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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

$ L\sp p$ continuity of pseudo-differential operators with discontinuous symbol


Authors: J. Alvarez and R. Durán
Journal: Proc. Amer. Math. Soc. 104 (1988), 165-168
MSC: Primary 42B20; Secondary 42B10, 47B05
MathSciNet review: 958060
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Abstract: A set $ \Gamma \subset {{\mathbf{R}}^n} \times {{\mathbf{R}}^n}$ is called conic if $ \left( {x,\xi } \right) \in \Gamma $ implies $ \left( {x,t\xi } \right) \in \Gamma $, for all $ t > 0$.

In localizing a partial differential equation, the following question arose: Let us consider a pseudo-differential operator $ L$ whose symbol is the characteristic function of a conic subset of $ {{\mathbf{R}}^n} \times {{\mathbf{R}}^n}$. Will $ L$ be a bounded operator in $ {L^2}\left( {{{\mathbf{R}}^n}} \right)$?

We present as a counterexample a pseudo-differential operator $ L$ whose symbol is the characteristic function of a conic subset of $ {{\mathbf{R}}^2} \times {{\mathbf{R}}^2}$, which is unbounded on $ {L^p}\left( {{{\mathbf{R}}^2}} \right)$ for $ 1 \leq p \leq 2$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0958060-X
PII: S 0002-9939(1988)0958060-X
Keywords: Conic set, pseudo-differential operator, Hilbert transform over a curve
Article copyright: © Copyright 1988 American Mathematical Society