<p>In this paper we study the behavior of dilation operators <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ D_\lambda :f \mapsto f(\lambda \,\cdot ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>λ</mi> </msub> <mo>:</mo> <mi>f</mi> <mo>↦</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mspace width="0.166667em" /> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ \lambda > 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in the context of Triebel-Lizorkin-Morrey spaces <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {E}}^{s}_{u,p,q}(\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="script">E</mi> </mrow> <mrow> <mi>u</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. For that purpose we prove upper and lower bounds for the operator (quasi-)norm <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left\| D_\lambda \; \vert \;{\mathcal {L}}\big ({\mathcal {E}}^s_{u,p,q}({\mathbb {R}}^d)\big ) \right\| $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="∥" open="∥"> <msub> <mi>D</mi> <mi>λ</mi> </msub> <mrow> <mspace width="0.277778em" /> <mo stretchy="false">|</mo> <mspace width="0.277778em" /> <mi mathvariant="script">L</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> </mrow> <msubsup> <mrow> <mi mathvariant="script">E</mi> </mrow> <mrow> <mi>u</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mfenced> </math></EquationSource> </InlineEquation>. We show that for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$s>\sigma _p $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <msub> <mi>σ</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> the operator (quasi-)norm <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\left\| D_\lambda \; \vert \;{\mathcal {L}}\big ({\mathcal {E}}^s_{u,p,q}({\mathbb {R}}^d)\big ) \right\| $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="∥" open="∥"> <msub> <mi>D</mi> <mi>λ</mi> </msub> <mrow> <mspace width="0.277778em" /> <mo stretchy="false">|</mo> <mspace width="0.277778em" /> <mi mathvariant="script">L</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> </mrow> <msubsup> <mrow> <mi mathvariant="script">E</mi> </mrow> <mrow> <mi>u</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mfenced> </math></EquationSource> </InlineEquation> up to constants behaves as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda ^{s - \frac{d}{u}} $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>λ</mi> <mrow> <mi>s</mi> <mo>-</mo> <mfrac> <mi>d</mi> <mi>u</mi> </mfrac> </mrow> </msup> </math></EquationSource> </InlineEquation>. For the borderline case <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ s = \sigma _{p} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <msub> <mi>σ</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> we observe a behavior of the form <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda ^{\sigma _p- \frac{d}{u}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>λ</mi> <mrow> <msub> <mi>σ</mi> <mi>p</mi> </msub> <mo>-</mo> <mfrac> <mi>d</mi> <mi>u</mi> </mfrac> </mrow> </msup> </math></EquationSource> </InlineEquation>, multiplied with logarithmic terms of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> that also depend on the fine index<i>q</i>. For <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$s< \sigma _{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo><</mo> <msub> <mi>σ</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$p \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> we find the relation <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\left\| D_\lambda \; \vert \;{\mathcal {L}}\big ({\mathcal {E}}^s_{u,p,q}({\mathbb {R}}^d)\big ) \right\| \sim \lambda ^{ - \frac{d}{u}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="∥" open="∥"> <msub> <mi>D</mi> <mi>λ</mi> </msub> <mrow> <mspace width="0.277778em" /> <mo stretchy="false">|</mo> <mspace width="0.277778em" /> <mi mathvariant="script">L</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> </mrow> <msubsup> <mrow> <mi mathvariant="script">E</mi> </mrow> <mrow> <mi>u</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mfenced> <mo>∼</mo> <msup> <mi>λ</mi> <mrow> <mo>-</mo> <mfrac> <mi>d</mi> <mi>u</mi> </mfrac> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. The case <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$s < \sigma _{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo><</mo> <msub> <mi>σ</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$p < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is investigated as well. Our proofs are mainly based on the Fourier analytic approach to Triebel-Lizorkin-Morrey spaces. As byproducts we show an advanced Fourier multiplier theorem for band-limited functions in the context of Morrey spaces and derive some new equivalent (quasi-)norms and characterizations of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">${\mathcal {E}}^{s}_{u,p,q}(\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="script">E</mi> </mrow> <mrow> <mi>u</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the Boundedness of Dilation Operators in the Context of Triebel-Lizorkin-Morrey Spaces

期刊论文
发表日期：

2026年5月27日

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Marc Hovemann,
Markus Weimar

摘要

In this paper we study the behavior of dilation operators $ D_\lambda :f \mapsto f(\lambda \,\cdot ) $ $D_{λ} : f \mapsto f (λ \cdot)$ with $ \lambda > 1 $ $λ > 1$ in the context of Triebel-Lizorkin-Morrey spaces ${\mathcal {E}}^{s}_{u,p,q}(\mathbb {R}^d)$ $E_{u, p, q}^{s} (R^{d})$ . For that purpose we prove upper and lower bounds for the operator (quasi-)norm $\left\| D_\lambda \; \vert \;{\mathcal {L}}\big ({\mathcal {E}}^s_{u,p,q}({\mathbb {R}}^d)\big ) \right\| $ $∥D_{λ}, | L (, E_{u, p, q}^{s}, (R^{d}), )∥$ . We show that for $s>\sigma _p $ $s > σ_{p}$ the operator (quasi-)norm $\left\| D_\lambda \; \vert \;{\mathcal {L}}\big ({\mathcal {E}}^s_{u,p,q}({\mathbb {R}}^d)\big ) \right\| $ $∥D_{λ}, | L (, E_{u, p, q}^{s}, (R^{d}), )∥$ up to constants behaves as $\lambda ^{s - \frac{d}{u}} $ $λ^{s - \frac{d}{u}}$ . For the borderline case $ s = \sigma _{p} $ $s = σ_{p}$ we observe a behavior of the form $\lambda ^{\sigma _p- \frac{d}{u}}$ $λ^{σ_{p} - \frac{d}{u}}$ , multiplied with logarithmic terms of $\lambda $ $λ$ that also depend on the fine indexq. For $s< \sigma _{p}$ $s < σ_{p}$ and $p \ge 1$ $p \geq 1$ we find the relation $\left\| D_\lambda \; \vert \;{\mathcal {L}}\big ({\mathcal {E}}^s_{u,p,q}({\mathbb {R}}^d)\big ) \right\| \sim \lambda ^{ - \frac{d}{u}}$ $∥D_{λ}, | L (, E_{u, p, q}^{s}, (R^{d}), )∥ \sim λ^{- \frac{d}{u}}$ . The case $s < \sigma _{p}$ $s < σ_{p}$ and $p < 1$ $p < 1$ is investigated as well. Our proofs are mainly based on the Fourier analytic approach to Triebel-Lizorkin-Morrey spaces. As byproducts we show an advanced Fourier multiplier theorem for band-limited functions in the context of Morrey spaces and derive some new equivalent (quasi-)norms and characterizations of ${\mathcal {E}}^{s}_{u,p,q}(\mathbb {R}^d)$ $E_{u, p, q}^{s} (R^{d})$ .