The trace inequality and eigenvalue estimates for Schrödinger operators
Annales de l'institut Fourier (1986)
- Volume: 36, Issue: 4, page 207-228
- ISSN: 0373-0956
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topKerman, R., and Sawyer, Eric T.. "The trace inequality and eigenvalue estimates for Schrödinger operators." Annales de l'institut Fourier 36.4 (1986): 207-228. <http://eudml.org/doc/74736>.
@article{Kerman1986,
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.},
author = {Kerman, R., Sawyer, Eric T.},
journal = {Annales de l'institut Fourier},
keywords = {distribution of eigenvalues of Schrödinger operators},
language = {eng},
number = {4},
pages = {207-228},
publisher = {Association des Annales de l'Institut Fourier},
title = {The trace inequality and eigenvalue estimates for Schrödinger operators},
url = {http://eudml.org/doc/74736},
volume = {36},
year = {1986},
}
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
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 4
SP - 207
EP - 228
AB - 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.
LA - eng
KW - distribution of eigenvalues of Schrödinger operators
UR - http://eudml.org/doc/74736
ER -
References
top- [1] D. R. ADAMS, A trace inequality for generalized potentials, Studia Math., 48 (1973), 99-105. Zbl0237.46037MR49 #1091
- [2] D. R. ADAMS, On the existence of capacitary strong type estimates in Rn, Ark. Mat., 14 (1976), 125-140. Zbl0325.31008MR54 #5822
- [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. Zbl0102.32401MR26 #1485
- [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. Zbl0575.42025MR87d:42027
- [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. Zbl0578.46024MR87d:42028
- [7] R. COIFMAN and C. FEFFERMAN, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241-250. Zbl0291.44007MR50 #10670
- [8] B. DAHLBERG, Regularity properties of Riesz potentials, Ind. U. Math. J., 28 (1979), 257-268. Zbl0413.31003MR80g:31004
- [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. Zbl0498.35042MR84i:35070
- [10] C. L. FEFFERMAN, The Uncertainty Principle, Bull. A.M.S., (1983), 129-206. Zbl0526.35080MR85f:35001
- [11] M. De GUZMAN, Differentiation of Integrals in Rn, Lecture Notes in Math., vol. 481, Springer-Verlag, Berlin and New York, 1975. Zbl0327.26010MR56 #15866
- [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. Zbl0564.35027MR86m:35126
- [14] V. G. MAZ'YA, On capacitary estimates of the strong type for the fractional norm, Zap. Sen. LOMI Leningrad, 70 (1977), 161 - 168. Zbl0433.46032
- [15] B. MUCKENHOUPT and R. L. WHEEDEN, Weighted norm inequalities for fractional integrals, Trans. A.M.S., 192 (1974), 251-275. Zbl0289.26010MR49 #5275
- [16] M. REED and B. SIMON, Methods of Mathematical Physics, Vol. I, Academic Press, New York and London, 1972. Zbl0242.46001
- [17] E. SAWYER, Weighted norm inequalities for fractional maximal operators, C.M.S. Conf. Proc., 1 (1980), 283-309. Zbl0546.42018MR83k:42020a
- [18] E. SAWYER, A characterization of a two-weight norm inequality for maximal operators, Studia Math., 75 (1982), 1-11. Zbl0508.42023MR84i:42032
- [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. Zbl0127.32002
- [20] E. M. STEIN, Singular Integrals and Differentiability Properties of Functions, 2nd edition, Princeton University Press, 1970. Zbl0207.13501MR44 #7280
- [21] J.-O. STRÖMBERG and R. L. WHEEDEN, Fractional integrals on weighted Hp and Lp spaces, Trans. Amer. Math., Soc., 287 (1985), 293-321. Zbl0524.42011
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- Vladimir Maz'ya, Igor Verbitsky, The form boundedness criterion for the relativistic Schrödinger operator
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- Mohammed El Aïdi, Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé.
- Donatella Danielli, Nicola Garofalo, Duy-Minh Nhieu, Trace inequalities for Carnot-Carathéodory spaces and applications
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