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In this chapter, we introduce the shearlet localization operator (SLO) and explore the microlocal behavior of functions using the continuous shearlet transform (CST). We extend existing results on the characterization of \(C^{\infty }\) and Sobolev wavefront sets through CST and examine the relationship between shearlet characterization of wavefront sets and the SLO. Finally, we discuss some boundedness properties of the localization operator within the CST framework. This brief review draws on the works of Kutyniok and Labate (2009, Trans. Am. Math. Soc. 376, 7353–7375), Han, Paul, and Shukla (2022, Constr. Approx. 55.2, 661–704), along with our recent work (Paul, Shearlet localization operator on Sobolev space).

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Shearlet Localization Operator and Microlocal Analysis

  • 会议论文

  • 发表日期：

    2025年

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  • Swaraj Paul

摘要

In this chapter, we introduce the shearlet localization operator (SLO) and explore the microlocal behavior of functions using the continuous shearlet transform (CST). We extend existing results on the characterization of \(C^{\infty }\) and Sobolev wavefront sets through CST and examine the relationship between shearlet characterization of wavefront sets and the SLO. Finally, we discuss some boundedness properties of the localization operator within the CST framework. This brief review draws on the works of Kutyniok and Labate (2009, Trans. Am. Math. Soc. 376, 7353–7375), Han, Paul, and Shukla (2022, Constr. Approx. 55.2, 661–704), along with our recent work (Paul, Shearlet localization operator on Sobolev space).