# ANNALES DE L'INSTITUT FOURIER

${L}^{p}$-inequalities for the laplacian and unique continuation
Annales de l'Institut Fourier, Volume 31 (1981) no. 3, p. 153-168

We prove an inequality of the type

$\parallel |x{|}^{{r}_{f}}{\parallel }_{{L}^{p}\left({\mathbf{R}}^{n}\right)}\le c\left(n,p,q,r\right)\parallel |x{|}^{\tau +\mu }\Delta f{\parallel }_{{L}^{q}\left({\mathbf{R}}^{n}\right)}.$

This is then used to derive the unique continuation property for the differential inequality $|\Delta f\left(x\right)|\le |v\left(x\right)|\phantom{\rule{4pt}{0ex}}|f\left(x\right)|$ under suitable local integrability assumptions on the function $v$.

Nous démontrons une inégalité de la forme

$\parallel |x{|}^{{r}_{f}}{\parallel }_{{L}^{p}\left({\mathbf{R}}^{n}\right)}\le c\left(n,p,q,r\right)\parallel |x{|}^{\tau +\mu }\Delta f{\parallel }_{{L}^{q}\left({\mathbf{R}}^{n}\right)}.$

Comme applications nous obtenons la propriété de prolongement unique pour l’inégalité différentielle $|\Delta f\left(x\right)|\le |v\left(x\right)|\phantom{\rule{4pt}{0ex}}|f\left(x\right)|$ si $v\in {L}_{Loc}^{p}$ avec $p>max\left(\frac{n}{2},n-2\right)\right)$.

@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},
publisher = {Imprimerie Louis-Jean},
volume = {31},
number = {3},
year = {1981},
pages = {153-168},
doi = {10.5802/aif.843},
mrnumber = {83g:35011},
zbl = {0468.35017},
language = {en},
url = {aif.centre-mersenne.org/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, Volume 31 (1981) no. 3, pp. 153-168. doi : 10.5802/aif.843. https://aif.centre-mersenne.org/item/AIF_1981__31_3_153_0/

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