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<p>We prove the acting by composition of nontrivial functions <i>f</i>: ℝ → ℝ (i.e., <i>T</i><sub><i>f</i></sub>: <i>g</i> → <i>f</i> ◦ <i>g</i>) on homogeneous Besov and Triebel-Lizorkin spaces realized as subspaces of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\cal{S}}^{\prime}(\mathbb{R}^{n})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mrow> <mi mathvariant="script">S</mi> </mrow> </mrow> <mrow> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> in case <i>s</i> = <i>n/p</i> &lt; 1 + 1/<i>p</i>, and <i>q</i> &gt; 1 (Besov space) and <i>p</i> &gt; 1 (Triebel-Lizorkin space). These subspaces are dilation invariant and endowed with quasi-seminorms such that ∥<i>g</i>∥ = 0 if and only if <i>g</i> is constant.</p>

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A note on nontrivial acting functions for homogeneous Besov and Triebel-Lizorkin spaces

  • 期刊论文

  • 发表日期：

    2025年10月21日

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  • Madani Moussai

摘要

We prove the acting by composition of nontrivial functions f: ℝ → ℝ (i.e., Tf: g → f ◦ g) on homogeneous Besov and Triebel-Lizorkin spaces realized as subspaces of \({\cal{S}}^{\prime}(\mathbb{R}^{n})\) S ′ ( R n ) in case s = n/p < 1 + 1/p, and q > 1 (Besov space) and p > 1 (Triebel-Lizorkin space). These subspaces are dilation invariant and endowed with quasi-seminorms such that ∥g∥ = 0 if and only if g is constant.