Let G be an infinite, compact abelian group, E be a homogeneous Banach space over G and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {L}\left( E\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> be the space of all continuous linear operators from E into itself equipped with the operator norm. Translation operators are isometries in E (by definition) and so the closed subalgebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathfrak {m}\left( E\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">m</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {L}\left( E\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> consisting of those operators which commute with all translations is well defined. It is shown that there exists a contractive projection <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \mathcal {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Q</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {L}\left( E\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> onto <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathfrak {m}\left( E\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">m</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> which is positivity preserving. Moreover, every operator <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \mathcal {Q}\left( T\right) \in \mathfrak {m}\left( E\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">Q</mi> <mfenced close=")" open="("> <mi>T</mi> </mfenced> <mo>∈</mo> <mi mathvariant="fraktur">m</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$T\in \mathcal {L}\left( E\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mfenced close=")" open="("> <mi>E</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, is induced by a unique Fourier multiplier function <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\hat{T}\in \ell ^{\infty }\left( \Gamma \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>T</mi> <mo stretchy="false">^</mo> </mover> <mo>∈</mo> <msup> <mi>ℓ</mi> <mi>∞</mi> </msup> <mfenced close=")" open="("> <mi mathvariant="normal">Γ</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is the dual group of G. In the setting of the homogeneous Banach spaces <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ L^{p}\left( G\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mfenced close=")" open="("> <mi>G</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$1\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and G an amenable group, these results are due to W. Arendt and J. Voigt.

Fourier Multipliers in Homogeneous Banach Spaces

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2026年7月2日

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Ben de Pagter,
Werner J. Ricker

摘要

Let G be an infinite, compact abelian group, E be a homogeneous Banach space over G and $\mathcal {L}\left( E\right) $ $L (E)$ be the space of all continuous linear operators from E into itself equipped with the operator norm. Translation operators are isometries in E (by definition) and so the closed subalgebra $\mathfrak {m}\left( E\right) $ $m (E)$ of $\mathcal {L}\left( E\right) $ $L (E)$ consisting of those operators which commute with all translations is well defined. It is shown that there exists a contractive projection $ \mathcal {Q}$ $Q$ of $\mathcal {L}\left( E\right) $ $L (E)$ onto $\mathfrak {m}\left( E\right) $ $m (E)$ which is positivity preserving. Moreover, every operator $ \mathcal {Q}\left( T\right) \in \mathfrak {m}\left( E\right) $ $Q (T) \in m (E)$ , with $T\in \mathcal {L}\left( E\right) $ $T \in L (E)$ , is induced by a unique Fourier multiplier function $\hat{T}\in \ell ^{\infty }\left( \Gamma \right) $ $\hat{T} \in ℓ^{\infty} (Γ)$ , where $\Gamma $ $Γ$ is the dual group of G. In the setting of the homogeneous Banach spaces $ L^{p}\left( G\right) $ $L^{p} (G)$ , for $1\le p<\infty $ $1 \leq p < \infty$ and G an amenable group, these results are due to W. Arendt and J. Voigt.