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Benjamin
Benjamin
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Cross-post from math.sx.

Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the operators $\eta(-i\nabla)$, which is a well-defined bounded and selfadjoint operator by the functional calculus, and $M_f$, which is the multiplication operator given by $f$, acting on $L^2(\mathbb R^n)$.

If $f$ is Lipschitz-continuous, I can prove the commutator is bounded by using properties of the Fourier transform. However, I would like to use weaker assumptions (for example locally Hölder continuous) on $f$, as my assumption on $\eta$ is rather strong. References, proofs and counterexamples are all warmly welcome.

Edit: I should stress that I especially do not have any integrability or boundedness conditions on $f$. Examples, I am thinking of, are usually diverging at infinity, such as $f(x)=|x|$ (which is an example of the Lipschitz case) or, say, $f(x)=x^2$.

Cross-post from math.sx.

Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the operators $\eta(-i\nabla)$, which is a well-defined bounded and selfadjoint operator by the functional calculus, and $M_f$, which is the multiplication operator given by $f$, acting on $L^2(\mathbb R^n)$.

If $f$ is Lipschitz-continuous, I can prove the commutator is bounded by using properties of the Fourier transform. However, I would like to use weaker assumptions (for example locally Hölder continuous) on $f$, as my assumption on $\eta$ is rather strong. References, proofs and counterexamples are all warmly welcome.

Cross-post from math.sx.

Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the operators $\eta(-i\nabla)$, which is a well-defined bounded and selfadjoint operator by the functional calculus, and $M_f$, which is the multiplication operator given by $f$, acting on $L^2(\mathbb R^n)$.

If $f$ is Lipschitz-continuous, I can prove the commutator is bounded by using properties of the Fourier transform. However, I would like to use weaker assumptions (for example locally Hölder continuous) on $f$, as my assumption on $\eta$ is rather strong. References, proofs and counterexamples are all warmly welcome.

Edit: I should stress that I especially do not have any integrability or boundedness conditions on $f$. Examples, I am thinking of, are usually diverging at infinity, such as $f(x)=|x|$ (which is an example of the Lipschitz case) or, say, $f(x)=x^2$.

Source Link
Benjamin
Benjamin
  • 245
  • 1
  • 5

Commutator of pseudodifferential operator and multiplication operator

Cross-post from math.sx.

Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the operators $\eta(-i\nabla)$, which is a well-defined bounded and selfadjoint operator by the functional calculus, and $M_f$, which is the multiplication operator given by $f$, acting on $L^2(\mathbb R^n)$.

If $f$ is Lipschitz-continuous, I can prove the commutator is bounded by using properties of the Fourier transform. However, I would like to use weaker assumptions (for example locally Hölder continuous) on $f$, as my assumption on $\eta$ is rather strong. References, proofs and counterexamples are all warmly welcome.