<p>Let <i>G</i> be a locally compact group, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2118_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Phi _1,~ \Phi _2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> <mo>,</mo> <mspace width="3.33333pt" /> <msub> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> be Young functions and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2118_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> be a moderate weight function on <i>G</i>. We introduce the weighted Orlicz amalgam spaces <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2118_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">\(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <msubsup> <mi>L</mi> <mrow> <mi>ω</mi> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </msub> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> defined on <i>G</i>, where the local component space is the Orlicz space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2118_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\Phi _1}(G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the global component is the weighted Orlicz space <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2118_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_{\omega }^{\Phi _2}(G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mi>ω</mi> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </msub> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We derive some properties of the spaces <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2118_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">\(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <msubsup> <mi>L</mi> <mrow> <mi>ω</mi> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </msub> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such as translation invariance, density and duality. We obtain an equivalent discrete type norm on <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2118_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">\(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <msubsup> <mi>L</mi> <mrow> <mi>ω</mi> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </msub> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. By using the equivalent norm, we characterize the Banach algebra <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2118_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">\(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <msubsup> <mi>L</mi> <mrow> <mi>ω</mi> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </msub> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with respect to convolution when the underlying group is an IN group. We show that <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2118_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">\(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <msubsup> <mi>L</mi> <mrow> <mi>ω</mi> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </msub> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> admits no bounded approximate identity under certain conditions.</p>

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Orlicz amalgam spaces and banach algebra structure

  • Büşra Arıs,
  • Serap Öztop

摘要

Let G be a locally compact group, \(\Phi _1,~ \Phi _2\) Φ 1 , Φ 2 be Young functions and \(\omega \) ω be a moderate weight function on G. We introduce the weighted Orlicz amalgam spaces \(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\) W ( L Φ 1 ( G ) , L ω Φ 2 ( G ) ) defined on G, where the local component space is the Orlicz space \(L^{\Phi _1}(G)\) L Φ 1 ( G ) and the global component is the weighted Orlicz space \(L_{\omega }^{\Phi _2}(G)\) L ω Φ 2 ( G ) . We derive some properties of the spaces \(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\) W ( L Φ 1 ( G ) , L ω Φ 2 ( G ) ) such as translation invariance, density and duality. We obtain an equivalent discrete type norm on \(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\) W ( L Φ 1 ( G ) , L ω Φ 2 ( G ) ) . By using the equivalent norm, we characterize the Banach algebra \(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\) W ( L Φ 1 ( G ) , L ω Φ 2 ( G ) ) with respect to convolution when the underlying group is an IN group. We show that \(W(L^{\Phi _1} (G),~ L_{\omega }^{\Phi _2} (G))\) W ( L Φ 1 ( G ) , L ω Φ 2 ( G ) ) admits no bounded approximate identity under certain conditions.