Sharp bounds for multilinear fractional integral operators on Morrey type spaces: the endpoint cases

In this paper, the Strichartz’s result of the exponential integrability of fractional integral operators is improved. Also, we establish the endpoint boundedness of the multilinear fractional integrals acting on the multi-Morrey spaces. The conclusions relax the restriction that pi≠1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{i}\ne 1$$\end{document} for all i=1,…,m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,m$$\end{document} and extend some known results.


Introduction
Let R n be the n-dimensional Euclidean space and (R n ) m = R n × . . . R n . The multilinear fractional operator is defined by where 0 < α < mn, m ∈ N.
The multilinear fractional integral I α,m is a natural generalization of the classical fractional integral. Kenig and Stein [9] as well as Grafakos and Kalton [4] considered the boundedness of a family of related fractional integrals. In 2012, Iida, Sato, Sawano and Tanaka in [6] obtained the boundedness property of the Adams type for multilinear fractional integral operators. Let 0 < α < mn, 1 < p 1 , p 2 , . . . , p m < ∞, 0 < p ≤ p 0 < n α , 0 < q ≤ q 0 < ∞. Suppose that Then there exists a constant C > 0 such that where M q 0 q stands for the Morrey spaces. The right-hand side of (1.2) is named by "the multi-Morrey norm", which is strictly smaller than m-fold product of the Morrey norms.
An interesting question arises. Can we obtain the similar results for the endpoint cases, that is p 0 = n α , p 0 > n α and p i = 1 for some i ∈ {1, . . . , m}. In this paper, we prove that the questions above have a affirmative solution.
To state the main results of this paper, we need first to recall some necessary notations and notion.
Morrey spaces seem to describe precisely the boundedness property of fractional integral operators. Morrey spaces describe local regularity more precisely than L p spaces and can be seen as a complement of L p . In fact, Recall that the fractional integral operator (or the Riesz potential) I α , 0 < α < n, is given by
They also established some endpoint estimates for the multilinear fractional integral.
Theorem (cf. [13]) Let m ∈ N, 0 < α < mn, 1/ p = 1/ p 1 + . . . + 1/ p m = α/n with 1 < p i < ∞ for i = 1, . . . , m. Let B be a ball of radius R in R n and let f j ∈ L p j (B) be supported in B. Then there exist constants k 1 , k 2 depending only on n, m, α, p and the p j such that For the case p 0 ≥ n/α, we also study the boundedness for multilinear fractional integrals on spaces as BMO space and Lipschitz spaces. Campanato spaces are a useful tool in the regularity theory of PDEs due to their better structures, which allows us to give an integral characterization of the spaces of Hölder continuous functions.

Definition 2
Let 0 < p < ∞ and −n/ p < β < n. A locally integrable function f is said to belong to Campanato space C β,q if there exists a constant C > 0 such that for any ball B ⊂ R n , The Lipschitz (Hölder) and Campanato spaces are related by the following equivalences: The equivalence can be found in [3] for q = 1, [7] for 1 < q < ∞ and [15] for 0 < q < 1. Specially, C 0,q = BMO, the spaces of bounded mean oscillation. The crucial property of BMO functions is the John-Nirenberg inequality [8], where c 1 and c 2 depend only on the dimension. A well-known immediate corollary of the John-Nirenberg inequality as follows: In fact, the equivalence also holds for 0 < q < 1. See, for example, the work of Strömberg [12](or [5] and [16] for the general case). In addition, we also proved in [14] that f ∈ BMO if and only if for 0 < q < ∞, The main result of this paper are stated as follows.
For the case n/α < p 0 < ∞, we obtain the result as follows. We remark that when p 0 = ∞ in Theorem 1.3, the conclusion also holds. Indeed, the proof is similar to the case p 0 < ∞, moreover the proof is simpler.
If there is at least p i is equiv to 1, one has the following weak type estimate. Theorem 1.4 Let m ∈ N, 0 < α < mn and I α,m be as in (1.1). Let P = ( p 1 , . . . , p m ), 1 ≤ p j < ∞ for j = 1, . . . , m, 1/ p = 1/ p 1 + . . . + 1/ p m , 1/q = 1/q 1 + . . . + 1/q m with 1/q = 1/ p − α/n and 1/q 0 = 1/ p 0 − α/n. If p 0 < n/α and there is at least one q i which is equal to 1, we have Let |E| denote the Lebesgue measure of a measurable set E ⊂ R n . Throughout this paper, the letter C denotes constants which are independent of main variables and may change from one occurrence to another. B(x, r ) denotes a ball centered at x, with side length r .
For F 1 , by a direct argument, we see that where M denotes the multilinear Hardy-Littlewood maximal function For any x ∈ B = B(x 0 , R), there exist a ballB = B(x 0 ,C R) withC > 0 such that the functions f 1 , . . . , f m are supported inB. We conclude that for any y i ∈B, |x − y i | ≤ |x − x 0 | + |x 0 − y i | ≤ (C + 1)R and this shows that Combining (2.1) and (2.2), we obtain for any x ∈ B and 0 < δ ≤ (C + 1) √ m R, In particular, the choice of δ = (C + 1) √ m R yields for all x ∈ B, Therefore, the election of If we use the notation By exponentiating (2.5), we get . . . , f m )(x)| ≥ 1} and B 2 = B\B 1 . By the inequality (2.6), p . On the other hand, Thus, adding the integrals above over B 1 and B 2 , where k 2 = (exp{k 1 } + C 2 )ω n and ω n = |B(0, 1)|.

Proof of Theorem 1.2
The inequality (1.5) implies that for 0 < q < ∞, f ∈ BMO if and only if there exists a constant c B related to the ball B, such that Then, we need only to prove that for any ball B = B(x 0 , r ), On the other hand, for x, z ∈ B, (y 1 , . . . , y m ) ∈ c (x 0 , 2 m r ), by the direct calculation, we get Using this observation, we see that The inequality (2.10) shows that Combining with the estimates (2.9) and (2.11), we obtain the inequality (2.8).

Proofs of Theorems 1.3-1.4
Proof of Theorem 1. 3 For any x 1 , x 2 ∈ R n , and r = |x 1 − x 2 |, we write (3.1) The same argument as (2.9), we have For the last term II 3 , we also obtain Combining with the estimates (3.1),(3.2) and (3.3), we arrive at Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.