发表机构
Cellular Products Research and Development; Antalya Bilim University（细胞产品研发公司; 安塔利亚比尔姆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为具有Morrey时间控制的Banach值演化类建立中间导数嵌入定理，通过有理频率分裂和K-泛函证明无需UMD等假设，并给出最优指标、迹公式及紧性结果。
AI 中文摘要
我们为具有Morrey时间控制的Banach值演化类发展了中间导数理论。设$E_0\hookrightarrow E$为Banach空间，$m\ge2$，$1<p<\infty$，且$0\le\lambda<1$。若$u\in\mathcal M^{p,\lambda}(E_0)$且$u^{(m)}\in\mathcal M^{p,\lambda}(E)$，则每个中间导数满足\\[ u^{(j)}\in\mathcal M^{p,\lambda}\bigl((E_0,E)_{j/m,\infty}\bigr),\qquad 0<j<m, \\]且无需UMD、Fourier类型或算子值乘子假设。证明基于$D^j$的精确有理频率分裂、Peetre $K$-泛函以及Morrey空间上的Hardy--Littlewood极大算子。在直线上，该估计具有尺度尖锐的乘法形式。在有限区间上，我们构造了图-Morrey延拓并获得相同的嵌入。弱精细指标对任意Banach偶是最优的。将导数定理与精确Morrey迹定理相结合，可为每个$u^{(j)}$导出精确的满射迹公式，连同混合Hölder/插值嵌入以及迹幂次的$p$-到-$q$ Morrey增益。若$E_0\hookrightarrow E$，这些嵌入在任意小的插值光滑性损失后，在同一Morrey范数下以及在导数-迹阈值之上的时间-Hölder空间中均成为紧嵌入。对于正自伴Hilbert尺度，我们进一步将弱插值目标改进为分数域$D(B^{1-j/m})$，并证明该指数是尖锐的。
英文摘要
We develop an intermediate-derivative theory for Banach-valued evolution classes with Morrey control in time. Let $E_0\hookrightarrow E$ be Banach spaces, $m\ge2$, $1<p<\infty$, and $0\leλ<1$. If $u\in\mathcal M^{p,λ}(E_0)$ and $u^{(m)}\in\mathcal M^{p,λ}(E)$, then every intermediate derivative satisfies \[ u^{(j)}\in\mathcal M^{p,λ}\bigl((E_0,E)_{j/m,\infty}\bigr),\qquad 0<j<m, \] with no UMD, Fourier type, or operator-valued multiplier assumption. The proof is based on an exact rational-frequency splitting of $D^j$, the Peetre $K$-functional, and the Hardy--Littlewood maximal operator on Morrey spaces. On the line the estimate has the scale-sharp multiplicative form. On finite intervals we construct a graph-Morrey extension and obtain the same embedding. The weak fine index is optimal for arbitrary Banach couples. Combining the derivative theorem with the exact Morrey trace theorem yields an exact onto trace formula for every $u^{(j)}$, together with mixed Hölder/interpolation embeddings and a trace-powered $p$-to-$q$ Morrey gain. If $E_0\hookc E$, these embeddings become compact both in the same Morrey norm after an arbitrarily small loss of interpolation smoothness and in time-Hölder spaces above the derivative-trace threshold. For positive selfadjoint Hilbert scales we further improve the weak interpolation target to the fractional domain $D(B^{1-j/m})$ and show that this exponent is sharp.
