Doctor of Philosophy (PhD)
Included with every oscillatory singular integral operator is a phase function and a kernel, both essential to the operator’s behavior over any function space. Studies on such operators are rooted in the deep theories developed by Fourier, Calder´on, Zygmund, and Stein. In this work we show results on a new oscillatory singular integral operator with Calder´on-Zygmund kernel and a generalized phase function. We use techniques such as Calder´on-Zygmund theory, dyadic decomposition, the method of rotations, and others to show boundedness of the new operator over the real ndimensional Hardy Space. Our work expands upon results stemming from Sj¨olin, Fan, Pan, Al-Qassem, and Cheng by demonstrating that the bounds continue to exhibit logarithmic behavior with regards to the coefficients associated to the phase function. These new conclusions allow room for further generalization of the phase function and for continued developments in bounding oscillatory singular integral operators beyond classical techniques.
Smiley, Danielle. "Estimates on Oscillatory Singular Integral Operators." PhD Diss., Bryn Mawr College, 2019.