  In this article we establish novel decompositions of Gaussian fields taking
values in suitable spaces of generalized functions, and then use these
decompositions to prove results about Gaussian multiplicative chaos.
  We prove two decomposition theorems. The first one is a global one and says
that if the difference between the covariance kernels of two Gaussian fields,
taking values in some Sobolev space, has suitable Sobolev regularity, then
these fields differ by a H\"older continuous Gaussian process. Our second
decomposition theorem is more specialized and is in the setting of Gaussian
fields whose covariance kernel has a logarithmic singularity on the diagonal --
or log-correlated Gaussian fields. The theorem states that any log-correlated
Gaussian field $X$ can be decomposed locally into a sum of a H\"older
continuous function and an independent almost $\star$-scale invariant field (a
special class of stationary log-correlated fields with 'cone-like' white noise
representations). This decomposition holds whenever the term $g$ in the
covariance kernel $C_X(x,y)=\log(1/|x-y|)+g(x,y)$ has locally
$H^{d+\varepsilon}$ Sobolev smoothness.
  We use these decompositions to extend several results that have been known
basically only for $\star$-scale invariant fields to general log-correlated
fields. These include the existence of critical multiplicative chaos, analytic
continuation of the subcritical chaos in the so-called inverse temperature
parameter $\beta$, as well as generalised Onsager-type covariance inequalities
which play a role in the study of imaginary multiplicative chaos.
