𝐿²-boundedness for pseudo-differential operators with unbounded symbols

Childs, Gary

doi:10.1090/S0002-9939-1978-0500300-9

$L^{2}$-boundedness for pseudo-differential operators with unbounded symbols
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Proc. Amer. Math. Soc. 72 (1978), 77-81 Request permission

Abstract:

Kato has proven ${L^2}$-boundedness if the symbol $a(x,z)$ is such that $|D_x^\beta D_z^\alpha a(x,z)|\; \leqslant ({\text {constant}}){(1 + |z|)^{(|\beta | - |\alpha |)\rho }}$ for $|\alpha | \leqslant [n/2] + 1,|\beta | \leqslant [n/2] + 2$ and $0 < \rho < 1$. In this paper, ${L^2}$-boundedness is shown for a corresponding Hölder continuity condition which requires slightly less smoothness for $a(x,z)$.

References

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