<p>Generalised Morrey (function) spaces enjoyed some interest recently and found applications to PDE. Here we turn our attention to their discrete counterparts. We define generalised Morrey sequence spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_443_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$m_{\varphi ,p}=m_{\varphi ,p}({\mathbb {Z}}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mrow> <mi>φ</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow> <mi>φ</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. They are natural generalisations of the classical Morrey sequence spaces <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_443_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$m_{u,p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_443_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$0<p\le u<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi>u</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, which were studied earlier. We consider some basic features of the spaces as well as embedding properties such as continuity, compactness and strict singularity.</p>

Generalised Morrey sequence spaces

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2025年5月31日

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Dorothee D. Haroske,
Leszek Skrzypczak

摘要

Generalised Morrey (function) spaces enjoyed some interest recently and found applications to PDE. Here we turn our attention to their discrete counterparts. We define generalised Morrey sequence spaces $m_{\varphi ,p}=m_{\varphi ,p}({\mathbb {Z}}^d)$ $m_{φ, p} = m_{φ, p} (Z^{d})$ . They are natural generalisations of the classical Morrey sequence spaces $m_{u,p}$ $m_{u, p}$ , $0<p\le u<\infty $ $0 < p \leq u < \infty$ , which were studied earlier. We consider some basic features of the spaces as well as embedding properties such as continuity, compactness and strict singularity.