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In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than , extending the well-posedness result of Molinet [20].
We summarize the main ideas in a series of papers ([], [], [], []) devoted to the construction of invariant measures and to the long-time behavior of solutions of the periodic Benjamin-Ono equation.
In this paper we propose a new method, based on R-C similar transformation method, to study classification for the magic squares of order 5. The R-C similar transformation is defined by exchanging two rows and related two columns of a magic square. Many new results for classification of the magic squares of order 5 are obtained by the R-C similar transformation method. Relationships between basic forms and R-C similar magic squares are discussed. We also propose a so called GMV (generating magic...
Let and let be pseudo-differential operators with symbols , where , and . Let , be weights in Muckenhoupt classes , for some . We establish a two-weight inequality for commutators generated by pseudo-differential operators with weighted BMO functions , namely, the commutator is bounded from into . Furthermore, the range of can be extended to the whole .
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