engleski

Boundedness of pseudodifferential operators on Lebesgue spaces with mixed norm

The continuity of pseudo-differential operators, with symbols in Hormander's class, on the Lebesgue spaces was studied by numerous authors decades ago, and complete results can be found in a number of standard monographs today. These results can be extended to pseudo-differential operators of non-zero order on Sobolev spaces, allowing for numerous applications in the theory of partial differential equations. However, in some problems arising in partial differential equations, like fine studies of elliptic regularity, or for evolution equations, results involving Sobolev spaces with mixed-norm would be of significant interest. These spaces are based on the Lebesgue spaces with mixed-norm, that were introduced by Benedek and Panzone (1961), while some further properties (including a version of the Marcinkiewicz interpolation theorem)were obtained by Russian school (S. M. Nikolskii and collaborators) a few years later. We shall prove the boundedness result for pseudo-differential operators with symbols in S^0_{; ; ; 1, \delta}; ; ; on the Lebesgue spaces with mixed norm, extending the classical proof based on Calderon-Zygmund decomposition. Furthermore, some applications to the partial differential equations will be discussed as well.

pseudodifferential ; mixed norm spaces

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