Suppose β, . Consider the fractional space and the Hölder space . In this section, we prove the following structure theorem.
Theorem 5 The norms of the spaces and are equivalent.
Proof Assume that . Let and be fixed. From formula (2) it follows that
(10)
Using equation (10), the triangle inequality, the following inequalities
(11)
(12)
estimates (10), (11), (12), and the definition of -norm, we obtain
(13)
Thus, it follows from estimate (13) that
(14)
Let and be fixed. Using equation (10), we can write
(15)
Now, we will estimate the right-hand side of equation (15). We consider two cases and , respectively. Let us first assume that . Furthermore, this situation will be considered in two cases: and . Let . From equation (15), the triangle inequality, the definition of -norm, the assumptions and , it follows that
(16)
We will estimate , , separately.
First, let us estimate . Clearly, by the assumption , we have
From estimates (3), (11), (12), and the assumption , it follows that
(18)
Estimates (3), (11), (12), and the assumption yield that
(19)
Combining estimates (17)-(19), we get
(20)
Now, let us consider the case . Then, using equation (10), we can write
(21)
From equation (21), the triangle inequality, the Lagrange theorem, the definition of -norm, and the assumption , it follows that for some between , , and between , ,
We will estimate , , separately. Let us start with . Clearly, we have
Using estimates (3), (4), (11), (12), we obtain
(23)
Note that by the triangle inequality and the fact that between , , we get
(24)
(25)
By inequalities (11), (12), estimates (4), (24), (25), and Hilbert’s inequality, we have
(26)
Similarly, we get
Combining estimates (22), (23), (26), (27), we obtain that for , ,
(28)
It follows from estimates (20) and (28) that for , ,
(29)
Next, let us assume that . By equation (10), we have
(30)
Equation (30), estimates (3), (11), Hilbert’s inequality, the triangle inequality, the definition of -norm, and the assumption yield that for ,
(31)
From estimates (29) and (31) it follows that
(32)
Combining estimates (14) and (32), we obtain
Now, we will prove that . By Theorem 4, A is a positive operator in the Banach space . Hence, for , we have
(33)
Let . It follows from formula (2) and equation (33) that
(34)
Using the triangle inequality, equation (34), estimate (3), and the definition of -norm, we obtain
Thus,
(35)
Let be fixed. From equation (34) it follows that
(36)
Now, we will estimate the right-hand side of equation (36). We consider two cases and . Let us first assume that . By equation (36), the triangle inequality, the Lagrange theorem, the definition of -norm, and the assumption , we have, for some between , , and between , , that
We will estimate , , separately. Using the triangle inequality, estimates (4), (11), (25), (24), (12), the Lagrange theorem, and the assumption , we obtain
From the triangle inequality, estimates (4), (11), (12), (24), (25), the Lagrange theorem, and the assumption , it follows that
Using the triangle inequality, estimates (3), (24), (25), (12), (11), the assumption , and the following estimate
we obtain
Thus, for , we have
(37)
Next, let us assume that . Using formula (36), estimate (3), the triangle inequality, Hilbert’s inequality, and the assumption , we get
Hence, for , we obtain
(38)
Combining estimates (37) and (38), we get
(39)
Estimates (35) and (39) yield that
This is the end of the proof of Theorem 5. □