The trace inequality and eigenvalue estimates for Schrödinger operators
Annales de l'Institut Fourier, Volume 36 (1986) no. 4, pp. 207-228.

Suppose Φ is a nonnegative, locally integrable, radial function on R n , which is nonincreasing in |x|. Set (Tf)(x)= R n Φ(x-y)f(y)dy when f0 and xR n . Given 1<p< and v0, we show there exists C>0 so that R n (Tf)(x) p v(x)dxC R n f(x) p dx for all f0, if and only if C >0 exists with Q T(x Q v)(x) p dxC Q v(x)dx< for all dyadic cubes Q, where p =p/(p-1). This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

Soit Φ une fonction radiale, non négative, localement intégrable sur R n , qui ne s’accroît pas en |x|. Posons (Tf)(x)= R n Φ(x-y)f(y)dyf0 et xR n . Étant donné 1<p< et v0, nous démontrons qu’il existe C>0 de sorte que R n (Tf)(x) p v(x)dxC R n f(x) p dx pour tout f0, si et seulement si, C >0 existe avec Q T(x Q v)(x) p dxC Q v(x)dx< pour tout cube dyadique Q, où p =p/(p-1).

On se sert de ce résultat pour raffiner des approximations récentes de la part de C.L. Fefferman et D.H. Phong de la distribution de valeurs propres d’opérateurs de Schrödinger.

@article{AIF_1986__36_4_207_0,
     author = {Kerman, R. and Sawyer, Eric T.},
     title = {The trace inequality and eigenvalue estimates for {Schr\"odinger} operators},
     journal = {Annales de l'Institut Fourier},
     pages = {207--228},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {36},
     number = {4},
     year = {1986},
     doi = {10.5802/aif.1074},
     mrnumber = {88b:35150},
     zbl = {0591.47037},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1074/}
}
TY  - JOUR
AU  - Kerman, R.
AU  - Sawyer, Eric T.
TI  - The trace inequality and eigenvalue estimates for Schrödinger operators
JO  - Annales de l'Institut Fourier
PY  - 1986
SP  - 207
EP  - 228
VL  - 36
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1074/
DO  - 10.5802/aif.1074
LA  - en
ID  - AIF_1986__36_4_207_0
ER  - 
%0 Journal Article
%A Kerman, R.
%A Sawyer, Eric T.
%T The trace inequality and eigenvalue estimates for Schrödinger operators
%J Annales de l'Institut Fourier
%D 1986
%P 207-228
%V 36
%N 4
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1074/
%R 10.5802/aif.1074
%G en
%F AIF_1986__36_4_207_0
Kerman, R.; Sawyer, Eric T. The trace inequality and eigenvalue estimates for Schrödinger operators. Annales de l'Institut Fourier, Volume 36 (1986) no. 4, pp. 207-228. doi : 10.5802/aif.1074. http://www.numdam.org/articles/10.5802/aif.1074/

[1] D. R. Adams, A trace inequality for generalized potentials, Studia Math., 48 (1973), 99-105. | MR | Zbl

[2] D. R. Adams, On the existence of capacitary strong type estimates in Rn, Ark. Mat., 14 (1976), 125-140. | MR | Zbl

[3] D. R. Adams, Lectures on Lp-potential theory (preprint), Univ. of Umeä, 2 (1981).

[4] N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. | Numdam | MR | Zbl

[5] S. Y. A. Chang, J. M. Wilson and T. H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv., 60 (1985), 217-246. | MR | Zbl

[6] S. Chanillo and R. L. Wheeden, Lp estimates for fractional integrals and Sobolev inequalities, with applications to Schrödinger operators, Comm. Partial Differential Equations, 10 (1985), 1077-1116. | MR | Zbl

[7] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241-250. | MR | Zbl

[8] B. Dahlberg, Regularity properties of Riesz potentials, Ind. U. Math. J., 28 (1979), 257-268. | MR | Zbl

[9] E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. in P.D.E., 7 (1982), 77-116. | MR | Zbl

[10] C. L. Fefferman, The Uncertainty Principle, Bull. A.M.S., (1983), 129-206. | MR | Zbl

[11] M. De Guzman, Differentiation of Integrals in Rn, Lecture Notes in Math., vol. 481, Springer-Verlag, Berlin and New York, 1975. | MR | Zbl

[12] K. Hansson, Continuity and compactness of certain convolution operators, Institut Mittage-Leffler, Report No. 9, (1982).

[13] R. Kerman and E. Sawyer, Weighted norm inequalities for potentials with applications to Schrödinger operators, Fourier transforms and Carleson measures, announcement in Bull. A.M.S., 12 (1985), 112-116. | MR | Zbl

[14] V. G. Maz'Ya, On capacitary estimates of the strong type for the fractional norm, Zap. Sen. LOMI Leningrad, 70 (1977), 161 - 168. | Zbl

[15] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. A.M.S., 192 (1974), 251-275. | MR | Zbl

[16] M. Reed and B. Simon, Methods of Mathematical Physics, Vol. I, Academic Press, New York and London, 1972. | Zbl

[17] E. Sawyer, Weighted norm inequalities for fractional maximal operators, C.M.S. Conf. Proc., 1 (1980), 283-309. | MR | Zbl

[18] E. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math., 75 (1982), 1-11. | MR | Zbl

[19] E. M. Stein, The characterization of functions arising as potentials I, Bull. Amer. Math. Soc., 67 (1961), 102-104, II (IBID), 68 (1962), 577-582. | Zbl

[20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, 2nd edition, Princeton University Press, 1970. | MR | Zbl

[21] J.-O. Strömberg and R. L. Wheeden, Fractional integrals on weighted Hp and Lp spaces, Trans. Amer. Math., Soc., 287 (1985), 293-321. | Zbl

Cited by Sources: