I am looking for generalized functions (including pseudodifferential operators) that satisfy the following identity ($x,y \in R^d$):
$f(h(x),h(y))=\left(\frac{1}{\det(D_xh) \det(D_yh)}\right)^{1/4} f(x,y)$
in a distributional sense for every diffeomorphism $h(x)$ and (preferably) $d = 4$.
The reason for this is that I am trying to construct a "square root" of the Dirac delta $\delta^{(4)}(x-y)$ and since it satisfies the above identity for ${1/2}$ instead of $1/4$ it seems like a natural consequence to consider the above.
My results so far are sadly rather modest:
- I have proved the above for the Dirac delta distribution (1/2-density instead of 1/4) using infinitesimal diffeomorphisms.
- I am rather sure by now that the f(x,y) I am looking for is a pseudodifferential operator, but that's all.
Of course I have searched pretty much the whole internet but couldn't find anything that brought me considerably further (fractional calculus, theory of fourier integral operators etc).
I would be very thankful for any help you can provide or pointers in some direction you might think can help.
Thank you very much in advance.