The trace inequality and eigenvalue estimates for Schrödinger operators

Kerman, R. ; Sawyer, Eric T.

Annales de l'Institut Fourier, Tome 36 (1986), p. 207-228 / Harvested from Numdam

Résumé

Soit $\Phi $ une fonction radiale, non négative, localement intégrable sur ${\bf R}^n$, qui ne s’accroît pas en $\vert x\vert $. Posons $ (Tf)(x)=\int _{{\bf R}^n}\Phi (x-y)f(y)dy$ où $f\ge 0$ et $x\in {\bf R}^n$. Étant donné $1< p< \infty $ et $v\ge 0$, nous démontrons qu’il existe $C>0$ de sorte que $\int _{{\bf R}^n}(Tf)(x)^pv(x)dx\le C\int _{{\bf R}^n}f(x)^pdx$ pour tout $f\ge 0$, si et seulement si, $C^{\prime }>0$ existe avec $ \int _{Q}T(x_ Qv)(x)^{p^{\prime }}dx \le C^{\prime }\int _{Q}v(x)dx< \infty $ pour tout cube dyadique $Q$, où $p^{\prime }=p/(p-1)$.

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.

Abstract

Suppose $\Phi $ is a nonnegative, locally integrable, radial function on ${\bf R}^n$, which is nonincreasing in $\vert x\vert $. Set $ (Tf)(x)=\int _{R^n}\Phi (x-y)f(y)dy $ when $f\ge 0$ and $x\in {\bf R}^n$. Given $1< p< \infty $ and $v\ge 0$, we show there exists $C>0$ so that $\int _{{\bf R}^n}(Tf)(x)^pv(x)dx\le C\int _{{\bf R}^n}f(x)^pdx$ for all $f\ge 0$, if and only if $C^{\prime }>0$ exists with $\int _{Q}T(x_ Qv)(x)^{p^{\prime }}dx\le C^{\prime }\int _{Q}v(x)dx< \infty $ for all dyadic cubes Q, where $p^{\prime }=p/(p-1)$. This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

Détail

Publié le : 1986-01-01

MR 88b:35150 | Zbl 0591.47037 | 3 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1074

Bibliographie

[1] D. R. Adams, A trace inequality for generalized potentials, Studia Math., 48 (1973), 99-105. | MR 49 #1091 | Zbl 0237.46037

[2] D. R. Adams, On the existence of capacitary strong type estimates in Rn, Ark. Mat., 14 (1976), 125-140. | MR 54 #5822 | Zbl 0325.31008

[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 26 #1485 | Zbl 0102.32401

[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 87d:42027 | Zbl 0575.42025

[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 87d:42028 | Zbl 0578.46024

[7] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241-250. | MR 50 #10670 | Zbl 0291.44007

[8] B. Dahlberg, Regularity properties of Riesz potentials, Ind. U. Math. J., 28 (1979), 257-268. | MR 80g:31004 | Zbl 0413.31003

[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 84i:35070 | Zbl 0498.35042

[10] C. L. Fefferman, The Uncertainty Principle, Bull. A.M.S., (1983), 129-206. | MR 85f:35001 | Zbl 0526.35080

[11] M. De Guzman, Differentiation of Integrals in Rn, Lecture Notes in Math., vol. 481, Springer-Verlag, Berlin and New York, 1975. | MR 56 #15866 | Zbl 0327.26010

[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 86m:35126 | Zbl 0564.35027

[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 0433.46032

[15] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. A.M.S., 192 (1974), 251-275. | MR 49 #5275 | Zbl 0289.26010

[16] M. Reed and B. Simon, Methods of Mathematical Physics, Vol. I, Academic Press, New York and London, 1972. | Zbl 0242.46001

[17] E. Sawyer, Weighted norm inequalities for fractional maximal operators, C.M.S. Conf. Proc., 1 (1980), 283-309. | MR 83k:42020a | Zbl 0546.42018

[18] E. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math., 75 (1982), 1-11. | MR 84i:42032 | Zbl 0508.42023

[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 0127.32002

[20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, 2nd edition, Princeton University Press, 1970. | MR 44 #7280 | Zbl 0207.13501

[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 0524.42011