Weyl product algebras and modulation spaces

Abstract

We discuss algebraic properties of the Weyl product acting on modulation spaces. For a certain class of weight functions ω we prove that M p,q (ω) is an algebra under the Weyl product if p ∈ [1, ∞] and 1 q min(p, p ). For the remaining cases p ∈ [1, ∞] and min(p, p ) < q ∞ we show that the unweighted spaces M p,q are not algebras under the Weyl product.

Multiplication properties in pseudo-differential calculus with small regularity on the symbols

“…For completeness we write down the following lemma, where the first part agrees with Lemma 4.4 in [37] and Lemma 2.1 in [23], and was fundamental in the proofs of [37, Theorem 4.1] and for the Weyl product results in [23]. The second part follows from the first one, (1.9), (1.10) and (1.20).…”

“…The most general result corresponds to Theorem 0.3 in [23], which concerns continuity for the Weyl product on modulation spaces of the form M p,q (ω) . Thereafter we use this result to establish continuity properties for the twisted convolution when acting on weighted Lebesgue spaces.…”

“…[19,23,25,32], and in Sect. 2 we use Theorem 0.3 in [23] to prove continuity results for modulation and Lebesgue spaces under the twisted convolution. In particular we obtain a weighted version of the fact that L 2 is an algebra under the twisted convolution.…”

“…Continuity properties for modulation spaces under the Weyl product have been investigated in, e.g. [19,23,25,32], and in Sect. 2 we use Theorem 0.3 in [23] to prove continuity results for modulation and Lebesgue spaces under the twisted convolution.…”

“…Important contributions in this context can be found in [19,23,25,32,37], where Theorem 0.3 in [23] seems to be the most general result so far (see also Theorem 2.2).…”