Complex interpolation with Dirichlet boundary conditions on the half line

Abstract

We prove results on complex interpolation of vector‐valued Sobolev spaces over the half‐line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space‐valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.

The heat equation with rough boundary conditions and holomorphic functional calculus

“…Therefore, the above isomorphism extends to all k ∈ N 0 . Therefore, by a standard retraction-coretraction argument (see [77,Theorem 1.2.4] and see [56,Lemma 5.3] The final assertion is clear for even k. For odd k = 2ℓ + 1 with ℓ ∈ N 0 by Proposition 2.3, Theorem 4.1 and the result in the even case we can write…”

“…Therefore, it suffices to approximate such functions f in the L q (R d , w) norm. To do so one can use a standard argument (see [56,Lemma 2.2]) by using a mollifier with compact support. The density result [48,Theorem 7.2] can be extended to the vector-valued setting:…”

Stochastic maximal regularity for rough time-dependent problems

“…Proof (1): This can be proved as in [80,Proposition 5.5] by using a suitable extension operator and a suitable extension of w| I to a weight on R. The following result follows from [89, Theorem 1.1] and standard arguments (see [1] for details). Knowing the optimal trace space is essential in the proof of Theorem 3.15.…”

Stability properties of stochastic maximal L-regularity

“…In this section we recall some basic fact about vector-valued Sobolev spaces and Bessel potential spaces with power weights. We refer to [31,36] It is of interest to note that w α belongs to the Muckenhoupt class A p if and only if α ∈ (−1, p − 1). For k ∈ N, let W k,p (I, w α ; X) denote the subspace of L p (I, w α ; X) of all functions for which ∂ j f ∈ L p (R, w α ; X) for j = 0, .…”

“…Let D(I; X) = C ∞ c (I; X) with the usual topology and let D ′ (I; X) = L (D(I), X) denote the X-valued distributions. To handle Bessel potential space on intervals we need the following standard result, which can be proved as in [31,Propositions 5.5 and 5.6], where the case I = R + was treated. In the case I = (0, T ) with T ∈ (0, ∞) it is possible to construct E k such that its norm is T -independent (see [35,Lemma 2.5]).…”