Besov spaces on domains in $\textbf {R}^ d$

Abstract:

We study Besov spaces $B_q^\alpha ({L_p}(\Omega ))$, $0 < p,q,\alpha < \infty$, on domains $\Omega$ in ${\mathbb {R}^d}$ . We show that there is an extension operator $\mathcal {E}$ which is a bounded mapping from $B_q^\alpha ({L_p}(\Omega ))$ onto $B_q^\alpha ({L_p}({\mathbb {R}^d}))$. This is then used to derive various properties of the Besov spaces such as interpolation theorems for a pair of $B_q^\alpha ({L_p}(\Omega ))$, atomic decompositions for the elements of $B_q^\alpha ({L_p}(\Omega ))$, and a description of the Besov spaces by means of spline approximation.