a'|,0,\dots,0,|a''|)$$ where $$c = \sqrt{Q(a)}$$ with $$a=(a',a'')\in \mathbb R^j \times \mathbb R^{n-j}$$ and $$Q(b) = |a'|^2 - |a''|^2$$.
After the reduction that lets us assume $$a\not\in\operatorname{Char} P$$, we either have $$Q(a)>0$$ or $$Q(a)<0$$, or equivalently $$|a'|^2>|a''|^2$$ or $$|a'|^2<|a''|^2$$. In the first case (the second entirely analogous) we can set $$c = \sqrt{Q(b)},\quad \theta = \cosh^{-1}|a'/c|$$ so that $$\tilde b = b/c = (\sinh \theta) e_1 + (\cosh \theta) e_n.$$ Then the "hyperbolic rotation" $$R(x_1,\dots,x_n) = (x_1\cosh\theta - x_n \sinh \theta, x_2,\dots, x_{n-1}, -x_1\sinh \theta + x_n \cosh \theta)$$ can be easily seen by a direct calculation to preserve $$Q$$ and map $$\tilde b$$ to $$e_n$$.
(The relationship to the differential operator $$a\cdot \nabla$$ comes by using the rotated function $$u_R(x)=u(R^Tx)$$, for which $$a\cdot\nabla u_R(x) = (\partial_n u)(R^Tx).$$)