Real analytic extension of functions on normal crossings

We consider a compact $C^{\omega }$ manifold $X$ and finitely many regular $C^{\omega }$ submanifolds $Y_1,\dots ,Y_q$ of $X$, which are closed subsets in $X$, such that the union of $Y_j$’s has only normal crossings. We show that every continuous function on the union which is of class $C^{\omega }$ on each $Y_j$ can be extended to a $C^{\omega }$ function on $X$. A crucial feature of our proof is to employ basic tools of real analytic geometry – Cartan Theorems A and B.