The trace inequality and eigenvalue estimates for Schrödinger operators

R. Kerman; Eric T. Sawyer

Annales de l'institut Fourier (1986)

  • Volume: 36, Issue: 4, page 207-228
  • ISSN: 0373-0956

Abstract

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Suppose Φ is a nonnegative, locally integrable, radial function on R n , which is nonincreasing in | x | . Set ( T f ) ( x ) = ∫ R n Φ ( x - y ) f ( y ) d y when f ≥ 0 and x ∈ R n . Given 1 < p < ∞ and v ≥ 0 , we show there exists C > 0 so that ∫ R n ( T f ) ( x ) p v ( x ) d x ≤ C ∫ R n f ( x ) p d x for all f ≥ 0 , if and only if C ' > 0 exists with ∫ Q T ( x Q v ) ( x ) p ' d x ≤ C ' ∫ Q v ( x ) d x < ∞ 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.

How to cite

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Kerman, 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&lt; p&lt; \infty $ and $v\ge 0$, we show there exists $C&gt;0$ 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 $C^\{\prime \}&gt;0$ exists with $\int _\{Q\}T(x_ Qv)(x)^\{p^\{\prime \}\}dx\le C^\{\prime \}\int _\{Q\}v(x)dx&lt; \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&lt; p&lt; \infty $ and $v\ge 0$, we show there exists $C&gt;0$ 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 $C^{\prime }&gt;0$ exists with $\int _{Q}T(x_ Qv)(x)^{p^{\prime }}dx\le C^{\prime }\int _{Q}v(x)dx&lt; \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

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Citations in EuDML Documents

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  1. Yuri V. Egorov, Sur des estimations des valeurs propres d'opérateurs elliptiques
  2. Ronan Pouliquen, Lower bounds for Schrödinger operators in H¹(ℝ)
  3. Zhongwei Shen, The magnetic Schrödinger operator and reverse Hölder class
  4. Vladimir Maz'ya, Igor Verbitsky, The form boundedness criterion for the relativistic Schrödinger operator
  5. Yves Rakotondratsimba, A two-weight inequality for the Bessel potential operator
  6. Mohammed El Aïdi, Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé.
  7. Donatella Danielli, Nicola Garofalo, Duy-Minh Nhieu, Trace inequalities for Carnot-Carathéodory spaces and applications

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