Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes |
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Authors: | Nicholas Michalowski |
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Institution: | a School of Mathematics and the Maxwell Institute of Mathematical Sciences, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom b Department of Mathematics and the Maxwell Institute of Mathematical Sciences, Heriot-Watt University, Colin Maclaurin Building, Edinburgh, EH14 4AS, United Kingdom |
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Abstract: | We prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables. The result is sharp, even for smooth amplitudes. Nevertheless, in the case when the amplitude contains the oscillatory factor ξ?ei|ξ|1−ρ, the result can be substantially improved. We extend the Lp-boundedness of pseudo-pseudodifferential operators to certain weights. End-point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable. Our weighted norm inequalities also yield the boundedness of commutators of these pseudodifferential operators with functions of bounded mean oscillation. |
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Keywords: | Weighted norm inequality Pseudodifferential operator Pseudo-pseudodifferential operator BMO commutator |
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