The inclusion relation between Sobolev and modulation spaces

Abstract

The inclusion relations between the L p -Sobolev spaces and the modulation spaces is determined explicitly. As an application, mapping properties of unimodular Fourier multiplier e i|D| α between L p -Sobolev spaces and modulation spaces are discussed.

Inclusion relations between Lp-Sobolev and Wiener amalgam spaces

“…due to Theorem 1.3 of [13]. Next we show that L p → W p,2 , p > 2 by proving the dual statement W p,2 → L p where 1 ≤ p ≤ 2.…”

“…In particular, the works of J. Toft [19], Sugimoto and Tomita [18] and Wang and Huang [22] gave a full picture on the inclusion relations between Besov spaces B p,q s and modulation spaces M p,q s . This naturally led to establishing the inclusion relations between L p -Sobolev spaces and modulation spaces M p,q s [13]. Kobayashi, Miyachi and Tomita determined the inclusion relation between modulation spaces M p,q s and local Hardy spaces h p for 0 < p ≤ 1, this work is done prior to [13].…”

“…This naturally led to establishing the inclusion relations between L p -Sobolev spaces and modulation spaces M p,q s [13]. Kobayashi, Miyachi and Tomita determined the inclusion relation between modulation spaces M p,q s and local Hardy spaces h p for 0 < p ≤ 1, this work is done prior to [13]. The purpose of this article is to establish the embedding relation between L p -Sobolev spaces and Wiener amalgam spaces W p,q , which also implies the inclusion relations between Besov spaces B p,q s and Wiener amalgam spaces W p,q .…”

“…The purpose of this article is to establish the embedding relation between L p -Sobolev spaces and Wiener amalgam spaces W p,q , which also implies the inclusion relations between Besov spaces B p,q s and Wiener amalgam spaces W p,q . Although, certain cases of these embedding relations could already be deduced from [13] by using the known embeddings…”

Excursions in Harmonic Analysis, Volume 4

“…There are several known embeddings between the Besov, Sobolev, and modulation spaces; see, for example, Okoudjou [40], Toft [51], Sugimoto-Tomita [48], and Kobayashi-Sugimoto [35].…”