A unified approach for Littlewood-Paley decomposition of abstract Besov spaces

Abstract

We apply spectral theoretic methods to obtain a Littlewood-Paley decomposition of abstract inhomogeneous Besov spaces in terms of “smooth” and “bandlimited” functions. Well-known decompositions in several contexts are shown as special examples and are unified under the spectral theoretic approach.

Identification of Besov spaces via Littlewood Paley-Stein $g$ functions

“…Therefore (11) represents a decomposition of f in terms of band-limited and smooth functions. In [22] we applied (11) to prove that for α > 1/2 and 1…”

“…The organization of this paper is as follows: After some historical comments in Section 2, in Section 3 we introduce some notations and definitions, including a short description of our previous results in [22]. In Section 4 we prove few key lemmas and the main result in Theorem 1.1.…”

“…In our previous paper [22], we proved the norm in (1) can be expressed in terms of "smooth" and "band-limited" functions using the Littlewood-Paley decomposition. In this paper we show this norm is also equivalent to a norm which can be expressed in terms of Littlewood-Paley-Stein type g-functions as we will explain.…”

“…For example, see [18,3,28] for a characterization of B α q (L p ) in terms of the Weierstrass or heat semigroup T t = e −Lt (convolution with the heat kernel) associated to a self-adjoint positive definite operator L on L 2 . The organization of this paper is as follows: After some historical comments in Section 2, in Section 3 we introduce some notations and definitions, including a short description of our previous results in [22]. In Section 4 we prove few key lemmas and the main result in Theorem 1.1.…”