Weighted Sobolev spaces and pseudodifferential operators with smooth symbols
HTML articles powered by AMS MathViewer
- by Nicholas Miller PDF
- Trans. Amer. Math. Soc. 269 (1982), 91-109 Request permission
Abstract:
Let ${u^\# }$ be the Fefferman-Stein sharp function of $u$, and for $1 < r < \infty$, let ${M_r}u$ be an appropriate version of the Hardy-Littlewood maximal function of $u$. If $A$ is a (not necessarily homogeneous) pseudodifferential operator of order $0$, then there is a constant $c > 0$ such that the pointwise estimate ${(Au)^\# }(x) \leqslant c{M_r}u(x)$ holds for all $x \in {R^n}$ and all Schwartz functions $u$. This estimate implies the boundedness of $0$-order pseudodifferential operators on weighted ${L^p}$ spaces whenever the weight function belongs to Muckenhoupt’s class ${A_p}$. Having established this, we construct weighted Sobolev spaces of fractional order in ${R^n}$ and on a compact manifold, prove a version of Sobolev’s theorem, and exhibit coercive weighted estimates for elliptic pseudodifferential operators.References
- A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 33–49. MR 0143037
- R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
- Reinhard Illner, A class of $L^{p}$-bounded pseudo-differential operators, Proc. Amer. Math. Soc. 51 (1975), 347–355. MR 383153, DOI 10.1090/S0002-9939-1975-0383153-9
- Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
- L. Nirenberg, Pseudo-differential operators, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 149–167. MR 0270217
- Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 269 (1982), 91-109
- MSC: Primary 47G05; Secondary 35S05, 46E35
- DOI: https://doi.org/10.1090/S0002-9947-1982-0637030-4
- MathSciNet review: 637030