  Soft-operators, loosely speaking, are operators which create or annihilate
zero energy massless particles on the celestial sphere in Minkowski space. The
Lorentz group acts on the celestial sphere by conformal transformation and the
soft-operators transform as conformal primary operators of various dimension
and spin. Working in space-time dimensions $D=4$ and $6$, we study some
properties of the conformal representations with the (leading) soft photon and
graviton as the highest weight vectors. Typically these representations contain
null-vectors. We argue, from the $S$-matrix point of view, that infinite
dimensional asymptotic symmetries and conformal invariance require us to set
some of these null-vectors to zero. As a result, the corresponding
soft-operator satisfies linear PDE on the celestial sphere. Curiously, these
PDEs are equations of motion of Euclidean gauge theories on the celestial
sphere with scalar gauge-invariance, i.e, the gauge parameter is a scalar field
on the sphere. These are probably related to large $U(1)$ and supertranslation
transformations at infinity. Now, the PDE satisfied by the soft-operator can be
converted into PDE for the $S$-matrix elements with the insertion of the
soft-operator. These equations can then be solved subject to appropriate
boundary conditions on the celestial sphere, provided by conformal invariance.
The solutions determine the soft $S$-matrix elements, for different helicities
of the soft-particle, in terms of a single scalar function. This makes the
Ward-identity for the asymptotic symmetry almost integrable. The result of the
integration, which we are not able to perform completely, should of course be
Weinberg's soft-theorem. Finally, we comment on the similarity between the
roles played by null-states in the context of asymptotic symmetry and in string
theory in relation to space-time gauge symmetry.
