<p>Let <i>G</i> be a locally compact group, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_696_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> be a Young function. In this paper, we study the amalgam space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_696_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(W(L^{\Phi },Y)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msup> <mo>,</mo> <mi>Y</mi> <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="11868_2025_696_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\Phi }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msup> </math></EquationSource> </InlineEquation> and the global component is a Banach space <i>Y</i> containing the characteristic function of any compact subset of <i>G</i>. We obtain an equivalent norm on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_696_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(W(L^{\Phi },Y)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msup> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. By using the equivalent norm, we investigate right translation invariance and completeness of the amalgam space <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_696_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(W(L^{\Phi },Y)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msup> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with a non translation invariant space <i>Y</i>, in general. We also present an example of a non translation invariant space <i>Y</i> such that <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_696_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(W(L^{\Phi },Y)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msup> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is right translation invariant in contrast to the literature.</p>

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Aspect of invariant properties of Orlicz amalgam spaces with respect to Banach spaces

  • Büşra Arıs,
  • Serap Öztop

摘要

Let G be a locally compact group, \(\Phi \) Φ be a Young function. In this paper, we study the amalgam space \(W(L^{\Phi },Y)\) W ( L Φ , Y ) defined on G, where the local component space is the Orlicz space \(L^{\Phi }\) L Φ and the global component is a Banach space Y containing the characteristic function of any compact subset of G. We obtain an equivalent norm on \(W(L^{\Phi },Y)\) W ( L Φ , Y ) . By using the equivalent norm, we investigate right translation invariance and completeness of the amalgam space \(W(L^{\Phi },Y)\) W ( L Φ , Y ) with a non translation invariant space Y, in general. We also present an example of a non translation invariant space Y such that \(W(L^{\Phi },Y)\) W ( L Φ , Y ) is right translation invariant in contrast to the literature.