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In this paper, we prove that any relative character (a.k.a. sphericalcharacter) of any admissible representation of a real reductive group withrespect to any pair of spherical subgroups is a holonomic distribution on thegroup. This implies that the restriction of the relative character to an opendense subset is given by an analytic function. The proof is based on anargument from algebraic geometry and thus implies also analogous results in thep-adic case.  As an application, we give a short proof of some results from [KO13,KS16] onboundedness and finiteness of multiplicities of irreducible representations inthe space of functions on a spherical space.  In order to deduce this application we prove relative and quantitativeanalogs of the Bernstein-Kashiwara theorem, which states that the space ofsolutions of a holonomic system of differential equations in the space ofdistributions is finite-dimensional. We also deduce that, for every algebraicgroup $G$ defined over $\mathbb{R}$, the space of $G(\mathbb{R})$-equivariantdistributions on the manifold of real points of any algebraic $G$-manifold $X$is finite-dimensional if $G$ has finitely many orbits on $X$.

Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs