In this paper, a new functional <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Vert f\Vert _{\Psi ,s^}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <mi mathvariant="normal">Ψ</mi> <mo>,</mo> <msup> <mi>s</mi> <mo>∗</mo> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation> will be introduced in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(L_{\Phi })^\prime $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> by the convex modular <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho (f).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> It is proved that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Vert f\Vert _{\Psi ,s^}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <mi mathvariant="normal">Ψ</mi> <mo>,</mo> <msup> <mi>s</mi> <mo>∗</mo> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation> is the norm formula for bounded linear functionals in dual spaces of Orlicz spaces <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_{\Phi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msub> </math></EquationSource> </InlineEquation> equipped with s-norms. Support functionals at any point of the unit sphere in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L_{\Phi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msub> </math></EquationSource> </InlineEquation> are described for s-norms. Next, these results are applied to get sufficient and necessary conditions for smooth points, very (strongly) smooth points of the unit sphere in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L_{\Phi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msub> </math></EquationSource> </InlineEquation> equipped with s-norms. As a result, criteria for smoothness, very (strongly) smoothness of Orlicz spaces <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L_{\Phi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi mathvariant="normal">Φ</mi> </msub> </math></EquationSource> </InlineEquation> equipped with s-norms are also obtained.

Smoothness and very (strongly) smoothness of Orlicz function spaces equipped with s-norms

期刊论文
发表日期：

2025年11月3日

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Xiaoyan Li,
Yunan Cui,
Ping Wang

摘要

In this paper, a new functional $\Vert f\Vert _{\Psi ,s^*}$ ${‖ f ‖}_{Ψ, s^{*}}$ will be introduced in $(L_{\Phi })^\prime $ ${(L_{Φ})}^{'}$ by the convex modular $\rho (f).$ $ρ (f) .$ It is proved that $\Vert f\Vert _{\Psi ,s^*}$ ${‖ f ‖}_{Ψ, s^{*}}$ is the norm formula for bounded linear functionals in dual spaces of Orlicz spaces $L_{\Phi }$ $L_{Φ}$ equipped with s-norms. Support functionals at any point of the unit sphere in $L_{\Phi }$ $L_{Φ}$ are described for s-norms. Next, these results are applied to get sufficient and necessary conditions for smooth points, very (strongly) smooth points of the unit sphere in $L_{\Phi }$ $L_{Φ}$ equipped with s-norms. As a result, criteria for smoothness, very (strongly) smoothness of Orlicz spaces $L_{\Phi }$ $L_{Φ}$ equipped with s-norms are also obtained.