Multiparameter singular integrals on the Heisenberg group: uniform estimates

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Vitturi, Marco
Wright, James
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American Mathematical Society
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We consider a class of multiparameter singular Radon integral operators on the Heisenberg group H-1 where the underlying submanifold is the graph of a polynomial. A remarkable difference with the euclidean case, where Heisenberg convolution is replaced by euclidean convolution, is that the operators on the Heisenberg group are always L-2 bounded. This is not the case in the euclidean setting where L-2 boundedness depends on the polynomial defining the underlying surface. Here we uncover some new, interesting phenomena. For example, although the Heisenberg group operators are always L-2 bounded, the bounds are not uniform in the coefficients of polynomials with fixed degree. When we ask for which polynomials uniform L-2 bounds hold, we arrive at the same class where uniform bounds hold in the euclidean case.
Double Hilbert transforms , Radon transforms , Polynomial surfaces , Harmonic analysis , Nilpotent groups , Kernels
Vitturi, M. and Wright, J. (2020) 'Multiparameter singular integrals on the Heisenberg group: uniform estimates', Transactions of the American Mathematical Society, 373, pp. 5439-5465. doi: 10.1090/tran/8079