${L}^{p}$ and Hölder estimates for pseudodifferential operators: sufficient conditions
Annales de l'Institut Fourier, Volume 29 (1979) no. 3, p. 239-260

Continuity in ${L}^{p}$ spaces and spaces of Hölder type is proved for pseudodifferential operators of order zero, under general conditions on the class of symbols. Applications to the regularity theory of some hypoelliptic operators are outlined.

La continuité des opérateurs pseudo-différentiels d’ordre zéro dans les espaces ${L}^{p}$ et dans des espaces de Hölder est démontrée, sous des conditions générales pour les symboles. On esquisse des applications à la théorie de régularité des opérateurs hypoelliptiques.

@article{AIF_1979__29_3_239_0,
author = {Beals, Richard},
title = {$L^p$ and H\"older estimates for pseudodifferential operators: sufficient conditions},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {29},
number = {3},
year = {1979},
pages = {239-260},
doi = {10.5802/aif.760},
zbl = {0387.35065},
mrnumber = {81c:47049},
language = {en},
url = {http://www.numdam.org/item/AIF_1979__29_3_239_0}
}

Beals, Richard. $L^p$ and Hölder estimates for pseudodifferential operators: sufficient conditions. Annales de l'Institut Fourier, Volume 29 (1979) no. 3, pp. 239-260. doi : 10.5802/aif.760. http://www.numdam.org/item/AIF_1979__29_3_239_0/

[1] R. Beals, Lp and Hölder estimates for pseudodifferential operators : necessary conditions, Amer. Math. Soc. Proc. Symp. Pure Math., to appear. | Zbl 0418.35085

[2] A.P. Calderon, Lebesgue space of differentiable functions and distributions, Amer. Math. Soc. Proc. Symp. Pure Math., 5 (1961), 33-49. | MR 26 #603 | Zbl 0195.41103

[3] C.-H. Ching, Pseudo-differential operators with non-regular symbols, J. Differential Equations, 11 (1972), 436-447. | MR 45 #5823 | Zbl 0248.35106

[4] R.R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, no. 242, Springer-Verlag, Berlin, 1971. | MR 58 #17690 | Zbl 0224.43006

[5] L. Hormander, The Weyl calculus of pseudodifferential operators, to appear.

[6] Y. Kannai, An unsolvable hypoelliptic differential operator, Israel J. Math., 9 (1971), 306-315. | MR 43 #2573 | Zbl 0211.40601

[7] H. Kumano-Go and K. Taniguchi, Oscillatory integrals of symbols of operators on Rn and operators of Fredholm type, Proc. Japan Acad., 49 (1973), 397-402. | MR 50 #8167 | Zbl 0272.47032

[8] K. Miller, Parametrices for a class of hypoelliptic operators, J. Differential Equations, to appear. | Zbl 0409.35022

[9] A. Nagel and E.M. Stein, A new class of pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A., 75 (1978), 582-585. | MR 58 #7222 | Zbl 0376.35053

[10] A. Unterberger, Symboles associés aux champs de repères de la forme symplectique, C.R. Acad. Sci., Paris, sér. A, 245 (1977), 1005-1008. | MR 58 #24411 | Zbl 0381.47025