Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces

Abstract

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L 2 of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [

Time-Frequency Analysis of Sjöstrand’s Class

“…s , the Bessel potential space; likewise, the Shubin class Q s can be identified as a modulation space [7,37]; and even S can be represented as an intersection of modulation spaces.…”

Localization Operators Via Time-Frequency Analysis

“…A localization operator can always be written as a Weyl operator. Namely, if W (ϕ 2 , ϕ 1 ) is the cross-Wigner distribution of ϕ 2 , ϕ 1 , and L σ is the Weyl transform with Weyl symbol σ, then we have (see [4,18])…”

“…, 4, be non-zero windows in S(R d ) (even rougher windows are allowed, with smoothness depending on the integer N ). If we consider any rough symbol a ∈ s≥0 M ∞ 1/vs (R 2d ) = S (R d ) and we choose a symbol b smooth enough to make the pointwise multiplication ab ∈ M ∞ 1/τ s (R 2d ) (e. Namely, the product is expressed in terms of a sum of localization operators having the first window Φ α made of the factor windows ϕ 1 , ϕ 3 , ϕ 4 and depending on the derivative order (for the detailed expression we refer to [5,8] Also operator norm estimates, involving the time-frequency distribution of symbols and windows can be easily provided. Here we end up recalling an important application of formula (23).…”

“…We shall start considering symbols in L p -spaces, with 1 ≤ p ≤ ∞, we go to potential Sobolev spaces and we end up with the the so-called modulation spaces. For symbols in L p -spaces, we recall results contained in [34] and we show how to get them by means of shorter proofs [4]. However, sometimes in the applications one needs rougher symbols, even tempered distributions rather than functions.…”

Gabor multipliers with varying lattices

“…For the present paper we choose the more standard context of operators on L 2 (R d ), but go for the case that the resulting Gabor multipliers are Hilbert Schmidt operators. Note, that in the terminology of these Anti-Wick calculus one has the following result (see [1,Theorem 3.1]). …”