Recently there are increasing interests on fractional differential equations due to their wide applications in viscoelasticity, dynamics of particles, economic and science et al. For more details we refer to  .
Many evolution equations can be rewritten as an abstract Cauchy problem, and then they can be studied in an unified way. For example, a heat equation with different initial data or boundary conditions can be written as a first order Cauchy problem, in which the governing operator generates a C0-semigroup, and then the solution is given by the operation of this semigroup on the initial data. See for instance  . Prüss  developed the theory of solution operators to research some abstract Volterra integral equations and it was Bajlekova  who first use solution operators to discuss the fractional abstract Cauchy problems. If the coefficient operator of a fractional abstract Cauchy problem generates a C0-semigroup, we can invoke an operator described by the C0-semigroup and a probability density function to solve this problem, for more details we refer to   . The vector-valued Laplace transform developed in  is an important tool in the theory of fractional differential equations.
There are some papers devoted to the fractional differential equations in many different respects: the connection between solutions of fractional Cauchy problems and Cauchy problems of first order ; the existence of solution of several kinds of fractional equations  ; the Hölder regularity for a class of fractional equations  ; the maximal regularity for fractional order equations ; the boundary regularity for the fractional heat equation ; the relation of continuous regularity for fractional order equations with semi-variations . In this paper we are mainly interested in the Hölder regularity for abstract Cauchy problems of fractional order.
Pazy  considered the regularity for the abstract Cauchy problem of first order:
where A is the infinitesimal generator of an analytic C0-semigroup. He showed that if for some , then is Hölder continuous with
exponent in ; if moreover , then u is Hölder continuous
with the same exponent in . If in addition f is Hölder continuous, then
Pazy showed that there are some further regularity of and . Li 
gave similar results for fractional differential equations with order . In this paper we will extend their results to fractional Cauchy problems with order in .
Our paper is organized as follows. In Section 2 there are some preliminaries on fractional derivatives, fractional Cauchy problems and fractional resolvent families. In Section 3 we give the regularity of the mild solution under the condition that . And some further continuity results are given in Section 4.
Let A be a closed densely defined linear operator on a Banach space X. In this paper we consider the following equation:
where u and f are X-valued functions, , and is the Caputo fractional derivative defined by
in which for ,
and is understood as the Dirac measure at 0. The convolution of two functions f and g is defined by
when the above integrals exist.
The classical (or strong) solution to (2.1) is defined as:
Definition 2.1. If , is called a solution of (2.1) if
3) u satisfies (2.1) on .
By integration (2.1) for α-times, we are able to define a kind of weak solutions.
Definition 2.2. If , is called a mild solution of (2.1) if for every and
And it is therefore natural to give the following definition of α-resolvent family for the operator A.
Definition 2.3. A family is called an α-resolvent family for the operator A if the following conditions are satisfied:
1) is continuous for every and ;
2) and for all and ;
3) the resolvent equation
holds for every .
If there is an α-times resolvent family for the operator A, then the mild solution of (2.1) is given by the following lemma.
Lemma 2.4.  Let A generate an α-times resolvent family and let . If (2.1) has a mild solution, then it is given by
For the strong solution of (2.1), we have
Lemma 2.5.  Let A generate an α-times resolvent family and let , . If , then the following statements are equivalent:
(a) (2.1) has a strong solution on .
(b) is differentiable on .
(c) for and is
continuous on .
If in addition, the α-times resolvent family admits an analytic extension to some sector , and for all , we will then denote it by .
If , then there exists constants C, and such that and
for each . The α-times resolvent family generated by A can be given by
is oriented counter-clockwise. And the corresponding operators are defined by
Lemma 2.6. Let and . We have
(1) for every and for ;
(2) for every , and for ;
(3) for , for any integer and for , where .
Proof. (1) By the definition of and (2.2),
taking , we can obtain that the above integral is bounded by
Analogously one can show the estimate
It thus follows the estimate for .
(2) By the identity , we have
since . Moreover,
By the closedness of the operator A, the assertion of (2) follows.
(3) By the proof of (2) and the closedness of A,
And the second part of (3) can be proved similarly. □
Remark 2.7. Similar results for were given in . It is obvious that
3. Regularity of the Mild Solutions
In this section we consider the mild solution of (2.1) with . Suppose that the operator A generates an analytic α-resolvent family, then by Lemma 2.4 and Remark 2.7 the mild solution of (2.1) is given by
Theorem 3.1. Let , , and with .
Then for every and , , where is given
by (3.1). If moreover such that , then .
Proof. Since is analytic, we only need to show that .
Let and , then
By Hölder’s inequality and Lemma 2.6,
We remark that the constant C here and in the sequel may be vary line by line, but not depending on t and h. Next, we estimate . For , , first assume that ,
since; if, then
from which it follows also that.
If with, then by [, Lemma 4.5] we have that is differentiable and thus Lipschitz continuous. □
If we put more conditions on, the regularity of can be raised.
Proposition 3.2. Let, and with. For every, we define the function by
If exists, then for every,. If moreover, then.
Proof. If satisfies the assumption, by [, Theorem 13.2] there exists a function such that. Thus
Since is analytic and bounded,. It is easy to see
that is Hölder continuous with index. This completes the
4. Regularity of the Classical Solutions
Motivated by the results in  for the C0-semigroups, we first give the following proposition.
Proposition 4.1. Let and. Assume, and. Then the mild solution of (2.1) is the classical solution.
Proof. By Lemma 2.5 we only need to show that for every and is continuous on. We decompose into two parts:
belongs to and is continuous. To prove that, we define the following functions:
for small enough. It is clear that as. Moreover.
for all, it follows from the fact that for a fix the map is a continuous mapping, we conclude that
By our assumption and Lemma 2.6 there exists a constant such that
consequently, the function is integrable. Hence by the closedness of A we obtain that
The continuity of the function follows directly from the fact
This completes the proof. □
We will then give the regularity of such classical solutions.
Lemma 4.2. Let with, and with. Define
Then for any, and.
Proof. For fixed, since we have
By the closedness of A, we obtain. Thus it remains to show that. For and we have the following decomposition:
Since we have
We can estimate as follows:
And it is easy to show that. Combining all above we have. □
The following theorem extends [, Theorem 4.4] to the case that.
Theorem 4.3. Let with, with, , and u is the classical solution of (2.1). The following assertions hold.
(1) For every, ,.
(2) If moreover, then and are continuous on.
(3) If, then and.
Proof. (1) If u is the classical solution of (2.1) on, then. It is only need to prove We decompose
By Lemma 4.2,. Let. If, then
thus we have
(2) We only need to show that is continuous at. Since and is continuous,
(3) We show that. Indeed, this follows from
In this paper, we proved the Hölder regularity of the mild and strong solutions to the α-order abstract Cauchy problem (2.1) with. Our results are complemental to the existing results of Pazy  for the case and Li  for the case that.