Interpolation Hilbert Spaces Between Sobolev Spaces

Abstract

Abstract. We explicitly describe all Hilbert function spaces that are interpolation spaces with respect to a given couple of Sobolev inner product spaces considered over R n or a halfspace in R n or a bounded Euclidean domain with Lipschitz boundary. We prove that these interpolation spaces form a subclass of isotropic Hörmander spaces. They are parametrized with a radial function parameter which is OR-varying at +∞ and satisfies some additional conditions. We give explicit examples of intermediate but not interpolation spaces.

Theorems on Isomorphisms for Some Parabolic Initial-Boundary-Value Problems in Hörmander Spaces: Limiting Case

“…According to the last embedding, the consistency conditions (14) are well-posed for an arbitrary vector We now consider problem (1), (2), (4) corresponding to the general boundary-value condition of the first kind. We associate this problem with a linear mapping…”

“…The space Q s−2,(s−2)/2,' Dis complete, i.e., Hilbert. This fact is known in the Sobolev case ' ⌘ 1 and follows from boundedness of the boundary operators in relation(14) in the corresponding anisotropic Sobolev spaces. In the general case where ' belongs to M 1 , the completeness of the space Q…”

“…X 0 . We need a criterion for a function 2 B to be the interpolation parameter [14] (Theorem 4.2). Proposition 1.…”

“…The role of the index of regularity of distributions for these spaces is played by a general weight function µ. These spaces are now extensively used in analysis and in the theory of partial differential equations [8][9][10][11][12][13][14][15][16].…”

Elliptic operators on refined Sobolev scales on vector bundles

“…Their theory [7] is supplemented in [22][23][24][25][26][27][28][29] for a more extensive class of Hörmander inner product spaces. The refined Sobolev scale and other classes of Hörmander spaces are applied to the spectral theory of elliptic differential operators on manifolds [7,Section 2.3], theory of interpolation of normed spaces [23,30], to some differential-operator equations [31], parabolic initial-boundary value problems [32][33][34][35], and in mathematical physics [36,37]. However, elliptic operators on vector bundles have not been covered by this theory.…”