$L^ p$ continuity of pseudo-differential operators with discontinuous symbol
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- by J. Alvarez and R. Durán PDF
- Proc. Amer. Math. Soc. 104 (1988), 165-168 Request permission
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 $1 \leq p \leq 2$.References
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- Ole G. Jørsboe and Leif Mejlbro, The Carleson-Hunt theorem on Fourier series, Lecture Notes in Mathematics, vol. 911, Springer-Verlag, Berlin-New York, 1982. MR 653477, DOI 10.1007/BFb0094072
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 165-168
- MSC: Primary 42B20; Secondary 42B10, 47B05
- DOI: https://doi.org/10.1090/S0002-9939-1988-0958060-X
- MathSciNet review: 958060