Research Focus
My research is concerned with Harmonic Analysis and the theory of singular integral operators. My main area of work is time-frequency analysis, initiated by Carleson with his celebrated result on the pointwise convergence of Fourier series for $ L^2 $ .
functions. Since then the field has been significantly developed and many deep and surprising connections have been found with Ergodic Theory, Additive Combinatorics, and, crucially, with the study of dispersive PDEs and SDEs. On the other hand, many fundamental questions in the area remain open and some are beyond the reach of currently developed techniques. Time-frequency behavior often arises when considering maximal or multilinear analogues of Calderón-Zygmund SIOs.
My PhD thesis (2016) was concerned with developing and applying outer measure Lebesgue space theory: a powerful and general functional-analytic framework allowing one to systematically deal with with a large class of time-frequency operators.
Publications
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Amenta, A. and Uraltsev, G. (2019) The bilinear Hilbert transform in UMD spaces, arXiv:1909.06416.
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Amenta, A. and Uraltsev, G. (2019) Banach-valued modulation invariant Carleson embeddings and outer- $ L^ p $ spaces: the Walsh case, arXiv:1905.08681.
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Di Plinio, F., Do, Y.Q. and Uraltsev, G. (2018) Positive Sparse Domination of Variational Carleson Operators, Annali della Scuola Normale Superiore di Pisa. Classe di scienze, 18(4), pp.1443-1458.
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Uraltsev, G. (2016) Variational Carleson embeddings into the upper 3-space, arXiv:1610.07657.
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C. Mantegazza , G. Mascellani, and G. Uraltsev (2014) On the distributional Hessian of the distance function, Pacific Journal of Mathematics 270.1: pp.151-166.
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PhD Thesis, Time-Frequency Analysis of the Variational Carleson Operator using outer-measure $L^p$ spaces.