Let me try this in n=1 for simplicities sake: Let $H'\sim H$. I want to know where $u_Lv_H\neq 0$ so I calculate using fourirtransformation and convolution: $$(u_{H'}v_H)^\wedge(y)=\int_{\mathbb{R}}\psi_{H'}(y-x) \psi_H(x) u(y-x) v(x) dx $$ Therefore $$supp((u_{H'}v_H)^\wedge)=(y: \exists x \ with \ H/2\leq x\leq2H \ and \ H'/2\leq y-x \leq 2H')=(y: (H+H')/2 \leq y \leq 2(H+H'))=(y: y \sim H)$$. This exact calculation worked for the case $L<<H$ but here it seems to to fail. And I'm unsure how that works with n>1 since I have to deal with $|y-x| $ instead of $y-x$ then.

I understand your first 2 points and your right about me not knowing enough about these general principles. I looked in your two references but they mostly concern Differentiation. I'm interested in a proof of your statement "Two high frequencies can interact to produce something of much lower frequencies, but not something with very high frequency." I could not find that in your references, as Principle A.5 is "just" an heuristic.

That does work for question 1 but if I use that on question 2 I get ( since $H \approx H'$): $$ P_L I^+(u_{H'}, v_H) \neq 0 \iff L \approx H (:\iff L\lesssim H \ or \ H \lesssim L)$$ but for the estimation in question 2 to work I would need $$ P_L I^+(u_{H'}, v_H) \neq 0 \iff L\lesssim H$$ Therefore I don't understand why he can ignore the sum over $H\lesssim L$ there.

1. I can see, that the Knapp counterexample gets weakened, the thing I don't get is why it is at all useful that you get more (q,r) which are admissable. How can that help in proving wellposedness? The procedures I know of only use $L^4$ or $L^2$ estimates. 2. I looked deeply into the two papers you directed me to, I see that this can be applied to KGE, however I can't see what the angular restrictions got to do with radial data. The thing is that the procedure outlined there (and used in most of the other papers) relies only on intersections and not on the specific attributes of initial data.

sorry but I don't understand how you get the expression exp(2πi(c(ρ)t±r+k/4)ρ). Maybe I have to understand why (8) [that is the idenity for the wave propagator with the Bessel function] is true first. This is taken for granted in all the papers I found. Do you have a proof/reference for that?