<p>We establish a boundary condition on the variable exponent <i>p</i>, for which the operators <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_469_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="144" /> </InlineMediaObject> <EquationSource Format="TEX">$U_z:f\mapsto (f\circ \varphi _z)\varphi '_z$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>U</mi> <mi>z</mi> </msub> <mo>:</mo> <mi>f</mi> <mo>↦</mo> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <msub> <mi>φ</mi> <mi>z</mi> </msub> <mo stretchy="false">)</mo> </mrow> <msubsup> <mi>φ</mi> <mi>z</mi> <mo>′</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation> are bounded in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_469_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$A^{p(\cdot )}(\textbf{D})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This boundary condition enables us to investigate the boundedness and compactness of Toeplitz operators <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_469_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$T_\varphi$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> with symbols <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_469_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi$</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_469_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$L^1(\textbf{D})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, via the functions <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_469_Article_IEq8.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="173" /> </InlineMediaObject> <EquationSource Format="TEX">$z\mapsto \Vert U_zT_\varphi U_z(\mathbbm {1})\Vert _{L^{p(\cdot )}(\textbf{D})}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi>z</mi> <mo>↦</mo> <mo stretchy="false">‖</mo> </mrow> <msub> <mi>U</mi> <mi>z</mi> </msub> <msub> <mi>T</mi> <mi>φ</mi> </msub> <msub> <mi>U</mi> <mi>z</mi> </msub> <msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mn mathvariant="double-struck">1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_469_Article_IEq9.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="173" /> </InlineMediaObject> <EquationSource Format="TEX">$z\mapsto \Vert U_zT_{\overline{\varphi }}U_z(\mathbbm {1})\Vert _{L^{p(\cdot )}(\textbf{D})}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi>z</mi> <mo>↦</mo> <mo stretchy="false">‖</mo> </mrow> <msub> <mi>U</mi> <mi>z</mi> </msub> <msub> <mi>T</mi> <mover> <mi>φ</mi> <mo>¯</mo> </mover> </msub> <msub> <mi>U</mi> <mi>z</mi> </msub> <msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mn mathvariant="double-struck">1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>.</p>

Toeplitz operators with symbols in $L^1(\textbf{D})$ on Bergman spaces with variable exponent

期刊论文
发表日期：

2025年10月3日

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Gerardo A. Chacón,
Gerardo R. Chacón,
Humberto Rafeiro

摘要

We establish a boundary condition on the variable exponent p, for which the operators $U_z:f\mapsto (f\circ \varphi _z)\varphi '_z$ $U_{z} : f \mapsto (f \circ φ_{z}) φ_{z}^{'}$ are bounded in $A^{p(\cdot )}(\textbf{D})$ $A^{p (\cdot)} (D)$ . This boundary condition enables us to investigate the boundedness and compactness of Toeplitz operators $T_\varphi$ $T_{φ}$ with symbols $\varphi$ $φ$ in $L^1(\textbf{D})$ $L^{1} (D)$ , via the functions $z\mapsto \Vert U_zT_\varphi U_z(\mathbbm {1})\Vert _{L^{p(\cdot )}(\textbf{D})}$ $z \mapsto ‖ U_{z} T_{φ} U_{z} {(1) ‖}_{L^{p (\cdot)} (D)}$ and $z\mapsto \Vert U_zT_{\overline{\varphi }}U_z(\mathbbm {1})\Vert _{L^{p(\cdot )}(\textbf{D})}$ $z \mapsto ‖ U_{z} T_{\bar{φ}} U_{z} {(1) ‖}_{L^{p (\cdot)} (D)}$ .

Toeplitz operators with symbols in \(L^1(\textbf{D})\) on Bergman spaces with variable exponent

摘要