## Rocky Mountain Journal of Mathematics

### Spectral theorems associated with the Riemann-Liouville-Wigner localization operators

#### Abstract

We introduce the notion of localization operators associated with the Riemann-Liouville-Wigner transform, and we give a trace formula for the localization operators associated with the Riemann-Liouville-Wigner transform as a bounded linear operator in the trace class from $L^{2}(d\nu _{\alpha })$ into $L^{2}(d\nu _{\alpha })$ in terms of the symbol and the two admissible wavelets. Next, we give results on the boundedness and compactness of localization operators associated with the Riemann-Liouville-Wigner transform on $L^{p}(d\nu _{\alpha })$, $1 \leq p \leq \infty$.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 1 (2019), 247-281.

Dates
First available in Project Euclid: 10 March 2019

https://projecteuclid.org/euclid.rmjm/1552186961

Digital Object Identifier
doi:10.1216/RMJ-2019-49-1-247

Mathematical Reviews number (MathSciNet)
MR3921876

Zentralblatt MATH identifier
07036627

#### Citation

Mejjaoli, Hatem; Trimeche, Khalifa. Spectral theorems associated with the Riemann-Liouville-Wigner localization operators. Rocky Mountain J. Math. 49 (2019), no. 1, 247--281. doi:10.1216/RMJ-2019-49-1-247. https://projecteuclid.org/euclid.rmjm/1552186961

#### References

• C. Baccar and N.B. Hamadi, Localization operators of the wavelet transform associated to the Riemann-Liouville operator, Int. J. Math. 27 (2016).
• C. Baccar, N.B. Hamadi and L.T. Rachdi, Inversion formulas for the Riemann-Liouville transform and its dual associated with singular partial differential operators, Int. J. Math. Math. Sci. 2006 (2006), 1–26.
• C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, New York, 1988.
• J.P. Calderon, Intermediate spaces and interpolation, The complex method, Stud. Math. 24 (1964), 113–190.
• E. Cordero and K. Gr öchenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205 (2003), 107–131.
• I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Info. Th. 34 (1988), 605–612.
• ––––, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Info. Th. 36 (1990), 961–1005.
• F. De Mari, H. Feichtinger and K. Nowak, Uniform eigenvalue estimates for time-frequency localization operators, J. London Math. Soc. 65 (2002), 720–732.
• F. De Mari and K. Nowak, Localization type Berezin-Toeplitz operators on bounded symmetric domains, J. Geom. Anal. 12 (2002), 9–27.
• J.A. Fawcett, Inversion of N-dimensional spherical means, SIAM. J. Appl. Math. 45 (1983), 336–341.
• G.B. Folland, Introduction to partial differential equations, Princeton University Press, Princeton 1995.
• K. Gr öchenig, Foundations of time-frequency analysis, Springer Sci. Bus. Media, New York, 2001.
• N.B. Hamadi, Localization operators for the windowed Fourier transform associated with singular partial differential operators, Rocky Mountain J. Math. 47 (2017), 2179–2195.
• N.B. Hamadi and L.T. Rachdi, Weyl transforms associated with the Riemann-Liouville operator, Int. J. Math. Math. Sci. 2006 (2006).
• H. Helesten and L.E. Andersson, An inverse method for the processing of synthetic aperture radar data, Inv. Prob. 3 (1987), 111–124.
• F. John, Plane waves and spherical means applied to partial differential equations, Interscience, New York, 1955.
• E.H. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys. 31 (1990), 594–599.
• L. Liu, A trace class operator inequality, J. Math. Anal. Appl. 328 (2007), 1484–1486.
• S. Omri and L. Rachdi, Heisenberg-Pauli-Weyl uncertainty principle for the Riemann-Liouville operator, J. Inequal. Pure Appl. Math. 9 (2008), 23.
• J. Ramanathan and P. Topiwala, Time-frequency localization via the Weyl correspondence, SIAM J. Math. Anal. 24 (1993), 1378–1393.
• E.M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492.
• K. Trimèche, Permutation operators and the central limit theorem associated with partial differential operators, in Probability measures on groups 10 (1991), 395–424.
• M.W. Wong, Wavelet transforms and localization operators, Springer Sci. Bus. Med. 136 (2002).