Illinois Journal of Mathematics

A spectral multiplier theorem for $H\sp 1$ spaces associated with Schrödinger operators with potentials satisfying a reverse Hölder inequality

Jacek Dziubański

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Abstract

Let $\{ T_t\}_{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator $-A=\Delta -V$ on $\mathbb{R}^d$, where $V$ is a nonnegative nonzero potential satisfying a reverse Hölder inequality, and let $\int_0^\infty \lambda \, dE_A(\lambda )$ be the spectral resolution of $A$. We say that a function $f$ is an element of $H_A^1$ if the maximal function $\mathcal{M}f(x)=\sup_{t>0} |T_tf(x)|$ belongs to $L^1$. We prove that if a function $F$ satisfies a Mihlin condition with exponent $\alpha >d/2$ then the operator $F(A)=\int_0^\infty F(\lambda )\, dE_A(\lambda )$ is bounded on $H_A^1$.

Article information

Source
Illinois J. Math., Volume 45, Number 4 (2001), 1301-1313.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138067

Digital Object Identifier
doi:10.1215/ijm/1258138067

Mathematical Reviews number (MathSciNet)
MR1895458

Zentralblatt MATH identifier
1038.42022

Subjects
Primary: 35J10: Schrödinger operator [See also 35Pxx]
Secondary: 35P05: General topics in linear spectral theory 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 47D08: Schrödinger and Feynman-Kac semigroups 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

Citation

Dziubański, Jacek. A spectral multiplier theorem for $H\sp 1$ spaces associated with Schrödinger operators with potentials satisfying a reverse Hölder inequality. Illinois J. Math. 45 (2001), no. 4, 1301--1313. doi:10.1215/ijm/1258138067. https://projecteuclid.org/euclid.ijm/1258138067


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