We show various estimates for Schrödinger operators on and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of and their gradients.
On montre des estimations pour des opérateurs de Schrödinger sur et leurs racines carrées. Le potentiel est dans une classe Hölder inverse améliorant les résultats de Shen. On s’appuie sur une inégalité de type Fefferman-Phong améliorée et des inégalités Hölder inverse pour des solutions faibles de et leurs gradients.
Keywords: Schrödinger operators, maximal inequalities, Riesz transforms, Fefferman-Phong inequality, reverse Hölder estimates
Mot clés : opérateurs de Schrödinger, inégalité maximale, transformée de Riesz, inégalité de Fefferman-Phong, inégalités Hölder inverse
@article{AIF_2007__57_6_1975_0, author = {Auscher, Pascal and Ben Ali, Besma}, title = {Maximal inequalities and {Riesz} transform estimates on $L^p$ spaces for {Schr\"odinger} operators with nonnegative potentials}, journal = {Annales de l'Institut Fourier}, pages = {1975--2013}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {6}, year = {2007}, doi = {10.5802/aif.2320}, zbl = {1161.35003}, mrnumber = {2377893}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2320/} }
TY - JOUR AU - Auscher, Pascal AU - Ben Ali, Besma TI - Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials JO - Annales de l'Institut Fourier PY - 2007 SP - 1975 EP - 2013 VL - 57 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2320/ DO - 10.5802/aif.2320 LA - en ID - AIF_2007__57_6_1975_0 ER -
%0 Journal Article %A Auscher, Pascal %A Ben Ali, Besma %T Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials %J Annales de l'Institut Fourier %D 2007 %P 1975-2013 %V 57 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2320/ %R 10.5802/aif.2320 %G en %F AIF_2007__57_6_1975_0
Auscher, Pascal; Ben Ali, Besma. Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials. Annales de l'Institut Fourier, Volume 57 (2007) no. 6, pp. 1975-2013. doi : 10.5802/aif.2320. http://archive.numdam.org/articles/10.5802/aif.2320/
[1] On estimates for square roots of second order elliptic operators on , Publ. Mat., Volume 48 (2004) no. 1, pp. 159-186 | MR | Zbl
[2] Boundedness of Banach space valued singular integral operators and applications to Hardy spaces (unpublished manuscript)
[3] Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights, Advances in Mathematics, Volume 212 (2007) no. 1, pp. 225-276 | DOI | MR | Zbl
[4] A semilinear equation in , Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 2 (1975) no. 4, pp. 523-555 | Numdam | MR | Zbl
[5] Semi-linear second-order elliptic equations in , J. Math. Soc. Japan, Volume 25 (1973), pp. 565-590 | DOI | MR | Zbl
[6] Pointwise multipliers for reverse Hölder spaces. II., Proc. Roy. Irish Acad. Sect. A 95 (1995) no. 2, pp. 193-204 | MR | Zbl
[7] Riesz transforms for , Trans. Amer. Math. Soc., Volume 351 (1999) no. 3, pp. 1151-1169 | DOI | MR | Zbl
[8] Some norm bounds and quadratic form inequalities for Schrödinger operators, J. Operator Theory, Volume 9 (1983) no. 1, pp. 147-162 | MR | Zbl
[9] Heat kernels and spectral theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989 | MR | Zbl
[10] Spectral theory and differential operators, Cambridge Tracts in Advanced Mathematics, 92, Cambridge University Press, Cambridge, 1995 | MR | Zbl
[11] Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana, Volume 15 (1999) no. 2, pp. 233-265 | MR | Zbl
[12] Endpoint estimates for Riesz transforms of magnetic Schrödinger operators, Arkiv for Mat., Volume 44 (2006) no. 2, pp. 261-275 | DOI | MR | Zbl
[13] Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal., Volume 142 (1996) no. 1, pp. 89-128 | DOI | MR | Zbl
[14] Sobolev spaces related to Schrödinger operators with polynomial potentials, Preprint, 2006
[15] The uncertainty principle., Bull. Amer. Math. Soc. (N.S.), Volume 9 (1983) no. 2, pp. 129-206 | DOI | MR | Zbl
[16] spaces of several variables, Acta Math., Volume 129 (1972), pp. 137-193 | DOI | MR | Zbl
[17] Resolution of a semilinear equation in , Proc. Roy. Soc. Edinburgh Sect. A , Volume 96 (1984) no. 3-4, pp. 275-288 Corrigenda: Proc. Roy. Soc. Edinburgh Sect. A 99 (1985), no. 3-4, 399 | DOI | MR | Zbl
[18] Classical and Modern Fourier Analysis, Pearson Education, New Jersey, 2004 | Zbl
[19] Inégalités maximales pour l’opérateur de Schrödinger, Université de Rennes 1 (1992) (Ph. D. Thesis)
[20] Inégalités maximales pour l’opérateur de Schrödinger, C. R. Acad. Sci. Paris Sér. I Math., Volume 316 (1993) no. 3, pp. 249-252 (French. English, French summary) [Maximal inequalities for Schrödinger operators] | Zbl
[21] A multiplier theorem for Schrödinger operators, Colloq. Math., Volume 60/61 (1990) no. 2, pp. 659-664 | MR | Zbl
[22] A. Hardy-Littlewood inequality for quasiregular mappings in certain domains in , Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 10 (1985), pp. 267-282 | MR | Zbl
[23] Change of variable results for -and reverse Hölder -classes, Trans. Amer.Math. Soc., Volume 328 (1991) no. 2, pp. 639-666 | DOI | MR | Zbl
[24] Schrödinger operators with singular potentials. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Israel J. Math., Volume 13 (1973), pp. 135-148 | DOI | MR | Zbl
[25] -theory of Schrödinger operators with a singular potential. Aspects of positivity in functional analysis (Tübingen, 1985), North-Holland Math. Stud. (1986), pp. 63-78 (122, North-Holland, Amsterdam) | DOI | MR | Zbl
[26] On square functions associated to sectorial operators, Bull. Soc. Math. France, Volume 132 (2004) no. 1, pp. 137-156 | Numdam | MR | Zbl
[27] Une inégalité (unpublished manuscript)
[28] An theory for Schrödinger operators with nonnegative potentials, J. Math. Soc. Japan , Volume 36 (1984) no. 4, pp. 675-688 | DOI | MR | Zbl
[29] Analysis of heat equations on domains, London Mathematical Society Monographs Series, 31, Princeton University Press, Princeton, N.J., 2005 (xiv+284 pp) | MR | Zbl
[30] Schrödinger operators with -potentials, Comm. Math. Phys., Volume 53 (1977) no. 3, pp. 277-284 | DOI | MR | Zbl
[31] estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), Volume 45 (1995), pp. 513-546 | DOI | Numdam | MR | Zbl
[32] Bounds of Riesz transforms on spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble), Volume 55 (2005), pp. 173-197 | DOI | Numdam | MR | Zbl
[33] Riesz transform, Gaussian bounds and the method of wave equation, Math. Z., Volume 247 (2004) no. 3, pp. 643-662 | DOI | MR | Zbl
[34] Hardy and Rellich inequalities in nonintegral dimension, J. Operator Theory, Volume 9 (1983) no. 1, pp. 143-146 | MR | Zbl
[35] Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30., Princeton University Press, Princeton, N.J., 1970 | MR | Zbl
[36] Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32., Princeton University Press, Princeton, N.J., 1971 | MR | Zbl
[37] Weighted Hardy spaces, Springer-Verlag, 1989 (Lecture Notes in Mathematics 1381) | MR | Zbl
[38] Real-variable methods in harmonic analysis, Pure and Applied Mathematics, 123, Academic Press and Inc., Orlando, FL, 1986 | MR | Zbl
[39] Absorption semigroups, their generators, and Schrödinger semigroups, J. Funct. Anal., Volume 67 (1986) no. 2, pp. 167-205 | DOI | MR | Zbl
[40] The Sobolev estimates for some Schrödinger type operators, Math. Sci. Res. Hot-Line, Volume 3 (1999) no. 8, pp. 1-48 (and Harmonic Analysis of some Schrödinger type Operators, PhD thesis, Princeton University, 1993) | MR | Zbl
Cited by Sources: