Let $GL(n,\mathbb{C})$ be the general linear group and let $U(n)$ be the unitary group in it, which is a maximal compact subgroup. I consider the loop group $\Lambda GL(n,\mathbb{C})$ of maps from $S^1$ to $GL(n,\mathbb{C})$ with some regularity. It has two important subgroups, namely $\Lambda U(n)$, subgroup of loop with values in $U(n)$, and $\Lambda_+ GL(n,\mathbb{C})$ the subgroup of loops with only nonnegative Fourier coefficients.

Then one has the Iwasawa decomposition: $$\Lambda GL(n,\mathbb{C}) = \Lambda U(n). \Lambda_+ GL(n,\mathbb{C}).$$ which is unique up to the group $U(n)$, seen as constant loops in $\Lambda U(n)$.

Writing $g = k b$ for such a decomposition, I would like to understand how big $k$ can be, relatively to $g$. More precisely, I am interested in $H^s$-regularity with $s > 1/2$. Viewing both $GL(n,\mathbb{C})$ and $U(n)$ in $\mathbb{C}^{n^2}$, one can speak of the $H^s$-norms of $g$ and $k$ and I want to estimate the norm of $k$ with respect to the norm of $g$.

It seems reasonable to me that the norm of $k$ is bounded by (a constant times) the norm of $g$ times the absolute value on the determinant of $g^{-1}$ but I can't prove it.

Thanks a lot.