Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations pp 195-211 | Cite as

# Products of Two-Wavelet Multipliers and Their Traces

## Abstract

Following Wong’s point of view in his book [12] (see Chapter 21) we give in this paper two formulas for the product of two two-wavelet multipliers \(
\psi T_\sigma \bar \phi \)
: *L* ^{2}(ℝ^{ n }) and \(
\psi T_\tau \bar \phi \)
: *L* ^{2}(ℝ^{ n })→*L* ^{2}(ℝ^{ n }), where σ and τ are functions in *L* ^{2}(ℝ^{ n }) and ϕ and ψ are any functions in *L* ^{2}(ℝ^{ n })∩*L* ^{∞}(ℝ^{ n }) such that \(
\parallel \phi \parallel _{L^2 (\mathbb{R}^n )} = \parallel \psi \parallel _{L^2 (\mathbb{R}^n )} = 1\). We also give a trace formula and an upper bound estimate on the trace class norm for such a product. Moreover we find sharp estimates on the norm in the trace class of two-wavelet multipliers *P* _{σ,ϕ,ψ}: *L* ^{2}(ℝ^{ n })→*L* ^{2}(ℝ^{ n }) in terms of the symbols σ and the admissible wavelets ϕ and ψ and also we give an inequality about products of positive trace class one-wavelet multipliers. Finally, we give an example of a two-wavelet multiplier which extends Wong’s result concerning the Landau-Pollak-Slepian operator from the one-wavelet case to the two-wavelet case (see Chapter 20 in the book [12] by Wong).

## Keywords

Admissible wavelet two-wavelet multiplier trace class operator## Mathematics Subject Classification (2000)

Primary 42B15 Secondary 47G10## Preview

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## References

- [1]V. Catană, Schatten-von Neumann norm inequalities for two-wavelet in
*Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis*. Editors: L. Rodino, B.-W. Schulze and M. W. Wong, Fields Institute Communications**52**, American Mathematical Society, 2007, 265–277.Google Scholar - [2]J. Du and M. W. Wong, A product formula for localization operators,
*Bull. Korean Math. Soc.***37**(1) (2000), 77–84.zbMATHMathSciNetGoogle Scholar - [3]J. Du and M. W. Wong and Z. Zhang, Trace class norm inequalities for localization operators,
*Integr. Equ. Oper. Theory***41**(2001), 497–503.zbMATHCrossRefMathSciNetGoogle Scholar - [4]I. Gohberg, S. Goldberg and N. Krupnik,
*Traces and Determinants of Linear Operators*, Birkhäuser, 2001.Google Scholar - [5]A. Grossmann, G. Loupias, E. M. Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space,—it Ann. Inst. Fourier (Grenoble)
**18**(1968), 343–368.zbMATHMathSciNetGoogle Scholar - [6]Z. He and M. W. Wong, Wavelet multipliers and signals,
*J. Austral. Math. Soc. Ser. B*,**40**(1999), 437–446.zbMATHCrossRefMathSciNetGoogle Scholar - [7]L. Liu, A trace class operator inequality,
*J. Math. Anal. Appl.***328**(2007), 1484–1486.zbMATHCrossRefMathSciNetGoogle Scholar - [8]J. C. T. Pool, Mathematical aspects of the Weyl correspondence,
*J. Math. Phys.***7**( (1966), 66–76.zbMATHCrossRefMathSciNetGoogle Scholar - [9]J. Toft, Regularizations, decompositions and lower bound problems in the Weyl calculus,
*Comm. Partial Differential Equations***25**(2000), 1201–1234.zbMATHCrossRefMathSciNetGoogle Scholar - [10]M. W. Wong,
*Weyl Transforms*, Springer-Verlag, 1998.Google Scholar - [11]M. W. Wong and Z. Zhang, Traces of two-wavelet multipliers,
*Integr. Equ. Oper. Theory***42**(2002), 498–503.zbMATHCrossRefMathSciNetGoogle Scholar - [12]M. W. Wong,
*Wavelet Transforms and Localization Operators*, Birkhäuser, 2002.Google Scholar - [13]M. W. Wong and Z. Zhang, Trace class norm inequalities for wavelet multipliers,
*Bull. London Math. Soc.***34**(2002), 739–744.zbMATHCrossRefMathSciNetGoogle Scholar - [14]M. W. Wong, Trace-class Weyl transforms, in
*Recent Advances in Operator Theory and its Applications*, Editors: M. A. Kaashoek, S. Seatzu and C. van der Mee, Operator Theory: Advances and Applications**160**, Birkhäuser, 469–478.Google Scholar