Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2107_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$T_{a}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> be a pseudo-differential operator with symbol a. When <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2107_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="162" /> </InlineMediaObject> <EquationSource Format="TEX">$a\in S^m_{\rho ,1},m=n(\rho -1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <msubsup> <mi>S</mi> <mrow> <mi>ρ</mi> <mo>,</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, it is well known that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2107_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$T_{a}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> is not always bounded on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2107_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">${L^1}({\mathbb {R}^n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. However, under extra assumptions on a, we prove that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2107_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$T_{a}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> is bounded on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2107_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">${L^p}({\mathbb {R}^n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2107_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \le p \le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2107_Article_IEq8.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="128" /> </InlineMediaObject> <EquationSource Format="TEX">$a \in {L^\infty }S_\rho ^{n(\rho - 1)}(\omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <msubsup> <mi>S</mi> <mi>ρ</mi> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

The endpoint estimates for pseudo-differential operators

期刊论文
发表日期：

2025年3月1日

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Guoning Wu,
Jie Yang

摘要

Let $T_{a}$ $T_{a}$ be a pseudo-differential operator with symbol a. When $a\in S^m_{\rho ,1},m=n(\rho -1)$ $a \in S_{ρ, 1}^{m}, m = n (ρ - 1)$ , it is well known that $T_{a}$ $T_{a}$ is not always bounded on ${L^1}({\mathbb {R}^n})$ $L^{1} (R^{n})$ . However, under extra assumptions on a, we prove that $T_{a}$ $T_{a}$ is bounded on ${L^p}({\mathbb {R}^n})$ $L^{p} (R^{n})$ for $1 \le p \le \infty $ $1 \leq p \leq \infty$ when $a \in {L^\infty }S_\rho ^{n(\rho - 1)}(\omega )$ $a \in L^{\infty} S_{ρ}^{n (ρ - 1)} (ω)$ .