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In this paper, we first prove some local estimates for bilinear operators(closely related to the bilinear Hilbert transform and similar singularoperators) with truncated symbol. Such estimates, in accordance with theHeisenberg uncertainty principle correspond to a description of``off-diagonal'' decay. In addition they allow us to prove global continuitiesin Lebesgue spaces for bilinear operators with spatial dependent symbol.

Local estimates and global continuities in Lebesgue spaces for bilinear operators