具有变指数的适配序列空间的实插值
Real interpolation for adapted sequence spaces with variable exponents
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中文总结 AI 辅助
本文研究了具有可变指数的适应序列空间的实插值，证明了相关空间的等价性并提供了拟范数的等价性。
中文摘要 AI 辅助
我们研究具有变指数的适配序列空间的实插值。设\((\Omega,\mathcal{F},\mathbb{P};(\mathcal{F}_n)_{n\geq 1})\)为一个带过滤的完备概率空间，\(p(\cdot)\in\mathcal{P}(\Omega)\)，\(0<q\leq\infty\)且\(0<\theta<1\)。我们证明了\(\left(L^{\mathrm{ad}}_{p(\cdot)},L^{\mathrm{ad}}_{\infty}\right)_{\theta,q}=L^{\mathrm{ad}}_{\widetilde{p}(\cdot),q}\)，其中\(\frac{1}{\widetilde{p}(\cdot)} = \frac{1 - \theta}{p(\cdot)}\)，且拟范数等价。\(L^{\mathrm{ad}}_{\widetilde{p}(\cdot),q}\)由适配序列\(f=(f_n)_{n\geq 1}\)组成，其平方函数\(\sigma(f)=\left(\sum_{n = 1}^{\infty}|f_n|^2\right)^{1/2}\)属于变Lorentz空间\(L_{\widetilde{p}(\cdot),q}\)。证明使用了保持适配性的分解并给出了相应\(K\)-泛函的上界估计。不需要变指数的连续性条件以及\(p(\cdot)\)与过滤之间的可测性关系。
英文摘要
We study real interpolation for adapted sequence spaces with variable exponents. Let $(Ω,\mathcal{F},\mathbb{P};(\mathcal{F}_n)_{n\geq 1})$ be a filtered complete probability space, let $p(\cdot)\in\mathcal{P}(Ω)$, and let $0<q\leq\infty$ and $0<θ<1$. We prove that \[ \left(L^{\mathrm{ad}}_{p(\cdot)},L^{\mathrm{ad}}_{\infty}\right)_{θ,q} = L^{\mathrm{ad}}_{\widetilde{p}(\cdot),q}, \qquad \frac{1}{\widetilde{p}(\cdot)} = \frac{1-θ}{p(\cdot)}, \] with equivalent quasi-norms. Here $L^{\mathrm{ad}}_{\widetilde{p}(\cdot),q}$ consists of adapted sequences $f=(f_n)_{n\geq 1}$ whose square function $σ(f)=\left(\sum_{n=1}^{\infty}|f_n|^2\right)^{1/2}$ belongs to the variable Lorentz space $L_{\widetilde{p}(\cdot),q}$. The proof uses a decomposition that preserves adaptedness and provides an upper estimate for the corresponding $K$-functional. No continuity condition on the variable exponent and no measurability relation between $p(\cdot)$ and the filtration are required.
