Anisotropic function spaces and elliptic boundary value problems (Q2883886)

There are anisotropic spaces that allow functions to have variable numbers of derivatives in each direction. The classical example occurs for solutions of the heat equation \(u_{xx} = u_t\), which has twice as much smoothness in \(x\) as in \(t\). The author wishes to consider a different type of anisotropy. The variables are broken up into two sets by writing \(\mathbb R^n = \mathbb R^{n_1} \times \mathbb R^{n_2}\) and assuming smoothness of some order \(s_1\) in the first \(n_1\) variables and \(s_1 + s_2\) in the remaining \(n_2\) variables. A similar decomposition is used for functions in an open subset \(\Omega = \Omega_1 \times \Omega_2\).NEWLINENEWLINELet the usual Triebel-Lizorkin spaces \(F^s_{p,q}\) and Besov spaces \(B^s_{p,q}\) be defined through a partition of unity in Fourier space, NEWLINE\[NEWLINE \sum_{j = 0}^{\infty} \widehat{\phi_j} =1, NEWLINE\]NEWLINE where \(\phi_j\) form a dyadic Littlewood-Paley partition of unity, and NEWLINE\[NEWLINE \| f \|_{F^s_{p,q}} = \| \{ 2^{ s j} f_j \}_{l^q_j} \|_{L^p}, \;\;\| f \|_{B^s_{p,q}} = \{ \| 2^{s j} f_j \|_{L^p} \}_{l^q_j}, NEWLINE\]NEWLINE and NEWLINE\[NEWLINEH^{s.p}(\mathbb R^n) = F^s_{p,2}(\mathbb R^n), \;\;B^{s,p} = B^s_{p,p}. NEWLINE\]NEWLINE Then with \(\mathbb R^n = \mathbb R^{n_1} \times \mathbb R^{n_2}\), let \(J^{s_2}_{(2)} f = \mathcal{F}^{-1} ( \langle \xi^{(2)} \rangle \mathcal{F} f) \) be the Bessel potential operator in the \(n_2\) variables. Here \(\langle \xi^{(2)} \rangle = (1 + \xi_{n_1 + 1}^2 + \dots + \xi_n^2)^{1/2}\). His anisotropic spaces are NEWLINE\[NEWLINE H^{(s_1, s_2), p}(\mathbb R^{n_1} \times \mathbb R^{n_2}) = \{ f \in S^{\prime}(\mathbb R^n): \| J^{s_2}_{(2)} f \|_{H^{s_1,p}} < \infty \}, NEWLINE\]NEWLINE and NEWLINE\[NEWLINE B^{(s_1, s_2), p}(\mathbb R^{n_1} \times \mathbb R^{n_2}) = \{ f \in S^{\prime}(\mathbb R^n): \| J^{s_2}_{(2)} f \|_{B^{s_1,p}} < \infty \}, NEWLINE\]NEWLINE in each case equipped with the obvious norm.NEWLINENEWLINEThe author shows the important properties of these spaces; relations with pseudo-differential operators, trace and extension theorems, multiplication theorems and interpolation theorems. Armed with these properties, he generalizes results of Seeley and Hörmander (in particular, Hörmander proved comparable results for \(p = 2\)) to general \(p\).