Uniform 𝐿²-weighted Sobolev inequalities

Chiarenza, Filippo; Ruiz, Alberto

doi:10.1090/S0002-9939-1991-1055768-5

Uniform $L^ 2$-weighted Sobolev inequalities
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by Filippo Chiarenza and Alberto Ruiz PDF

Proc. Amer. Math. Soc. 112 (1991), 53-64 Request permission

Abstract:

We prove the inequality (1) for weights $w$ in a class which contains the class ${J_p},p > (n - 1)/2$, introduced by C. Fefferman and D. H. Phong in studying eigenvalues of Schrödinger operators. In our case, $C$ is independent of the lower order terms of $P$. As a consequence we prove unique continuation theorem for solutions of $\Delta + V,V$ in the same class.

References

Unique continuation for

and the C. Fefferman-Phong class

Sagun Chanillo and Richard L. Wheeden, $L^p$-estimates for fractional integrals and Sobolev inequalities with applications to Schrödinger operators, Comm. Partial Differential Equations 10 (1985), no. 9, 1077–1116. MR 806256, DOI 10.1080/03605308508820401
Charles Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44–52. MR 320624, DOI 10.1007/BF02771772
C. Fefferman and D. H. Phong, Lower bounds for Schrödinger equations, Conference on Partial Differential Equations (Saint Jean de Monts, 1982), Soc. Math. France, Paris, 1982, pp. Conf. No. 7, 7. MR 672274
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
Carlos E. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation, Harmonic analysis and partial differential equations (El Escorial, 1987) Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 69–90. MR 1013816, DOI 10.1007/BFb0086794
C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), no. 2, 329–347. MR 894584, DOI 10.1215/S0012-7094-87-05518-9
Douglas S. Kurtz and Richard L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343–362. MR 542885, DOI 10.1090/S0002-9947-1979-0542885-8

Factorization theory and

weights

106

Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 358216, DOI 10.1090/S0002-9904-1975-13790-6