Fractional integral operators on Herz spaces for supercritical indices

We consider the boundedness of fractional integral operators Iβ on Herz spaces K q (R ), where q ≥ n/β . We introduce a new function space that is a variant of Lipschitz space. Our results are optimal.


Introduction
Since Beurling [1] introduced the Beurling algebras and Herz [5] generalized these spaces, many studies have been done for Herz spaces (see, for example, [3], [4] and [9]).Li and Yang [7] considered the boundedness of fractional integral operators on Herz spaces.In this paper we consider the boundedness of fractional integral operators on Herz spaces for supercritical indices and we shall show our results are optimal.
The following notation is used: For a set E ⊂ R n we denote the Lebesgue measure of E by |E|.We denote the characteristic function of E by χ E .We write a ball of radius R centered at x 0 by B(x 0 , R) = {x; |x − x 0 | < R} , and write First we define homogeneous Herz spaces and fractional integral.Let 0 < p < ∞, 1 ≤ q < ∞ and α ∈ R 1 .Definition 1. Kα,p q (R n ) = {f ∈ L q loc (R n \ {0}); f Kα,p q < ∞}, where Definition 2.
The next proposition is well-known (see, for example, [14]).
Li and Yang [7] proved the following.

Lip
where the supremum is taken over all balls Q ⊂ R n .

We denote Lip
We consider the boudedness of I β on Kα,p q where q ≥ n/β .Lu and Yang [8] (see also [6]) proved the following.Proposition 5. Assume that q 1 ≥ n/β and 0 < p < n/β .Then However this result does not imply Proposition 4, because Proposition 5 is not applicable to the case p = q 1 .Note that K 0,q q = L q .In this paper we consider the theorem that is an extension of Proposition 4. By a simple observation we obtain the following result.We define CM O (central mean oscillation) introduced by García-Cuerva [3].Let 0 ≤ ε < 1.

CM O
The spaces CM O β−n/q−α are independent of p, but the operator norms I β Kα,p q →CMO β−n/q−α depend on p.Note that the modified fractional integrals I β are well-defined on Kα,p q where β − n/q − 1 < α.Compare with the condition (1).However this result is insufficient, because when α = 0 and p = q = n/β it says that Since BM O CM O 0 , it does not imply Proposition 4.
In the following, we consider the theorem that is consistent with Proposition 4 when α = 0 and p = q .
In Section 4, we shall show that I β is not bounded from K α,p q to Lip β−n/q in general, therefore we need to consider other function spaces.

Main Theorem
For our purpose we consider a variant of Lipschitz space.Our definition is the following.Let 0 ≤ ε < 1 and λ ∈ R 1 .Definition 6.
Such function spaces are introduced by Nakai et al. [10], [11], [2].They consider more general function spaces, that is, generalized Campanato spaces: Before we state our results, we investigate some properties of , Theorem 1 is obtained from Theorems 2 and 3 below.The following property is easily obtained from the definition.( 4) This observation is important for our theorems (see Remark below).We give some examples.
Our main result is the following.
Remark .The spaces Lip −α β−n/q are defined only for β−n/q < 1 .We shall show that the index β − n/q is optimal in Section 4, and also show that if α < 0 then I β is not bounded from Kα,p q to Lip β−n/q−α (see (5)).
Since L q ⊂ K 0,p q when q ≤ p, Corollary is an extension of Proposition 4.

Proofs
In order to prove Theorem 2 we prepare two lemmas.Lemma 1. Assume that 1 ≥ 1/s > max(1/q + α/n, 1/q).Then (6) Throughout this paper, C is a positive constant which is independent of essential parameters and not necessarily same at each occurrence.
Proof.We write Proof.We write Now we are in a position to prove Theorem 2. Proof.
Let k be the least integer such that Q ⊂ B(0, 2 k ).Note that ( 8) We consider three cases:

The case (i) or (ii).
Note that |Q| ≥ C2 kn in both cases.We write

First we estimate I
and obtain q by Lemma 2, and we have The case (iii).We write First we estimate By the same estimate as ( 9), we have Next we estimate I β f 2 .Let c 2 = I β f 2 (x 0 ).By the same estimate as (10) and Lemma 1 for s = 1, it follows that for any x ∈ Q , Finally we estimate I β f 3 .Let c 3 = I β f 3 (x 0 ).By the same estimate as f 2 , it follows that for any x ∈ Q , and we obtain The following result is known, see e.g., [2], [11] and [13].
Also we need the following Definition 7.
where the supremum is taken over all balls Q ⊂ R n .Now, we prove Theorem 3.
Proof.[Proof of Theorem 3.] It suffices to show that for α ≥ 0, 1/t = 1/q + α/n and 1 < s < t, since then by Proposition 6 we obtain For any ball Q , let k be the least integer such that Q ⊂ B(0, 2 k ).
We calculate the mean oscillation of I β f j on intervals B j = (2 j − 1, 2 j ): the center is 2 j − 1/2 and the radius is 1/2 .Since it follows that for x ∈ B j , and we have Therefore