L p -inequalities for the laplacian and unique continuation
Amrein, W. O. ; Berthier, A. M. ; Georgescu, V.
Annales de l'Institut Fourier, Tome 31 (1981), p. 153-168 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous démontrons une inégalité de la forme

∥|x|rf∥Lp(Rn)≤c(n,p,q,r)∥|x|τ+μΔf∥Lq(Rn).

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

We prove an inequality of the type

∥|x|rf∥Lp(Rn)≤c(n,p,q,r)∥|x|τ+μΔf∥Lq(Rn).

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},
     volume = {31},
     year = {1981},
     pages = {153-168},
     doi = {10.5802/aif.843},
     mrnumber = {83g:35011},
     zbl = {0468.35017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 153-168. doi : 10.5802/aif.843. http://gdmltest.u-ga.fr/item/AIF_1981__31_3_153_0/

[1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. | MR 56 #9247 | Zbl 0314.46030

[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 81a:35072 | Zbl 0454.35070

[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 0047.05302

[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 17,626c | Zbl 0067.07503

[7] L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963. | Zbl 0108.09301

[8] M. Schechter and B. Simon, Unique Continuation for Schrödinger Operators with Unbounded Potentials, J. Math. Anal. Appl., 77 (1980), 482-492. | MR 83b:35031 | Zbl 0458.35024

[9] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. | MR 46 #4102 | Zbl 0232.42007

[10] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. | Zbl 0387.46032