Fully measurable small Lebesgue spaces

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• 作者：Giuseppina Anatriello^a ; ^{anatriello@unina.it" class="auth_mail" title="E-mail the corresponding author} ; Maria Rosaria Formica^b ; ^{mara.formica@uniparthenope.it" class="auth_mail" title="E-mail the corresponding author} ; Raffaella Giova^b ; ^{raffaella.giova@uniparthenope.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Banach function spaces ; Grand and small Lebesgue spaces ; Measurable exponent ; Hö ; lder-type inequality
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：550-563
• 全文大小：364 K

文摘

We build a new class of Banach function spaces, whose function norm is where ρ_p(x)${\rho }_{p(x)}$ denotes the norm of the Lebesgue space of exponent p(x)$p(x)$ (assumed measurable and possibly infinite), constant with respect to the variable of f, and δ   is measurable, too. Such class contains some known Banach spaces of functions, among which are the classical and the small Lebesgue spaces, and the Orlicz space L(log⁡L)^α$L{(\log L)}^{\alpha }$, α>0$\alpha >0$.

Furthermore we prove the following Hölder-type inequality

where ρ_{p[⋅]),δ[⋅]}(f)${\rho }_{p[\cdot ]),\delta [\cdot ]}(f)$ is the norm of fully measurable grand Lebesgue spaces introduced by Anatriello and Fiorenza in [2]. For suitable choices of p(x)$p(x)$ and δ(x)$\delta (x)$ it reduces to the classical Hölder's inequality for the spaces EXP_1/α$EX{P}_{1/\alpha }$ and L(log⁡L)^α$L{(\log L)}^{\alpha }$, α>0$\alpha >0$.