Weighted multilinear p-adic Hardy operators and commutators

Definition 1.1

([13]). Let 1 ≤ q < ∞ and λ ≥ −1/q. The p-adic Morrey space 𝓛^q,λ( Qpn ) is defined by

where

Remark 1.2

It is clear that Lq,−1/q(Qpn)=Lq(Qpn),Lq,0(Qpn)=L∞(Qpn).

In 2017, Wu and Fu [14] proved sufficient and necessary conditions of weighted functions, for which the weighted p-adic Hardy operators are bounded on p-adic central Morrey spaces.

The p-adic central Morrey space is defined as follows.

Definition 1.3

Let λ ∈ ℝ and 1 < q < ∞ . The p-adic central Morrey space B^q,λ( Qpn ) is defined by

where B_γ = B_γ(0). It is clear that Bq,−1q(Qpn)=Lq(Qpn), when λ < −1/q, the space B^q,λ( Qpn ) reduces to {0}, therefore, we can only consider the case λ ≥ −1/q. If 1 ≤ q₁ ≤ q₂ < ∞, by Hölder’s inequality

λ ∈ ℝ.

Definition 1.4

([10]). Let 1 ≤ q < ∞. A function f∈Llocq(Qpn) is said to be CMO^q( Qpn ), if

where

The study of multilinear averaging operators is traced to the multilinear singular integral operator theory [15], and motivated not only the generalization of the theory of linear ones but also their natural appearance in analysis. For a more complete account on multilinear opeartors, we refer to [16, 17, 18, 19] and the references therein.

In this paper, we consider the multilinear version of weighted p-adic Hardy operators in the p-adic fields. Firstly, we introduce the weighted multilinear p-adic Hardy operators as follows.

Definition 1.5

Let m ∈ ℕ, x ∈ Qpn , and φ be a nonnegative integrable function on Zp∗×Zp∗×⋯×Zp∗. The weighted multilinear p-adic Hardy operator Hφ,mp is defined as

where f⃗: = (f₁,…,f_m), t⃗: = (t₁,…,t_m), dt⃗: = dt₁⋯ dt_m, and f_i (i = 1,…,m) are measurable functions on Qpn . When m = 1, Hφ,mp is reduced to the weighted p-adic Hardy operators Hφp .

The outline of the paper is as follows. In Section 2, we furnish sharp estimate of weighted multilinear p-adic Hardy operator on the product of p-adic Lebesgue spaces, and then the result is extended to the product of p-adic central Morrey spaces, the product of p-adic Morrey spaces, respectively. In Section 3, we present the boundedness of commutators of the weighted multilinear p-adic Hardy operators.

Theorem 2.1

Let 1 < q,q_i < ∞, i = 1,…,m and 1q=1q1+⋯+1qm. Then Hφ,mp is bounded from Lq1(Qpn)×Lq2(Qpn)×⋯×Lqm(Qpn)toLq(Qpn) if and only if

Moreover,

Proof

Without loss of generality, we consider only the situation when m = 2. Actually, a similar procedure works for all m ∈ ℕ.

Suppose that (1) holds. Using Minkowski’s inequality yields

By Hölder’s inequality with 1q=1q1+1q2, we see that

Thus, Hφ,2p maps the product of p-adic Lebesgue spaces Lq1(Qpn)×Lq2(Qpn) to Lq(Qpn) and

To see the necessity, for any 0 < ε < 1 and |ε|_p > 1, we take

An elementary calculation gives that

Consequently, we have

Therefore,

Now take ε = p^−k, k = 1,2,⋯. Then |ε|_p = p^k > 1. Letting k approach to ∞, then ε approaches to 0 and |ε|pεq2=pkq2pk approaches to 1. Then by Fatou’s Lemma, we obtain

and

Combining (2) and (4) then finishes the proof. □

Next, we extend the result in Theorem 2.1 to the product of p-adic central Morrey spaces.

Theorem 2.2

Let 1 < q < q_i < ∞, 1q=1q1+⋯+1qm, λ = λ₁+⋯+λ_m and −1/q_i ≤ λ_i < 0, i = 1,…,m.

1. If

   A~m:=∫(Zp∗)m∏i=1m|ti|pnλiφ(t→)dt→<∞. (5)

   Then, Hφ,mp is bounded from Bq1,λ1(Qpn)×Bq2,λ2(Qpn)×⋯×Bqm,λm(Qpn)toBq,λ(Qpn) with its operator norm not more that 𝓐͠_m.

2. Assume that λ₁q₁ = ⋯ = λ_mq_m. In the case the condition (5) is also necessary for the boundedness of Hφ,mp:Bq1,λ1(Qpn)×Bq2,λ2(Qpn)×⋯×Bqm,λm(Qpn)→Bq,λ(Qpn). Moreover,

   ∥Hφ,mp∥Bq1,λ1(Qpn)×Bq2,λ2(Qpn)×⋯×Bqm,λm(Qpn)→Bq,λ(Qpn)=A~m.

Proof

By similarity, we only give the proof in the case m = 2.

When −1/q_i = λ_i, i = 1,2, then Theorem 2.2 is just Theorem 2.1.

Next we consider the case that −1/q_i < λ_i < 0, i = 1,2. Let γ ∈ ℤ, t_iB_γ = B(0,|t_i|_pp^γ) and 𝓐͠₂ < ∞. Since 1/q = 1/q₁+1/q₂, by Minkowski’s inequality and Hölder’s inequality, we see that, for all balls B = B(0,p^γ),

This means that

For the necessity when λ₁q₁ = λ₂q₂, let f1(x)=|x|pnλ1 and f2(x)=|x|pnλ2 for all x ∈ Qpn ∖{0}, and f₁(0) = f₂(0): = 0. Then for any B = B(0,p^γ), we have

where the series converge due to λ_i > −1/q_i. Then f_i ∈ B^q_i,λ_i( Qpn ). Since λ = λ₁+λ₂ and −1/q_i ≤ λ_i < 0, 1 < q < q_i < ∞, i = 1,2, we have

since λ₁q₁ = λ₂q₂. Then,

Combining (6) and (7) then concludes the proof. This finishes the proof of the Theorem 2.2.

We remark that Theorem 2.2 when m = 1 goes back to [14] Theorem 2.3. □

Next, we give sharp estimate of weighted multilinear p-adic Hardy operator on the product of p-adic Morrey spaces.

Theorem 2.3

Let 1 < q < q_i < ∞, 1q=1q1+⋯+1qm, λ = λ₁ + ⋯ + λ_m and −1/q_i < λ_i < 0, i = 1, …, m.

1. If

   Bm:=∫(Zp∗)m∏i=1m|ti|pnλiφ(t→)dt→<∞. (8)

   Then, Hφ,mp is bounded from Lq1,λ1(Qpn)×Lq2,λ2(Qpn)×⋯×Lqm,λm(Qpn)toLq,λ(Qpn) with its operator norm not more that 𝓑_m.

2. Assume that λ₁q₁ = ⋯ = λ_mq_m. In the case the condition (8) is also necessary for the boundedness of Hφ,mp:Lq1,λ1(Qpn)×Lq2,λ2(Qpn)×⋯×Lqm,λm(Qpn)→Lq,λ(Qpn). Moreover,

   ∥Hφ,mp∥Lq1,λ1(Qpn)×Lq2,λ2(Qpn)×⋯×Lqm,λm(Qpn)→Lq,λ(Qpn)=Bm.

Proof

By similarity, we only give the proof in the case m = 2. Suppose 𝓑₂ < ∞. Since 1/q = 1/q₁ + 1/q₂, by Minkowski’s inequality and Hölder’s inequality, we see that

This means that

For the necessity when λ₁q₁ = λ₂q₂, let f1(x)=|x|pnλ1 and f2(x)=|x|pnλ2 for all x ∈ Qpn \ {0}, and f₁(0) = f₂(0): = 0. Then for any B = B(a,p^γ), we need to show that f_i ∈ 𝓛^q_i,λ_i( Qpn ). Considering the following two cases.

1. If |a|_p > p^γ and x ∈ B_γ(a), then |x|_p = max{|x − a|_p, |a|_p} > p^γ. Since −1/q_i ≤ λ_i < 0, we have

   1|Bγ(a)|H1+λiqi∫Bγ(a)|x|pnλiqidx<1|Bγ(a)|H1+λiqi∫Bγ(a)pγnλiqidx=1.

2. If |a|_p ≤ p^γ and x ∈ B_γ(a), then |x|_p = max{|x − a|_p, |a|_p} ≤ p^γ. Therefore, x ∈ B_γ(a). Recall that two balls in Qpn are either disjoint or one is contained in the other [20]. So we have B_γ(a) = B_γ, thus

   1|Bγ(a)|H1+λiqi∫Bγ(a)|x|pnλiqidx=1|Bγ|H1+λiqi∫Bγ|x|pnλiqidx=1−p−n1−p−n(1+λiqi).

   From the previous discussion, we can see that f_i ∈ 𝓛^q_i,λ_i( Qpn ). By the similar estimates to the method of Theorem 2.2, we have

   B2≤∥Hφ,2p∥Lq1,λ1(Qpn)×Lq2,λ2(Qpn)→Lq,λ(Qpn)<∞. (10)