Characterization of nonlinear Besov spaces

Abstract

The canonical generalizations of two classical norms on Besov spaces are shown to be equivalent even in the case of non-linear Besov spaces, that is, function spaces consisting of functions taking values in a metric space and equipped with some Besov-type topology. The proofs are based on atomic decomposition techniques and metric embeddings. Additionally, we provide embedding results showing how non-linear Besov spaces embed into non-linear p-variation spaces and vice versa. We emphasize that we neither assume the UMD property of the involved spaces nor their separability.

The Universal Approximation Property

“…Nevertheless, certain spaces are outside the scope of Theorem 2.2. Those spaces include certain non-linear spaces, such as the ones studied in [50,51], as well as non-separable function spaces such as L 1 loc . We now summarize the contributions made in this article.…”

On Sobolev rough paths

“…Note that there is also a canonical way to introduce the inhomogeneous Sobolev distance analogously to the integral definition of the homogeneous Sobolev norm (2.2), which is expected to be equivalent to the discretely defined inhomogeneous Sobolev distance (2.4). However, already in the case of homogeneous Sobolev norms, it was a challenging task to show the equivalence of the Sobolev norm via integrals (2.2) and the discretely defined Sobolev norm (2.3), see [LPT20a].…”

“…As a first step, we demonstrate that these spaces lead to rough path integration with its known properties and to standard stability results as usually offered by rough path theory. Our approach is based on a novel discrete characterization of (non-linear) Sobolev spaces (see [LPT20a]) in combination with classical estimates from rough path theory and Sobolev-variation embedding theorems (see also [FV06]).…”

“…One would expect that the inhomogeneous mixed Hölder-variation distance ρ Ṽ α p is not needed for the continuity statement of Theorem 5.1 and that ρ Ṽ α p is dominated by the inhomogeneous Sobolev distance ρW α p as one can observe for the homogeneous Sobolev norms. However, at least if one wants to follow a similar approach as developed in the present work, this would require an extensive study of the inhomogeneous distances: First, it seems to require to generalize the Sobolev-variation embedding theorem for functions f : [0, T ] → E provided in [FV06] to functions f : ∆ → E. Second, similar generalizations seem to be needed for the characterization of non-linear Sobolev spaces [LPT20a] as well as for the embedding results in [FP18]. These generalizations are outside the scope of the present article.…”

Optimal Extension to Sobolev Rough Paths

“…Our first contribution is to prove that every path of Sobolev regularity α and integrability p can be lifted to a Sobolev rough path provided 1/p < α < 1, which represents a Lyons-Victoirs extension theorem for Sobolev paths with arbitrary low regularity. For this purpose, our approach combines ideas from [17] with a discrete characterization of fractional Sobolev spaces as recently provided by [13]. Furthermore, our Sobolev extension theorem provides the existence of a joint Sobolev rough path extension given, for example, two Sobolev rough paths taking values in lower dimensional free nilpotent groups.…”