Banach frames for multivariate α-modulation spaces

Abstract

The α-modulation spaces M s,α p,q (R d ), α ∈ [0, 1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for α-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate α-modulation spaces can be completely characterized by the Banach frames constructed.

Trace ideals for pseudo-differential operators and their commutators with symbols in α-modulation spaces

“…The precise definition of them will be given later in Section 2. The following is our main theorem: for all σ ∈ M (1,1), (1,1) (αn/2,αn/2),(α,α) (R n × R n ).…”

“…More precisely, the symbol σ ∈ M (∞,∞), (1,1) (αn/2,αn/2),(α,α) , which means σ(x, ξ) belongs to M ∞,1 αn/2,α in both x and ξ, generates the L 2 (R n )-bounded pseudo-differential operator. Especially in the case α = 0 (resp.…”

“…The L 2 -boundedness of the commutator was discussed by Calderón [3], Coifman-Meyer [4] and Marschall [19], where σ belongs to Hörmander's class S ρ ρ,δ (δ ≤ ρ, 0 ≤ δ < 1). In [17], we have generalized the result with ρ = δ = 0 to the case when σ ∈ M (∞,∞), (1,1) (αn/2,αn+1),(α,α) . We can again expect the trace property of the commutator if we assume σ ∈ M (1,1), (1,1) (αn/2,αn+1),(α,α) instead, replacing ∞ by 1.…”

“…(1,1), (1,1) (αn/2,αn/2),(α,α) . We remark that the notion of α-modulation spaces was introduced by Gröbner [11], and developed by the works of Feichtinger-Gröbner [7], Borup-Nielsen [1,2] and Fornasier [9].…”

“…On the other hand, Theorem 1.1 with α = 1 states the trace property of the operators with symbols in the Besov space B (1,1), (1,1) (n/2,n/2) , but there seem to be few literature mentioning this fact. We remark that the spaces M 1,1 and B…”