Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Unique continuation for $ \Delta+v$ and the C. Fefferman-Phong class


Authors: Sagun Chanillo and Eric Sawyer
Journal: Trans. Amer. Math. Soc. 318 (1990), 275-300
MSC: Primary 35J10; Secondary 35B99
DOI: https://doi.org/10.1090/S0002-9947-1990-0958886-6
MathSciNet review: 958886
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the strong unique continuation property holds for the inequality $ \left\vert {\Delta u} \right\vert \leq \left\vert \upsilon \right\vert\left\vert u \right\vert$, 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 [Enhancements On Off] (What's this?)

  • [BKRS] B. Barcelo, C. E. Kenig, A. Ruiz and C. D. Sogge, Weighted Sobolev inequalities and unique continuation for the Laplacian plus lower order terms, Preprint. MR 945861 (89h:35048)
  • [CW] S. Chanillo and R. Wheeden, $ {L^p}$ estimates for fractional integrals and Sobolev inequalities with applications to Schrodinger operators, Comm. Partial Differential Equations 10 (1985), 1077-1116. MR 806256 (87d:42028)
  • [ChR] S. Chiarenza and A. Ruiz,
  • [CR] R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), 249-254. MR 565349 (81b:42067)
  • [F] C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 129-206. MR 707957 (85f:35001)
  • [GS] I. Gelfand and G. Shilov, Generalized functions, Academic Press, 1964. MR 0166596 (29:3869)
  • [JK] D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrodinger operators, Ann. of Math. (2) (1985), 463-494. MR 794370 (87a:35058)
  • [K] C. E. Kenig, Lecture notesat conference in harmonic analysis at El Escorial, 1987.
  • [KRS] C. E. Kenig, A. Ruiz and C. Sogge, Sobolev inequalities and unique continuation for second order constant coefficient differential equations, Duke Math. J. 55 (1987), 329-348. MR 894584 (88d:35037)
  • [KS] R. Kerman and E. Sawyer, The trace inequality and eigenvalue estimates for Schràdinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), 207-228. MR 867921 (88b:35150)
  • [K] D. Kurtz, Littlewood-Paley and multiplier theorems on weighted $ {L^p}$ spaces, Trans. Amer. Math. Soc. 259 (1980), 235-254. MR 561835 (80f:42013)
  • [R] G. Roberts, Uniqueness in the Cauchy problem for characteristic operators of Fuchsian type, J. Differential Equations 38 (1980), 374-392. MR 605056 (82i:35006)
  • [Sa] E. Sawyer, Unique continuation for Schràdinger operators in dimension three or less, Ann. Inst. Fourier (Grenoble) 34 (1984), 189-200. MR 762698 (86i:35034)
  • [St$ _{1}$] E. Stein, Appendix to unique continuation, Ann. of Math. (2) 121 (1985), 489-494. MR 794370 (87a:35058)
  • [St$ _{2}$] -, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482-492. MR 0082586 (18:575d)
  • [T] P. Tomas, Restriction theorems for the Fourier transform, Proc. Sympos. Pure Math., vol. 35, part 1, Amer. Math. Soc., Providence, R.I., pp. 111-114. MR 545245 (81d:42029)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35J10, 35B99

Retrieve articles in all journals with MSC: 35J10, 35B99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0958886-6
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society