To prove new regularity theorems for abstract Cauchy problems \[ u'(t)= \Lambda u(t)+ f(t),\quad t\geq 0,\quad \nu(0)= u_0,\tag{1} \] where \(\Lambda\) is generator of a strongly continuous semigroup (and even only a Hille-Yosida operator), the author uses a regularizing property of certain interpolation spaces, the theory of Hille-Yosida operators, a method of homogenization and the theory of extrapolation spaces. If \((E,\|\cdot\|)\) is a Banach space and \(\Lambda: D(\Lambda)\subset E\to E\) is a generator of the strongly continuous semigroup \(e^{\Lambda t}\), the Favard class of \(e^{\Lambda t}\) is defined by \[ F:= \text{Fav}(e^{\Lambda t}):= \{x\in E: [x]_F:= \sup_{t>0} (\| e^{\Lambda t}x- x\|/ t)< \infty\} \] with the norm \(\| x\|_F= \| x\|+ [x]_F\). The first result states that if \((F_*,\|\cdot\|_*)\) is a Banach space with \(F_*\hookrightarrow F\), \(f\in L^1(F_*)\) and \(\nu_0\in D(\Lambda)\), then there exists a solution of (1), \(u\in C(D(\Lambda))\) differentiable for \(t\geq 0\) a.e. Moreover, if \(f\in L^1(F_*)\cap C(E)\) then the solution is unique in the class \(C(D(\Lambda))\cap C^1(E)\). On the other hand, if we set \(\Lambda_F: D(\Lambda_F)\subset F\to F\) with \(D(\Lambda_F)= \{x\in D(\Lambda):\Lambda x\in F\}\), then even if \(\Lambda_F\) is only a Hille-Yosida operator we obtain regularity results in the following cases: (i) \(f\in W^{1,1}(F)\), \(u_0\in D(\Lambda)\), \(\Lambda u_0\in F\), (ii) \(f(t)\in D(\Lambda)\), \(t\geq 0\) a.e.; \(f\), \(\Lambda f\in L^1(F)\), \(u_0\in D(\Lambda^2)\), (iii) \(f(t)\in D(\Lambda)\), \(t\geq 0\) a.e.; \(f\in C(F)\); \(\Lambda f\in L^1(F)\), \(u_0\in D(\Lambda^2)\). A method is used for reducing the nonhomogeneous (1) to a homogeneous one in a suitable product space to obtain regularity results. Finally, the author recalls the definitions for the extrapolation space \(X_{-1}\) of the generator \(\Lambda: D(\Lambda)\subset X\to X\) of a bounded semigroup \(T(t)\) and respectively for an appropriate extrapolated semigroup \(T_{-1}\) and he proves a regularity result for the problem \[ u'(t)= \Lambda_{-1}u(t)+ f(t),\quad t\geq 0,\quad \nu(0)= u_0, \] where \(\Lambda_{-1}: D(\Lambda_{-1})\subset X_{-1}\to X_{-1}\) is a generator of \(T_{-1}\).