Let \(\Omega\) be a bounded open subset in \({\mathbb{R}}^ n\) with boundary \(\partial \Omega\) of class \(C^{\infty}\); if \(\alpha\in {\mathbb{R}}\), \(1\leq p,q\leq +\infty\), consider the Besov space \(B^{\alpha}_{p,q}(\Omega)\). Let \(B=(B_ k)_{k=1,...,m}\) be a normal system of boundary operators on \(\partial \Omega\), with \(C^{\infty}\) coefficients. Define (roughly speaking) \(B^{\alpha}_{p,q,B}(\Omega)\) as the space of the elements u of \(B^{\alpha}_{p,q}(\Omega)\) such that \(B_ ku=0\) for any k such that \(\alpha\) is not less than the order of \(B_ k\) plus \(p^{-1}\). If \((.,.)_{[\theta]}\) \((0<\theta <1)\) is the complex interpolation functor, \(\alpha_ 0<\alpha_ 1\), \(1\leq p\leq +\infty\), \(1\leq q<+\infty\), \((B^{\alpha_ 0}_{p,q}(\Omega), B^{\alpha_ 1}_{p,q,B}(\Omega))_{[\theta]}= B^{\alpha}_{p,q,B}(\Omega),\) with \(\alpha =(1-\theta)\alpha_ 0+\theta \alpha_ 1\). The result is proved by adapting a method due to M. Seeley [Stud. Math. 44, 47-60 (1972; Zbl 0237.46041)]. Next, essentially using abstract properties of the real and complex interpolation methods, it is proved that, if \(\alpha_ 0<\alpha_ 1\), \(0<\theta <1\), \(1\leq p,q_ 0,q_ 1,q\leq +\infty\), the real interpolation space \((B^{\alpha_ 0}_{p,q_ 0}(\Omega), B^{\alpha_ 1}_{p,q_ 1,B}(\Omega))_{\theta,q}\) coincides with the space \(B^{\alpha}_{p,q,B}(\Omega)\), with \(\alpha =(1- \theta)\alpha_ 0+\theta \alpha_ 1\). Applying the previous results, one can show the following: let A(x,\(\partial)\) be a strongly elliptic partial differential operator of order 2m with \(C^{\infty}\) coefficients, giving rise with the boundary conditions \(B_ k\) to a regular elliptic problem in \(\Omega\). Set \(A_{NpB}\) the realization of the problem with homogeneous boundary conditions in \(W^{N,p}(\Omega)\) \((N\in {\mathbb{N}}_ 0\), \(1\leq p\leq +\infty)\) and indicate with \(D(A_{NpB})\) the domain of \(A_{NpB}\). Then, if \(0<\theta <1\), \(1\leq q\leq +\infty\), \((W^{N,p}(\Omega), D(A_{NpB}))_{\theta,q}= B^{\alpha}_{p,q,B}(\Omega)\), with \(\alpha =N+2m\theta\). These results are new in the case \(p=1.\)
In the final section extensions of the previous results to open subsets and problems with are not necessarily so regular are discussed.