L p -inequalities for the laplacian and unique continuation
Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 153-168.

Nous démontrons une inégalité de la forme

| x | r f L p ( R n ) c ( n , p , q , r ) | x | τ + μ Δ f L q ( R n ) .

Comme applications nous obtenons la propriété de prolongement unique pour l’inégalité différentielle |Δf(x)||v(x)||f(x)| si vL Loc p avec p>maxn 2 ,n-2).

We prove an inequality of the type

| x | r f L p ( R n ) c ( n , p , q , r ) | x | τ + μ Δ f L q ( R n ) .

This is then used to derive the unique continuation property for the differential inequality |Δf(x)||v(x)||f(x)| under suitable local integrability assumptions on the function v.

@article{AIF_1981__31_3_153_0,
     author = {Amrein, W. O. and Berthier, A. M. and Georgescu, V.},
     title = {$L^p$-inequalities for the laplacian and unique continuation},
     journal = {Annales de l'Institut Fourier},
     pages = {153--168},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {31},
     number = {3},
     year = {1981},
     doi = {10.5802/aif.843},
     mrnumber = {83g:35011},
     zbl = {0468.35017},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.843/}
}
TY  - JOUR
AU  - Amrein, W. O.
AU  - Berthier, A. M.
AU  - Georgescu, V.
TI  - $L^p$-inequalities for the laplacian and unique continuation
JO  - Annales de l'Institut Fourier
PY  - 1981
SP  - 153
EP  - 168
VL  - 31
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.843/
DO  - 10.5802/aif.843
LA  - en
ID  - AIF_1981__31_3_153_0
ER  - 
%0 Journal Article
%A Amrein, W. O.
%A Berthier, A. M.
%A Georgescu, V.
%T $L^p$-inequalities for the laplacian and unique continuation
%J Annales de l'Institut Fourier
%D 1981
%P 153-168
%V 31
%N 3
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.843/
%R 10.5802/aif.843
%G en
%F AIF_1981__31_3_153_0
Amrein, W. O.; Berthier, A. M.; Georgescu, V. $L^p$-inequalities for the laplacian and unique continuation. Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 153-168. doi : 10.5802/aif.843. http://archive.numdam.org/articles/10.5802/aif.843/

[1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. | MR | Zbl

[2] A.M. Berthier, Sur le spectre ponctuel de l'opérateur de Schrödinger, C.R. Acad. Sci., Paris 290 A, (1980), 393-395 ; On the Point Spectrum of Schrödinger Operators, Ann. Sci. Ecole Normale Supérieure (to appear). | Numdam | MR | Zbl

[3] N. Dunford and J.T. Schwartz, Linear Operators, Part I, Interscience, New York, 1957.

[4] V. Georgescu, On the Unique Continuation Property for Schrödinger Hamiltonians, Helv. Phys. Acta, 52 (1979), 655-670.

[5] G.H. Hardy, J.E. Littelewood and G. Polya, Inequalities, Cambridge University Press, 1952. | Zbl

[6] E. Heinz, Über die Eindeutigkeit beim Cauchy'schen Anfangswert-problem einer elliptischen Differentialgleichung zweiter Ordnung, Nachr. Akad.-Wiss. Göttingen, II (1955), 1-12. | MR | Zbl

[7] L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963. | Zbl

[8] M. Schechter and B. Simon, Unique Continuation for Schrödinger Operators with Unbounded Potentials, J. Math. Anal. Appl., 77 (1980), 482-492. | MR | Zbl

[9] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. | MR | Zbl

[10] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. | Zbl

Cité par Sources :