# Which functions are spherical derivatives?

Mathematics Asked by Giuseppe Negro on December 22, 2020

Let us define the differential operator
$$Z=x_1 partial_{x_2} – x_2 partial_{x_1},$$
where $$(x_1, x_2, x_3)$$ are the standard Cartesian coordinates on $$mathbb R^3$$. I would like to characterize the functions $$hin C^infty(mathbb S^{2})$$ such that
$$tag{1} h=Zf, qquad text{for some }fin C^infty(mathbb S^{2}).$$

The operator $$Z$$ is one of the generators of the rotation group $$SO(3)$$, in the sense that
$$tag{2} Zf(x)=lim_{epsilonto 0}frac1epsilon big( f(R_epsilon x)-f(x)big),$$
where $$R_epsilon$$ is the matrix
$$R_epsilon=begin{bmatrix} cos epsilon & -sin epsilon & 0 \ sin epsilon & cos epsilon & 0 \ 0 & 0 & 1 end{bmatrix}.$$
The formula (2) implies that
$$int_{mathbb S^{2}} Zf(x), dS(x)=0,$$
where $$dS$$ denotes the standard Lebesgue measure on the sphere. Thus, a necessary condition for (1) to hold is that
$$tag{3} int_{mathbb S^{2}} h, dS = 0.$$
Also, since $$Z$$ vanishes at $$(0,0, pm 1)$$, another necessary condition is
$$tag{4} h(0,0,pm 1)=0.$$
Are these two last conditions also sufficient?

No. A stronger condition is needed. Since the orbints of $$Z$$ are the circles on constant lattitude, $$Zf$$ must integrate to zero over each such circle. This can be seen by working in standard spherical coordinates $$(theta,varphi):[0,2pi)times(0,pi)to S^2$$. In these coordinates, $$Z=partial_theta$$, so we can apply the fundamential theorem of calculus along the lattitudes, obtaining $$f(theta,varphi)=f(0,varphi)+int_0^theta Zf(theta',varphi)dtheta'$$ Since $$f(0,varphi)=lim_{thetato 2pi^-}f(theta,varphi)$$, we must have $$int_0^{2pi} Zf(theta',varphi)dtheta'=0$$. If you modify your set of conditions, you can check sufficiency by checking if the above expression gives a smooth function on $$S^2$$ for some choice of $$f(0,varphi)$$.