Unique continuation for Δ+𝑣 and the C. Fefferman-Phong class

Chanillo, Sagun; Sawyer, Eric

doi:10.1090/S0002-9947-1990-0958886-6

Unique continuation for $\Delta +v$ and the C. Fefferman-Phong class
HTML articles powered by AMS MathViewer

by Sagun Chanillo and Eric Sawyer PDF

Trans. Amer. Math. Soc. 318 (1990), 275-300 Request permission

Abstract:

We show that the strong unique continuation property holds for the inequality $\left | {\Delta u} \right | \leq \left | \upsilon \right |\left | u \right |$, where the potential $\upsilon (x)$ satisfies the C. Fefferman-Phong condition in a certain range of $p$ values. We also deal with the situation of $u(x)$ vanishing at infinity. These are all consequences of appropriate Carleman inequalities.

References

B. Barceló, C. E. Kenig, A. Ruiz, and C. D. Sogge, Weighted Sobolev inequalities and unique continuation for the Laplacian plus lower order terms, Illinois J. Math. 32 (1988), no. 2, 230–245. MR 945861, DOI 10.1215/ijm/1255989128
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
R. R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), no. 2, 249–254. MR 565349, DOI 10.1090/S0002-9939-1980-0565349-8
Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
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
David Jerison and Carlos E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121 (1985), no. 3, 463–494. With an appendix by E. M. Stein. MR 794370, DOI 10.2307/1971205
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
R. Kerman and E. Sawyer, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 207–228 (English, with French summary). MR 867921, DOI 10.5802/aif.1074
Douglas S. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc. 259 (1980), no. 1, 235–254. MR 561835, DOI 10.1090/S0002-9947-1980-0561835-X
Gary B. Roberts, Uniqueness in the Cauchy problem for characteristic operators of Fuchsian type, J. Differential Equations 38 (1980), no. 3, 374–392. MR 605056, DOI 10.1016/0022-0396(80)90014-5
Eric T. Sawyer, Unique continuation for Schrödinger operators in dimension three or less, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 3, 189–200 (English, with French summary). MR 762698, DOI 10.5802/aif.982
David Jerison and Carlos E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121 (1985), no. 3, 463–494. With an appendix by E. M. Stein. MR 794370, DOI 10.2307/1971205
Elias M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492. MR 82586, DOI 10.1090/S0002-9947-1956-0082586-0
Peter A. Tomas, Restriction theorems for the Fourier transform, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–114. MR 545245