Multilinear strongly singular integral operators on non-homogeneous metric measure spaces

Let (X,d,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(X,d,\mu )$\end{document} be a non-homogeneous metric measure space satisfying the geometrically and upper doubling measure conditions. In this paper, the boundedness in Lebesgue spaces for multilinear strongly singular integral operators on non-homogeneous metric measure spaces is proved. As an application, the boundedness in Morrey spaces for multilinear strongly singular integral operators is also obtained.


Introduction and main results
It is well known that a metric measure space (X, d, μ) equipped with a non-negative doubling measure μ is called a space of homogeneous type. μ is said to satisfy the doubling condition if there exists a constant C > 0 such that μ(B(x, 2r)) ≤ Cμ(B(x, r)) for all x ∈ supp μ and r > 0. In the case of non-doubling measures, a non-negative measure μ only should satisfy the polynomial growth condition, i.e., for all x ∈ R n and r > 0, there exists a constant C 0 > 0 and k ∈ (0, n] such that μ B(x, r) ≤ C 0 r k , (1.1) where B(x, r) = {y ∈ R n : |y -x| < r}. This breakthrough brings rapid development in harmonic analysis (see [14,15,31,34,35,37,38] and their therein). And the analysis of nondoubling measures has important applications in solving the long-standing open Painlevé problem (see [35]).
Hytönen [17] stated that the measure satisfying (1.1) does not include the doubling measure as a special case. He introduced non-homogeneous metric measure spaces (X, d, μ), satisfying the geometrically and upper doubling measure conditions (see Definition 1.1 and 1.2). The highlight of this kind of spaces is that it includes both the homogeneous and metric spaces with polynomial growth measures as special cases. From then on, some results on non-homogeneous metric measure spaces were obtained. Hytönen et al. [20] and Bui and Duong [3] independently introduced the atomic Hardy space H 1 (μ) and proved that the dual space of H 1 (μ) is RBMO(μ). In [3], the authors also proved that the Calderón-Zygmund operator and commutators of the Calderón-Zygmund operators and RBMO functions are bounded in L p (μ) for 1 < p < +∞. Recently, some equivalent characterizations have been established by Liu et al. [29] for the boundedness of Carderón-Zygmund operators on L p (μ) for 1 < p < +∞. In [9], Fu et al. established boundedness of multilinear commutators of the Calderón-Zygmund operators on the Orlicz spaces on nonhomogeneous spaces. More results on non-homogeneous metric measure spaces have also been obtained in [4,5,10,11,[18][19][20][21][22][23][24] and the references therein.
Some researchers considered the theory of multilinear singular integral operators; for example, in [7], Coifman and Meyers firstly established the theory of bilinear Calderón-Zygmund operators. Later, Grafakos and Torres [12,13] demonstrated the boundedness of multilinear singular integral on the product Lebesgue spaces and Hardy spaces. The boundedness of multilinear singular integrals and commutators on non-doubling measures spaces (R n , μ) was established by Xu [38,39]. Weighted norm inequalities for multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces were also constructed in [16]. The boundedness for commutators of multilinear Calderón-Zygmund operators and multilinear fractional integral operators on non-homogeneous metric measure spaces was also established in [11,36].
The introduction of the strongly singular integral operator is motivated by a class of multiplier operators whose symbol is given by e i|ξ |α /|ξ |β away from the origin, where 0 < α < 1 and β > 0. Fefferman and Stein [8] enlarged the multiplier operators onto a class of convolution operators. Coifman [6] also considered a related class of operators for n = 1. The strongly singular non-convolution operators were introduced and researched by Alvarez and Milman [1,2], whose properties are similar to those of Calderón-Zygmund operators, but the kernel is more singular near the diagonal than those of the standard case. Furthermore, Lin and Lu [25][26][27][28] obtained the boundedness for the strongly singular integral and its commutators on Lebesgue spaces, Morrey spaces, and Hardy spaces.
In this paper, we first introduce the multilinear strongly singular integral operators on non-homogeneous metric spaces. Then we will also prove that it is bounded in m-multiple Lebesgue spaces, provided that multilinear strongly singular integrals are bounded from m-multiple L 1 (μ)×· · ·×L 1 (μ) to L 1/m,∞ (μ), where L p (μ) and L p,∞ (μ) denote the Lebesgue space and weak Lebesgue space, respectively. As an application, the boundedness in Morrey spaces for multilinear strongly singular integral on non-homogeneous metric spaces is obtained. A variant of sharp maximal operator M , Kolmogorov's theorem and some good properties of the dominating function λ (see Definition 1.2) are the main tools for proving the results in this paper.
Before stating the main results of this paper, we first recall some notations and definitions.  ([17]) A metric measure space (X, d, μ) is said to be upper doubling if μ is a Borel measure on X and there exists a dominating function λ : X × (0, +∞) → (0, +∞), and a constant C λ > 0 such that for each x ∈ X, r − → (x, r) is non-decreasing, and for all x ∈ X, r > 0, (1.2) Remark 1.3 (i) A space of homogeneous type is a special case of upper doubling spaces, where one can take the dominating function λ(x, r) = μ (B(x, r)). On the other hand, a metric space (X, d, μ) satisfying the polynomial growth condition (1.1) (in particular, (X, d, μ) = (R n , | · |, μ) with μ satisfying (1.1) for some k ∈ (0, n])) is also an upper doubling measure space if we take λ(x, r) = Cr k .
(ii) Let (X, d, μ) be an upper doubling space and λ be a dominating function on X × (0, +∞) as in Definition 1.2. In [20], it was showed that there exists another dominating functionλ such that for all x, y ∈ X with d(x, y) ≤ r, Thus, in this paper, we suppose that λ always satisfies (1.3).
Remark 1.5 As pointed out in Lemma 2.3 in [3], there exist plenty of doubling balls with small radii and with large radii. For the rest of this paper, unless α and β are specified otherwise, by an (α, β) doubling ball, we mean a (6, β 0 )-doubling with a fixed number β 0 > max{C 3 log 2 6 λ , 6 n }, where n = log 2 N 0 is viewed as a geometric dimension of the space.
, (1.4) where x B and r B are center and radius of B, respectively.
, then there exists a constant C > 0 such that As an application of the main result in this paper, we obtain the following result. (1.8). Assume that 1 < q 1 ≤ p 1 < +∞, 1 < q 2 ≤ p 2 < +∞,

Theorem 1.12 Let T be defined by
Throughout the paper, C denotes a positive constant independent of the main parameters involved, but it may be different in different places.

Proof of main results
To prove Theorem 1.11, we first give some notations and lemmas.
Let f ∈ L 1 loc (μ), the sharp maximal operator is defined as follows And the non-centered doubling maximal operator is denoted as follows For any 0 < δ < 1, we define that We can obtain that for any f ∈ L 1 loc (μ), for μa.e. x ∈ X (see [36]). Let ρ > 1 and 1 < p < ∞, the non-centered maximal operator M (ρ) f is defined as follows The operator M (ρ) f is bounded on L p (μ) for ρ ≥ 5 and p > 1 (see [3]). (2.4)

Lemma 2.2 ([38])
Let (X, d, μ) be probability measure spaces and let 0 < s < t < +∞, then there exists a constant C such that for any measurable function f , y). Next, we consider two cases for proving the result. Case 1: l(B) = l ≥ 1. As in the proof of Theorem 9.1 in [34], to obtain (2.6), it suffices to show that holds for any x and ball B with x ∈ B, and where B is an arbitrary ball, and Q is a doubling ball. For any ball B, we denote Since First, we estimate I 1 . Applying Lemma 2.2 with s = δ (0 < δ < 1/2) and t = 1/2, we get For I 2 , let z, y ∈ B, z 1 ∈ 6 5 B and z 2 ∈ X\ 6 5 By Definition 1.7 and the properties of λ, we deduce Therefore, Similar to estimate I 2 , we immediately obtain Let us move on to I 4 estimate. By Definition 1.7, we have To estimate I 41 , we take into account the properties of λ and the fact of r ≥ 1 and 0 < α < 1, so we have For I 42 , by the properties of λ, we have Thus, By the above estimate, we obtain (2.7). Next, we prove (2.8). Consider two balls B ⊂ Q with x ∈ B, where B is an arbitrary ball, and Q is a doubling ball. Let N = N B,Q + 1, then we obtain For the estimate J 1 , by the condition (1.5) in Definition 1.7 and the properties of λ, we obtain T(f 1 χ 6 5 B , f 2 χ 6 N 6 Also, Then, Let us estimate J 2 . By the condition (1.5) in Definition 1.7 and the properties of λ, we have For J 22 , since z ∈ B and z i ∈ 6 × 6 5 B\ 6 5 B, then 1 5 r B ≤ d(z, z i ) ≤ 41 5 r B for i = 1, 2. Therefore, For J 21 , by the properties of λ, we have For J 3 , similar to estimate I 1 , we have For J 4 and J 5 , similar to estimate I 2 , we have By a similar method to estimate J 1 , we can obtain that By a similar method to estimate J 2 , we also obtain that Hence, (2.8) is proved. Thus, Lemma 2.3, in this case, is proved. Case 2: 0 < l(B) = l < 1. Assume that B 0 and Q 0 are concentric with B and Q, respectively, and l(B 0 ) = l(B) α , l(Q 0 ) = l(Q) α . As in the proof of Theorem 9.1 in [34], to obtain (2.6), it suffices to show that holds for any x and ball B with x ∈ B, and where we split each f i as Since We first estimate L 1 . Applying Lemma 2.2 with s = δ (0 < δ < 1/2) and t = 1/2, we get For L 2 , let z, y ∈ B, z 1 ∈ 6 5 B 0 and z 2 ∈ X\ 6 5 B 0 , then max 1≤i≤2 d(z, z i ) ≥ d(z, z 2 ) ≥ Cl(B 0 ) = Cl(B) α ≥ Cd(z, y) α . By Definition 1.7 and the properties of λ, we obtain Therefore, Similar to estimate L 2 , we also obtain that Let us turn to estimate L 4 . Write To estimate L 41 , by the properties of λ, we obtain For L 42 , by the properties of λ, we have Thus, By the above estimate, we obtain (2.9).
Next, we prove (2.10). Consider two balls B ⊂ Q with x ∈ B, where B is an arbitrary ball, and Q is a doubling ball. Denote N = N B,Q + 1. Recall that B 0 and Q 0 are concentric with B and Q, respectively, and l(B 0 ) = l(B) α , l(Q 0 ) = l(Q) α . Then, To estimate M 1 , by the condition (1.5) in Definition 1.7 and the properties of λ, we obtain Also, Then, Let us estimate M 2 . By the condition (1.5) in Definition 1.7 and the properties of λ, we obtain For M 22 , since z ∈ B 0 and z i ∈ 6 × 6 5 B 0 \ 6 5 B 0 , then 1 5 r B 0 ≤ d(z, z i ) ≤ 41 5 r B 0 for i = 1, 2. Therefore, For M 21 , by the properties of λ, we obtain For M 3 , similar to estimate L 1 , we have For M 4 and M 5 , similar to estimate L 2 , we have By a similar method to estimate M 1 , we also obtain that By a similar method to estimate M 2 , we also obtain that Hence, (2.10) is proved. Thus, the proof of Lemma 2.3 is completed.
Next, let us prove Theorem 1.12. We first prove the following lemma.
Now we give the proof of Theorem 1.12.
Proof of Theorem 1.12 Fixing a ball B ∈ X, we have The proof of Theorem 1.12 is finished.