Range decreasing group homomorphisms and holomorphic maps between generalized loop spaces

Let $mathcal{G}$ resp. $M$ be a positive dimensional Lie group resp. connected complex manifold without boundary and $V$ a finite dimensional $C^{infty}$ compact connected manifold, possibly with boundary. Fix a smoothness class $mathcal{F}=C^{infty}$, Holder $C^{k, alpha}$ or Sobolev $W^{k, p}$. The space $mathcal{F}(V, mathcal{G})$ resp. $mathcal{F}(V, M)$ of all $mathcal{F}$ maps $V to mathcal{G}$ resp. $V to M$ is a Banach/Frechet Lie group resp. complex manifold. Let $mathcal{F}^0(V, mathcal{G})$ resp. $mathcal{F}^{0}(V, M)$ be the component of $mathcal{F}(V, mathcal{G})$ resp. $mathcal{F}(V, M)$ containing the identity resp. constants. A map $f$ from a domain $Omega subset mathcal{F}_1(V, M)$ to $mathcal{F}_2(W, M)$ is called range decreasing if $f(x)(W) subset x(V)$, $x in Omega$. We prove that if $dim_{mathbb{R}} mathcal{G} ge 2$, then any range decreasing group homomorphism $f: mathcal{F}_1^0(V, mathcal{G}) to mathcal{F}_2(W, mathcal{G})$ is the pullback by a map $phi: W to V$. We also provide several sufficient conditions for a range decreasing holomorphic map $Omega$ $to$ $mathcal{F}_2(W, M)$ to be a pullback operator. Then we apply these results to study certain decomposition of holomorphic maps $mathcal{F}_1(V, N) supset Omega to mathcal{F}_2(W, M)$. In particular, we identify some classes of holomorphic maps $mathcal{F}_1^{0}(V, mathbb{P}^n) to mathcal{F}_2(W, mathbb{P}^m)$, including all automorphisms of $mathcal{F}^{0}(V, mathbb{P}^n)$.