We find necessary and sufficient conditions on the function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> for the inequality<Equation ID="Equ94"> <EquationSource Format="TEX">$\begin{aligned} \Big |\int \limits _\Omega \Phi (K*f)\Big |\lesssim \Vert f\Vert _{L_1(\mathbb {R}^d)}^p \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> <munder> <mo movablelimits="false">∫</mo> <mi mathvariant="normal">Ω</mi> </munder> <mi mathvariant="normal">Φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mrow /> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> <mo>≲</mo> <msubsup> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> <mi>p</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>to be true. Here K is a positively homogeneous of order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha - d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>-</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>, possibly vector valued, kernel, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> is a p-homogeneous function, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p=d/(d-\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>d</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The domain <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is either bounded with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$C^{1,\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>β</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> smooth boundary for some <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\beta > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> or a half-space in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>. As a corollary, we describe the positively homogeneous of order <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$d/(d-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> functions <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Phi :\mathbb {R}^d \rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> that are suitable for the bound<Equation ID="Equ95"> <EquationSource Format="TEX">$\begin{aligned} \Big |\int \limits _\Omega \Phi (\nabla u)\Big |\lesssim \int \limits _\Omega |\Delta u|. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> <munder> <mo movablelimits="false">∫</mo> <mi mathvariant="normal">Ω</mi> </munder> <mi mathvariant="normal">Φ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">|</mo> </mrow> <mo>≲</mo> <munder> <mo movablelimits="false">∫</mo> <mi mathvariant="normal">Ω</mi> </munder> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Bibliography:16 titles.

MAZ’YA’S $\phi $-INEQUALITIES ON DOMAINS

期刊论文
发表日期：

2025年12月1日

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D. Stolyarov

摘要

We find necessary and sufficient conditions on the function $\Phi $ $Φ$ for the inequality $\begin{aligned} \Big |\int \limits _\Omega \Phi (K*f)\Big |\lesssim \Vert f\Vert _{L_1(\mathbb {R}^d)}^p \end{aligned}$ $\begin{matrix} | \int_{Ω} Φ (K * f) | ≲ {‖ f ‖}_{L_{1} (R^{d})}^{p} \end{matrix}$ to be true. Here K is a positively homogeneous of order $\alpha - d$ $α - d$ , possibly vector valued, kernel, $\Phi $ $Φ$ is a p-homogeneous function, and $p=d/(d-\alpha )$ $p = d / (d - α)$ . The domain $\Omega \subset \mathbb {R}^d$ $Ω \subset R^{d}$ is either bounded with $C^{1,\beta }$ $C^{1, β}$ smooth boundary for some $\beta > 0$ $β > 0$ or a half-space in $\mathbb {R}^d$ $R^{d}$ . As a corollary, we describe the positively homogeneous of order $d/(d-1)$ $d / (d - 1)$ functions $\Phi :\mathbb {R}^d \rightarrow \mathbb {R}$ $Φ : R^{d} \to R$ that are suitable for the bound $\begin{aligned} \Big |\int \limits _\Omega \Phi (\nabla u)\Big |\lesssim \int \limits _\Omega |\Delta u|. \end{aligned}$ $\begin{matrix} | \int_{Ω} Φ (\nabla u) | ≲ \int_{Ω} | Δ u | . \end{matrix}$ Bibliography:16 titles.

MAZ’YA’S \(\phi \)-INEQUALITIES ON DOMAINS

摘要