The study of pseudodifferential operators with symbols in the exotic classes S~m_1.1, has received a lot of attention. These are operators of the form (a fomulae (elli.)). The interest in such operators is due in part to the role they play in the paradifferential calculus of Bony [l]. The fact that not all such operators of order zero are bounded on L~2 complicates their study. Nevertheless, the exotic pseudodifferential operators do preserve spaces of smooth functions. See, for example, Meyer [12J, Paivarinta [14], Bourdaud [2], as well as Stein [16] and the references therein. The continuity results are often obtained by making use of the so-called sin- gular integral realization of the operators. This involves proving estimates on the Schwariz kernels of the pseudodifferential operators similar to those of the ker- nels of Calderon-Zygmund operators. There is. however, an altemative approach working directly with the symbols of the pseudodifforential operators. This ap- proach has been pursued by Hormander in [9] and [10] for L~2-based Sobolev spaces. The ideas in those papers combined with wavelets techniques were later extended by Torres [l 7] to Ln -based Soboley spaces and other more general spaces of smooth functions.
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