We study the boundedness of rough Fourier integral and pseudodifferential operators, defined by general rough Hörmander class amplitudes, on Banach and quasi-Banach L p spaces. Thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for Hörmander class bilinear pseudodifferential operators and certain classes of bilinear (as well as multilinear) Fourier integral operators. For these classes of amplitudes, the boundedness of the aforementioned Fourier integral operators turn out to be sharp. Furthermore we also obtain results for rough multilinear operators.
Abstract. We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in S m 1,0 (n, 2) and non-degenerate phase functions, from. This is a bilinear version of the classical theorem of Seeger-Sogge-Stein concerning the L p boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of non-Banach target spaces.
Extending work of Pichorides and Zygmund to the d-dimensional setting, we show that the supremum of L p -norms of the Littlewood-Paley square function over the unit ball of the analytic Hardy spaces H p A pT d q blows up like pp´1q´d as p Ñ 1`. Furthermore, we obtain an L log d L-estimate for square functions on H 1 A pT d q. Euclidean variants of Pichorides's theorem are also obtained. S T d pf qpxq "`ÿ k1,¨¨¨,k d PZ |∆ k1,¨¨¨,k d pf qpxq| 2˘1 {2 , 2010 Mathematics Subject Classification. 42B15, 42B25, 42B30.
ABSTRACT. In this paper we establish the boundedness of bilinear paraproducts on local BMO spaces. As applications, we also investigate the boundedness of bilinear Fourier integral operators and bilinear Coifman-Meyer multipliers on these spaces and also obtain a certain end-point result concerning Kato-Ponce type estimates.
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