本文研究带无限时滞的二阶退化微分方程$(Mu)‘‘(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds+f(t),\ (t\in\mathbb{R})$和$(Mu‘)‘(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds+f(t),\ (t\in\mathbb{R})$在向量值H"older连续函数空间上的最大正则性，其中$A:D(A)\rightarrow X$和$M:D(M)\rightarrow X$是Banach空间$X$上的闭线性算子，$D(A)\cap D(M)\neq\{0\}$，并且$a\in L^1(\mathbb{R}_{+})\cap L^1(\mathbb{R}_{+};t^{\alpha}dt)$。运用向量值H"older连续函数空间上的算子值Fourier乘子理论、以Carleman变换为基础的新变换和估计以及寻找特殊解方法，本文给出了上述方程在向量值H"older连续函数空间上具有最大正则性的充要条件。本文研究带有限时滞的二阶退化微分方程$(Mu)‘‘(t)=Au(t)+Fu_t+Gu‘_t+f(t),\ (t\in\mathbb{R})$在Lebesgue-Bochner空间上的最大正则性，其中$A: D(A)\rightarrow X$和$M:D(M)\rightarrow X$是Banach空间$X$上的闭线性算子，对于某个固定的常数$\tau >0$，$F,G\in B(L^p([-\tau,0]; X);X)$，$u_t$和$u_t‘$定义为$u_t(s):=u(t+s),\ u‘_t(s):=u‘(t+s),\ (t\in\mathbb{R},s\in [-\tau, 0])$。运用Lebesgue-Bochner空间上的算子值Fourier乘子定理及相关技巧、Carleman变换及相关定理以及寻找特殊解方法，并建立上述方程在Lebesgue-Bochner空间上的解与对应加权空间上的解的对应关系，本文给出了上述方程在Lebesgue-Bochner空间上具有最大正则性的充要条件。本文研究一阶和二阶微分方程$u‘(t)=Au(t)+f(t),\ (t\in\mathbb{R})$和$u‘‘(t)+\gamma u‘(t)=Au(t)+f(t),\ (t\in\mathbb{R})$在向量值Besov空间上的最大正则性，其中$\gamma \in \mathbb{R}$，算子$A$是Banach空间$X$上的闭线性算子。运用向量值Besov空间上的算子值Fourier乘子定理及相关技巧、Carleman变换及相关定理以及寻找特殊解方法，并建立上述方程在向量值Besov空间上的解与对应加权空间上的解的对应关系，本文给出了上述方程在向量值Besov空间上具有最大正则性的充要条件。在类似的向量值Triebel-Lizorkin空间上也得到了对应结论。本文还了给出以上结论的一些应用，如研究对应的半线性微分方程、中立型微分方程、反问题以及一些具体例子。

Using operator-valued $\dot{C}^{\alpha}$-Fourier multiplier results on vector-valued H"older continuous function spaces and new transforms and estimates based on the Carleman transform, we characterize the maximal regularity of second order degenerate differential equations with infinite delay: $(Mu)‘‘(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)\mathrm{d}s+f(t),\ (t\in\mathbb{R})$ and $(Mu‘)‘(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)\mathrm{d}s+f(t),\ (t\in\mathbb{R})$ on $C^{\alpha}(\mathbb{R};X)$, where $A:D(A)\rightarrow X$ and $M:D(M)\rightarrow X$ are closed linear operators in a Banach space $X$, and $a\in L^1(\mathbb{R}_{+})\cap L^1(\mathbb{R}_{+};t^{\alpha}dt)$. These results are used to study the maximal regularity of the associated semilinear differential equations and concrete partial differential equations.Using operator-valued $L^p$-Fourier multiplier results on Lebesgue-Bochner spaces, weighted Sobolev spaces and the Carleman transform, we characterize the maximal regularity of second-order degenerate differential equations with finite delay $(Mu)‘‘(t)=Au(t)+Fu_t+Gu‘_t+f(t),\ (t\in\mathbb{R})$ on $L^p(\mathbb{R}; X)$, where $A: D(A)\rightarrow X$ and $M: D(M)\rightarrow X$ are closed linear operators defined on a Banach space $X$, the operators $F$ and $G$ are in $B(L^p([-\tau,0]; X);X)$ for some fixed $\tau > 0$, and $u_t(s) = u(t+s),\ u‘_t(s) = u‘(t+s)$ when $t\in\mathbb{R}$ and $s\in [-\tau, 0]$. These results are used to study the maximal regularity of the associated second-order neutral degenerate differential equations and associated inverse problem.Using operator-valued $B^{s}_{p,q}$-Fourier multiplier results on vector-valued Besov spaces (resp. vector-valued Triebel–Lizorkin spaces), weighted Besov spaces $B^{s,\omega}_{p,q}(\mathbb{R}; X)$ (resp. weighted Triebel-Lizorkin spaces $F^{s,\omega}_{p,q}(\mathbb{R}; X)$) and the Carleman transform, we characterize the maximal regularity of differential equations $u‘(t)=Au(t)+f(t),\ (t\in\mathbb{R})$ and $u‘‘(t)+\gamma u‘(t)=Au(t)+f(t),\ (t\in\mathbb{R})$ on $B^{s}_{p,q}(\mathbb{R};X)$ (resp. $F^{s}_{p,q}(\mathbb{R};X)$), where $\gamma \in \mathbb{R}$, $s>0$, $1\leq p,q\leq \infty$ (resp. $1\leq p,q<\infty$) and $A$ is a closed linear operator defined on a Banach space $X$. These results are used to study the maximal regularity of the associated neutral degenerate differential equations and semilinear differential equations.