Weighted Lebesgue and central Morrey estimates for p-adic multilinear Hausdorff operators and its commutators

In this paper, we establish the sharp boundedness of p-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of p-adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained.


Introduction
The p-adic analysis in the past decades has received a lot of attention due to its important applications in mathematical physics as well as its necessity in sciences and technologies (see e.g. [2,3,6,12,22,23,24,33,34,35,36] and references therein). It is well known that the theory of functions from Q p into C play an important role in p-adic quantum mechanics, the theory of p-adic probability in which real-valued random variables have to be considered to solve covariance problems. In recent years, there is an increasing interest in the study of harmonic analysis and wavelet analysis over the p-adic fields (see e.g. [1,6,10,20,21,24]).
It is crucial that the Hausdorff operator is one of the important operators in harmonic analysis. It is closely related to the summability of the classical Fourier series (see, for instance, [13], [15], [17], and the references therein). Let Φ be a locally integrable function on R n . The matrix Hausdorff operator H Φ,A associated to the kernel function Φ is then defined in terms of the integral form as follows where A(y) is an n×n invertible matrix for almost everywhere y in the support of Φ. It is worth pointing out that if the kernel function Φ is chosen appropriately, then the Hausdorff operator reduces to many classcial operators in analysis such as the Hardy operator, the Cesàro operator, the Riemann-Liouville fractional integral operator and the Hardy-Littlewood average operator.
In 2010, Volosivets [37] introduced the matrix Hausdorff operator on the p-adic numbers field as follows where ϕ(t) is a locally integrable function on Q n p and A(t) is an n×n invertible matrix for almost everywhere t in the support of ϕ. It is easy to see that if ϕ(t) = ψ(t 1 )χ Z * p n (t) and A(t) = t 1 .I n (I n is an identity matrix), for t = (t 1 , t 2 , ..., t n ), where ψ : Q p → C is a measurable function, H ϕ,A then reduces to the p-adic weighted Hardy-Littlewood average operator due to Rim and Lee [31]. In recent years, the theory of the Hardy operators, the Hausdorff operators over the p-adic numbers field has been significantly developed into different contexts, and they are actually useful for p-adic analysis (see e.g. [7], [8], [16], [38]). It is known that the authors in [9] also introduced and studied a general class of multilinear Hausdorff operators on the real field defined by 3) for f = (f 1 , ..., f m ) and A = (A 1 , ..., A m ).
Motivated by above results, in this paper we shall introduce and study a class of p-adic multilinear (matrix) Hausdorff operators defined as follows. .., f m be measurable complex-valued functions on Q n p . The p-adic multilinear Hausdorff operator is defined by for f = f 1 , ..., f m .
Let b be a measurable function. We denote by M b the multiplication operator defined by M b f (x) = b(x)f (x) for any measurable function f . If H is a linear operator on some measurable function space, the commutator of Coifman-Rochberg-Weiss type formed by M b and H is defined by [M b , H]f (x) = (M b H − HM b )f (x). Analogously, let us give the definition for the commutators of Coifman-Rochberg-Weiss type of p-adic multilinear Hausdorff operator. Definition 1.2. Let Φ, A be as above. The Coifman-Rochberg-Weiss type commutator of p-adic multilinear Hausdorff operator is defined by .., b m and b i are locally integrable functions on Q n p for all i = 1, ..., m.
The main purpose of this paper is to study the p-adic multilinear Hausdorff operators and its commutators on the p-adic numbers field. More precisely, we obtain the necessary and sufficient conditions for the boundedness of H p Φ, A and H p Φ, A, b on the product of Lebesgue and central Morrey spaces with weights on p-adic field. In each case, we estimate the corresponding operator norms. Moreover, the boundedness of H p Φ, A, b on the such spaces with symbols in central BMO space is also established. It should be pointed out that all our results are new even in the case of p-adic linear Hausdorff operators.
Our paper is organized as follows. In Section 2, we present some notations and preliminaries about p-adic analysis as well as give some definitions of the Lebesgue and central Morrey spaces associated with power weights and Muckenhoupt weights. Our main theorems are given and proved in Section 3 and Section 4.

Some notations and definitions
For a prime number p, let Q p be the field of p-adic numbers. This field is the completion of the field of rational numbers Q with respect to the non-Archimedean p-adic norm | · | p . This norm is defined as follows: if x = 0, |0| p = 0; if x = 0 is an arbitrary rational number with the unique representation x = p α m n , where m, n are not divisible by p, α = α(x) ∈ Z, then |x| p = p −α . This norm satisfies the following properties: (i) |x| p ≥ 0, ∀x ∈ Q p , and |x| p = 0 ⇔ x = 0; (ii) |xy| p = |x| p |y| p , ∀x, y ∈ Q p ; (iii) |x + y| p ≤ max(|x| p , |y| p ), ∀x, y ∈ Q p , and when |x| p = |y| p , we have |x + y| p = max(|x| p , |y| p ).
It is also well-known that any non-zero p-adic number x ∈ Q p can be uniquely represented in the canonical series where α = α(x) ∈ Z, x k = 0, 1, ..., p − 1, x 0 = 0, k = 0, 1, .... This series converges in the p-adic norm since |x k p k | p ≤ p −k . The space Q n p = Q p × · · · × Q p consists of all points x = (x 1 , ..., x n ), where x i ∈ Q p , i = 1, ..., n, n ≥ 1. The p-adic norm of Q n p is defined by Let A be an n × n matrix with entries a ij ∈ Q p . For x = (x 1 , ..., x n ) ∈ Q n p , we denote a nj x j .
By Lemma 2 in paper [38], the norm of A, regarded as an operator from Q n p to Q n p , is For simplicity of notation, we write k A = log p A p . It is clear to see that k A ∈ Z. It is easy to show that |Ax| p ≤ A p .|x| p for any x ∈ Q n p . In addition, if A is invertible, by estimating as Lemma 3.1 in paper [29], then we get Let B α (a) = x ∈ Q n p : |x − a| p ≤ p α be a ball of radius p α with center at a ∈ Q n p . Similarly, denote by S α (a) = x ∈ Q n p : |x − a| p = p α the sphere with center at a ∈ Q n p and radius p α . If B α = B α (0), S α = S α (0), then for any x 0 ∈ Q n p we have x 0 + B α = B α (x 0 ) and x 0 + S α = S α (x 0 ). Since Q n p is a locally compact commutative group under addition, it follows from the standard theory that there exists a Haar measure dx on Q n p , which is unique up to positive constant multiple and is translation invariant. This measure is unique by normalizing dx such that where |B| denotes the Haar measure of a measurable subset B of Q n p . By simple calculation, it is easy to obtain that |B α (a)| = p nα , |S α (a)| = p nα (1 − p −n ) ≃ p nα , for any a ∈ Q n p . For f ∈ L 1 loc (Q n p ), we have In particular, if f ∈ L 1 (Q n p ), we can write For a more complete introduction to the p-adic analysis, we refer the readers to [22,36] and the references therein.
Let ω(x) be a weighted function, that is a non-negative locally integrable measurable function on Q n p . The weighted Lebesgue space L q ω (Q n p ) (0 < q < ∞) is defined to be the space of all measurable functions f on Q n p such that The space L q ω, loc (Q n p ) is defined as the set of all measurable functions f on Q n p satisfying K |f (x)| q ω(x)dx < ∞, for any compact subset K of Q n p . The space L q ω,loc (Q n p \ {0}) is also defined in a similar way as the space L q ω,loc (Q n p ). Throught the whole paper, we denote by C a positive geometric constant that is independent of the main parameters, but can change from line to line. We also write a b to mean that there is a positive constant C, independent of the main parameters, such that a ≤ Cb. The symbol f ≃ g means that f is equivalent to g (i.e. C −1 f ≤ g ≤ Cf ). For any real number ℓ > 1, denote by ℓ ′ conjugate real number of ℓ, i.e. 1 ℓ + 1 for λ ∈ R. Remark that if ω(x) = |x| α p for α > −n, then we have Next, let us give the definition of weighted λ-central Morrey spaces on p-adic numbers field as follows. (2.5) Remark that . B q,λ ω (Q n p ) is a Banach space and reduces to {0} when λ < − 1 q .
Let us recall the definition of the weighted central BMO p-adic space.
Definition 2.2. Let 1 ≤ q < ∞ and ω be a weight function. The weighted central bounded mean oscillation space CMO q ω (Q n p ) is defined as the set of all functions f ∈ L q ω,loc (Q n p ) such that The theory of A ℓ weight was first introduced by Benjamin Muckenhoupt on the Euclidean spaces in order to characterise the boundedness of Hardy-Littlewood maximal functions on the weighted L ℓ spaces (see [28]). For A ℓ weights on the p-adic fields, more generally, on the local fields or homogeneous type spaces, one can refer to [11,18] for more details. Let us now recall the definition of A ℓ weights.
It is said that a weight ω ∈ A 1 (Q n p ) if there is a constant C such that for all balls B, we get 1 |B| We denote by A ∞ (Q n p ) = 1≤ℓ<∞ A ℓ (Q n p ). Let us give the following standard result related to the Muckenhoupt weights.
is the reverse Hölder condition. If there exist r > 1 and a fixed constant C such that for all balls B ⊂ Q n p , we then say that ω satisfies the reverse Hölder condition of order r and write ω ∈ RH r (Q n p ). According to Theorem 19 and Corollary 21 in [19], ω ∈ A ∞ (Q n p ) if and only if there exists some r > 1 such that ω ∈ RH r (Q n p ). Moreover, if ω ∈ RH r (Q n p ), r > 1, then ω ∈ RH r+ε (Q n p ) for some ε > 0. We thus write r ω = sup{r > 1 : ω ∈ RH r (Q n p )} to denote the critical index of ω for the reverse Hölder condition.
An important example of A ℓ (Q n p ) weight is the power function |x| α p . By the similar arguments as Propositions 1.4.3 and 1.4.4 in [26], we obtain the following properties of power weights.
. Let us give the following standard characterization of A ℓ weights which it is proved in the similar way as the real setting (see [14,32] for more details).

Let us recall the definition of the Hardy-Littlewood maximal operator
Mf (x) = sup It is useful to remark that the Hardy-Littlewood maximal operator M is bounded on L ℓ ω (Q n p ) if and only if ω ∈ A ℓ (Q n p ) for all ℓ > 1. Finally, we introduce a new maximal operator which is used in the sequel, that is

The main results about the boundness of H p Φ, A
Let us now assume that q and q i ∈ [1, ∞), α, α i are real numbers such that α i ∈ (−n, ∞), for i = 1, 2, ..., m and 1 In this section, we will investigate the boundedness of multilinear Hausdorff operators on weighted Lebesgue spaces and weighted central Morrey spaces associated to the case of matrices having the important property as follows: for almost everywhere y ∈ Q n p . Thus, by the property of invertible matrice, it is easy to show that and Our first main result is the following.
Firstly, we will prove for the sufficient condition of the theorem. By applying the Minkowski inequality and the Hölder inequality, we have By making the change of variables, we get Thus, Note that, by (2.3) and (3.2), we have This shows that Next, to prove the necessary condition of this theorem, for i = 1, ..., m and r ∈ Z + , let us now take By a simple calculation, we have Next, we define two sets as follows In fact, by letting y ∈ U r , we have A i (y) p .|x| p ≥ 1, for all x ∈ Q n p \ B r−1 . Thus, by applying the condition (3.1), one has which confirms the relation (3.7). Now, by taking x ∈ Q n p \ B r−1 and using the relation (3.7), we get From this, by (3.3), one has . By estimating as (3.6) above, we also have As a consequence above, by (3.6), we find that Therefore, because H p Φ, A is bounded from L q 1 ω 1 (Q n p ) × · · · × L qm ωm (Q n p ) to L q ω (Q n p ), there exists M > 0 such that A r ≤ M, for r sufficiently big. On the other hand, by letting r sufficiently large, y ∈ U r and by (3.1), we get From this, by the dominated convergence theorem of Lebesgue, we obtain which finishes the proof of the theorem.
Theorem 3.2. Let 1 ≤ q * , ζ < ∞ and ω ∈ A ζ with the finite critical index r ω for the reverse Hölder and ω(B γ ) 1, for all γ ∈ Z. Assume that q > q * ζr ω /(r ω − 1), δ ∈ (1, r ω ) and the following condition holds: Proof. For any R ∈ Z, by the Minkowski inequality, we have From the inequality q > q * ζr ω /(r ω − 1), there exists r ∈ (1, r ω ) such that q = ζq * r ′ . Then, by the Hölder inequality and the reverse Hölder condition, we have Next, by making the Hölder inequality and the change of variables formula, and applying Proposition 2.7, we have where k A i (y) = log p A i (y) p . Thus, by On the other hand, by q > q * ≥ 1 and ω(B R ) 1 for all R ∈ Z, we imply that Next, for i = 1, ..., m, by using Proposition 2.6, we have Hence, by letting R → +∞ and applying the monotone convergence theorem of Lebesgue, we obtain that which completes the proof of the theorem. (α + n)λ = (α 1 + n)λ 1 + · · · + (α m + n)λ m . (3.10) We start with the proof for the sufficient condition of the theorem. For γ ∈ Z, by estimating as (3.4) and (3.5) above, we have This implies that On the other hand, by hypothesis (3.10), we immediately get

Hence, by (3.11), one has
. For i = 1, ..., m, let us choose the functions as follows Then, by (2.4), it is not difficult to show that and similarly, we also have (3.12) Next, by choosing f i 's and using (3.3) and (3.10), we have Thus, by (3.12), it follows that This gives that C 3 < ∞. Hence, the theorem is completely proved.

The main results about the boundness of H
Before stating our next results, we introduce some notations which will be used throughout this section. Let q, q i ∈ [1, ∞), and let α, α i , r i be real numbers such that r i ∈ (1, ∞), α i ∈ (−n, nr i r ′ i ), i = 1, 2, ..., m. Denote , for all i = 1, ..., m. Then, for any γ ∈ Z, we have Proof. By the Minkowski inequality and the Hölder inequality, for any γ ∈ Z, we get (4.1) To prove this lemma, we need to show that the following inequality holds , for all i = 1, ..., m.

(4.2)
We put By the definition of the space CMO r i ω i (Q n p ) and the estimation (2.4), we have To estimate I 2,i , we deduce that where k A i (y) = log p A i (y) p . Note that, by the formula for change of variables, we get Thus, by using the Hölder inequality and (2.4), it is clear to see that By using the formula for change of variables again, one has This leads to . Therefore, by (4.5) and (4.7), we have Next, we consider the term I 3,i . We have Fix y ∈ Q n p . We set As mentioned above, we have Combining the Hölder inequality, the definition of the space CMO r i ω i (Q n p ) and (2.4), one has In addition, by the Hölder inequality again and (4.6), we get . Consequently, by (4.10)-(4.12), it follows that . This together with (4.3), (4.4) and (4.9) follow us to have the proof of the inequality (4.2). Finally, by estimating as (4.8), we immediately have ) . In view of (4.1) and (4.2), the proof of this lemma is ended.
This together with (4.17) and (4.21) yields that the inequality (4.15) is finished. In other words, by estimating as (4.20) above, we also get Hence, by (4.14) and (4.15), we conclude that the proof of this lemma is finished.

Theorem 4.3. Let the assumptions of Lemma 4.1 hold and
Then, for any γ ∈ B γ , we have H p Φ, A, b is bounded from L q 1 ω 1 (Q n p )×· · ·×L qm ωm (Q n p ) to L q ω (B γ ). Proof. For any γ ∈ Z, by using Lemma 4.1, we infer , which shows that the proof of this theorem is completed.
Proof. In view of Lemma 4.2, for any R ∈ Z, it is clear to see that Next, by having inequality 1 and assuming ω(B R ) 1 for any .
. spaces with the Muckenhoupt weights, we have that This together with (4.25) yields that the proof of this theorem is completed.
In what follows, we set Proof. Firstly, we prove the part (i) of the theorem. For any R ∈ Z, by Lemma 4.1, we get     Note that, by the hypothesis (4.26), we see that |log p A i (y) p | ≥ 1, for all y ∈ supp(Φ).

Hence, one has
Therefore, since H p Φ, A, b is bounded from . B q 1 ,λ 1 ω 1 (Q n p ) × · · · × . B qm,λm ωm (Q n p ) to . B q,λ ω (Q n p ), it implies that C 8 < ∞. This leads to that the theorem is completely proved. Now, we consider A i (y) = s i (y).I n , for i = 1, ..., m. By the similar arguments, we then obtain the following useful result.
From this, by (3.14), we have , which implies that the proof of this theorm is finished.