Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> be an N-function whose Matuszewska-Orlicz indices satisfy <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$1<\alpha _\Phi \le \beta _\Phi <\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <msub> <mi>α</mi> <mi mathvariant="normal">Φ</mi> </msub> <mo>≤</mo> <msub> <mi>β</mi> <mi mathvariant="normal">Φ</mi> </msub> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Using these indices, we introduce “interpolation friendly” classes of Fourier multipliers <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$M_{[\Phi ]}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mrow> <mo stretchy="false">[</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">]</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M_{\langle \Phi \rangle }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mrow> <mo stretchy="false">⟨</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">⟩</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M_{[\Phi ]}\subset M_{\langle \Phi \rangle }\subset M_\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mrow> <mo stretchy="false">[</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mo>⊂</mo> <msub> <mi>M</mi> <mrow> <mo stretchy="false">⟨</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">⟩</mo> </mrow> </msub> <mo>⊂</mo> <msub> <mi>M</mi> <mi mathvariant="normal">Φ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M_\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi mathvariant="normal">Φ</mi> </msub> </math></EquationSource> </InlineEquation> is the Banach algebra of all Fourier multipliers on the reflexive Orlicz sequence space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell ^\Phi (\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mi mathvariant="normal">Φ</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Applying the Gohberg-Krupnik localisation in the corresponding Calkin algebra, the study of Fredholmness of the discrete Wiener-Hopf operator T(a) with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$a\in M_{\langle \Phi \rangle }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <msub> <mi>M</mi> <mrow> <mo stretchy="false">⟨</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">⟩</mo> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> is reduced to that of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$T(a_\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>τ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for certain, potentially easier to study, local representatives <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$a_\tau \in M_{[\Phi ]}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>τ</mi> </msub> <mo>∈</mo> <msub> <mi>M</mi> <mrow> <mo stretchy="false">[</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">]</mo> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> of a at all points <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\tau \in [-\pi ,\pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>-</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

GOHBERG-KRUPNIK LOCALISATION FOR DISCRETE WIENER-HOPF OPERATORS ON ORLICZ SEQUENCE SPACES

期刊论文
发表日期：

2025年10月3日

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Oleksiy Karlovych,
Sandra Mary Thampi

摘要

Let $\Phi $ $Φ$ be an N-function whose Matuszewska-Orlicz indices satisfy $1<\alpha _\Phi \le \beta _\Phi <\infty $ $1 < α_{Φ} \leq β_{Φ} < \infty$ . Using these indices, we introduce “interpolation friendly” classes of Fourier multipliers $M_{[\Phi ]}$ $M_{[Φ]}$ and $M_{\langle \Phi \rangle }$ $M_{⟨ Φ ⟩}$ such that $M_{[\Phi ]}\subset M_{\langle \Phi \rangle }\subset M_\Phi $ $M_{[Φ]} \subset M_{⟨ Φ ⟩} \subset M_{Φ}$ , where $M_\Phi $ $M_{Φ}$ is the Banach algebra of all Fourier multipliers on the reflexive Orlicz sequence space $\ell ^\Phi (\mathbb {Z})$ $ℓ^{Φ} (Z)$ . Applying the Gohberg-Krupnik localisation in the corresponding Calkin algebra, the study of Fredholmness of the discrete Wiener-Hopf operator T(a) with $a\in M_{\langle \Phi \rangle }$ $a \in M_{⟨ Φ ⟩}$ is reduced to that of $T(a_\tau )$ $T (a_{τ})$ for certain, potentially easier to study, local representatives $a_\tau \in M_{[\Phi ]}$ $a_{τ} \in M_{[Φ]}$ of a at all points $\tau \in [-\pi ,\pi )$ $τ \in [- π, π)$ .