Theorem 1. Let α be a complex number with 0 < Re(α) < n, 1/r = 1/q − Re(α)/n with 1 ≤ q < r < ∞, λ ≥ 0, λ − n/q + Re(α) < σ < n(1 − 1/q), 0 < l ≤ s ≤ ∞. If q > 1, then, there exists a constant C > 0 such that

Moreover, if q = 1, then, there exists a constant C > 0 such that

Remark 2. The definitions of the homogeneous Morrey-Herz spaces $$ and weak homogeneous Morrey-Herz spaces $$ will be given in Section 2, here, we point out that $$ for 1 ≤ r < ∞. We observe that Theorem 1 is counterparts of famous Hardy-Littlewood-Sobolev theorem on p-adic Morrey-Herz spaces. In addition, Theorem 1 coordinates with those on Morrey-Herz spaces in the setting of ℝⁿ, which was proved by Lu and Xu [20].

Moreover, let b = (b₁, b₂, ⋯, b_m), where $$ for 1 ≤ i ≤ m, m ∈ ℕ. Then, multilinear commutator $$ generated by b and $$ can be defined as follows:

When m = 1, Wu and Fu [9] obtained the λ-central BMO estimates for commutator $$ on p-adic central Morrey spaces. Furthermore, Mo et al. [13] considered multilinear case for m ≥ 1 and established the boundedness of $$ with symbols in Campanato spaces on p-adic generalized Morrey spaces. Motivated by the works, our second goal is to show the λ-central BMO estimates for the multilinear commutator $$ on Morrey-Herz spaces over the p-adic fields.

Theorem 3. Let α be a complex number with 0 < Re(α) < n, 1/r = 1/q₁ + 1/q₂ + ⋯+1/q_m + 1/q − Re(α)/n, where r > n/(n − Re(α)) and 1 < q, q₁, ⋯, q_m < ∞. λ ≥ 0, 0 < l ≤ s ≤ ∞, ν = ν₁ + ν₂ + ⋯+ν_m with 0 ≤ ν₁, ν₂, ⋯, ν_m < 1/n, nν + λ − n/q + Re(α) < σ₁ < n(1 − 1/q) + λ, σ₂ = σ₁ − n(1/q₁ + 1/q₂+⋯+1/q_m + ν). If $$, then, there exists a constant C > 0 such that

We notice that the important particular case of λ-central BMO spaces $$ is $$ defined in [14] when λ = 0. Hence, if let ν₁ = ν₂ = ⋯ = ν_m = 0 in Theorem 3, we obtain the following conclusion.

Theorem 4. Let α be a complex number with 0 < Re(α) < n, 1/r = 1/q₁ + 1/q₂ + ⋯+1/q_m + 1/q − Re(α)/n, where r > n/n − Re(α) and 1 < q, q₁, ⋯, q_m < ∞. λ ≥ 0, 0 < l ≤ s ≤ ∞, λ − n/q + Re(α) < σ₁ < n(1 − 1/q) + λ, σ₂ = σ₁ − n(1/q₁ + 1/q₂+⋯+1/q_m). If $$, then, there exists a constant C > 0 such that

On the other hand, it is eminent that $$ and $$. Therefore, by letting λ = 0 in Theorem 1, 3, and 4, respectively, we will get counterparts on p-adic Herz spaces.

Corollary 5. Let α be a complex number with 0 < Re(α) < n, 1/r = 1/q − Re(α)/n with 1 ≤ q < r < ∞, Re(α) − n/q < σ < n(1 − 1/q), 0 < l ≤ s ≤ ∞. If q > 1, then, there exists a constant C > 0 such that

Moreover, if q = 1, then, there exists a constant C > 0 such that

Corollary 6. Let α be a complex number with 0 < Re(α) < n, 1/r = 1/q₁ + 1/q₂ + ⋯+1/q_m + 1/q − Re(α)/n, where r > n/n − Re(α) and 1 < q, q₁, ⋯, q_m < ∞. 0 < l ≤ s ≤ ∞, ν = ν₁ + ν₂ + ⋯+ν_m with 0 ≤ ν₁, ν₂, ⋯, ν_m < 1/n, nν − n/q + Re(α) < σ₁ < n(1 − 1/q), σ₂ = σ₁ − n(1/q₁ + 1/q₂+⋯+1/q_m + ν). If $$, then, there exists a constant C > 0 such that

Corollary 7. Let α be a complex number with 0 < Re(α) < n, 1/r = 1/q₁ + 1/q₂ + ⋯+1/q_m + 1/q − Re(α)/n, where r > n/n − Re(α) and 1 < q, q₁, ⋯, q_m < ∞. 0 < l ≤ s ≤ ∞, Re(α) − n/q < σ₁ < n(1 − 1/q), σ₂ = σ₁ − n(1/q₁ + 1/q₂+⋯+1/q_m). If $$, then, there exists a constant C > 0 such that

Throughout this paper, for t > 0 and α ∈ ℂ, we always assume that t^α takes the principal branch, that is

Thus, we have

The letter C will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.

Definition 8 (see [15].)Let σ ∈ ℝ and 0 < s, r ≤ ∞. The homogeneous Herz space $$ is defined by

where

for s < ∞, and the usual modifications should be made when s = ∞.

Definition 9 (see [15].)Let σ ∈ ℝ, 0 < s, r ≤ ∞, and λ be a nonnegative real number. Then, the homogeneous Morrey-Herz space $$ is defined by

where for s < ∞, and the usual modifications should be made when s = ∞.

Here, we introduce the p-adic weak Herz spaces and p-adic weak Morrey-Herz spaces.

Definition 10. Let σ ∈ ℝ, 0 < s, r ≤ ∞. The weak homogeneous Herz space $$ defined by

where for s < ∞, and the usual modifications should be made when s = ∞.

Definition 11. Let σ ∈ ℝ, 0 < s, r ≤ ∞, and λ be a nonnegative real number. The weak homogeneous Morrey-Herz space $$ defined by

where for s < ∞, and the usual modifications should be made when s = ∞.

Definition 12 (see [9].)Given λ < 1/n, 1 < q < ∞, the λ -central bounded mean oscillation space $$ is defined as the set of all functions $$ such that

where

Lemma 13 (see [10].)Suppose that 0 ≤ λ < 1/n, 1 < q < ∞, $$ and j, k ∈ ℤ. Then

Lemma 14 (see [8].)Let α be a complex number with 0 < Re(α) < n, 1/r = 1/q − Re(α)/n with 1 ≤ q < r < ∞.If q > 1, then, there exists a constant C > 0 such that

Moreover, if q = 1, there is a constant C > 0 such that

Lemma 15. Let α be a complex number with 0 < Re(α) < n, 1/r = 1/q − Re(α)/n with 1 ≤ q < r < ∞. Then. there exists a constant C > 0 such that and

for any integers j and k.

Proof. Obviously, by the properties of |·|_p, it is not difficult to check that

for any x ∈ S_k, y ∈ S_j. Thus, by applying the Hölder’s inequality, we have

Therefore, Lemma 15 is completely proved.

Proof of Theorem 16. Suppose that $$, we decompose f in the following way

First, we prove that for q > 1. By the decomposition of f above and the fact l ≤ s, we get

If k − 1 ≤ j ≤ k + 1, by the L^r-boundedness of $$ (see Lemma 14), we can obtain the estimates of $$ by

If j ≤ k − 2, applying Lemma 15, Jensen’s inequality, Hölder’s inequality, and the fact σ < n(1 − 1/q) = n/q^′, we can get the estimates for $$ by

If j ≥ k + 2, Lemma 15 shows the estimates for $$ by

For $$, using similar methods as that for $$, by the fact σ > λ − n/q + Re(α) ≥ −n/q + Re(α), Jensen’s inequality, and Hölder’s inequality, we obtain

For $$, one can see from the definition of Morrey-Herz space that

Here, by the estimates (43) and the fact σ > λ − n/q + Re(α) ≥ −n/q + Re(α), we obtain

Therefore, by a combination for the estimates of $$, $$ and $$, we can conclude that $$.

Next, for q = 1, we will consider the weak type boundedness of $$. Note that $$ for l < s, so we only need to prove that in the case l = s. For any μ > 0, the decomposition of f shows that

For $$, applying the weak type property (1, r) of the operator $$ (see Lemma 14) and the estimates (43). Notice that λ − n + Re(α) < σ < 0 when q = 1, λ ≥ 0, so we can conclude that

For $$, by the Chebychev inequality, Lemma 15, the estimates (43), and the fact λ − n + Re(α) < σ < 0, we have

Hence, by a combination for the estimates of $$, and $$, we can get the desired inequality $$.

Therefore, we complete the proof of Theorem 16.

Proof of Theorem 17. Similarly to the proof of Theorem 16, let $$ and decompose f into

When m = 1, denote $$ by $$, we consider

Let us first estimate $$, note that

Suppose that 1/u = 1/q − Re(α)/n, 1/t = 1/q₁ + 1/q, then, it is easy to see 1/r = 1/u + 1/q₁ = 1/t − Re(α)/n for q > 1 and t > 1. Applying the fact σ₂ = σ₁ − n(1/q₁ + ν₁), Hölder’s inequality, Minkowski inequality, the boundedness of $$ from $$ to $$ and from $$ to $$, we have

Therefore, we get

Now, let us turn to the estimates of $$ and $$. If x ∈ S_k, we can easily deduce that

Then, using the facts 1/r = 1/q₁ + 1/q − Re(α)/n, σ₂ = σ₁ − n(1/q₁ + ν₁), |j − k| ≥ 2, Lemma 13, and Hölder’s inequality, we can get

Thus, the fact nν₁ + λ − n/q + Re(α) < σ₁ < n(1 − 1/q) + λ and (43) imply

Combining the estimates for $$, $$ and $$, which completes the proof for Theorem 3 of the case m = 1.

Now, we consider the case m ≥ 2. In order to simplify the proof, for positive integer m and 1 ≤ i ≤ m, we denote by $$ the family of all finite subsets θ = {θ₁, θ₂, ⋯, θ_m} of {1, 2, ⋯, m} of i different elements, let θ^c = {1, 2, ⋯, m}\θ for any $$. For b = (b₁, b₂, ⋯, b_m), let $$, $$ and $$ denote the integral average of the function b_i over the set B_k, then

where $$ and $$.

We write

Let us first estimate $$, we can obtain

For $$, taking 1/s = 1/q − Re(α)/n, then, 1/r = 1/q₁ + 1/q₂ + ⋯+1/q_m + 1/s. Applying Hölder’s inequality, the boundedness of $$ from $$ to $$ and the fact σ₂ = σ₁ − n(1/q₁+⋯+1/q_m + ν₁+⋯+ν_m), we obtain

For $$, taking 1/h = 1/q₁ + ⋯+1/q_m + 1/q, then, 1/r = 1/h − Re(α)/n for h > 1. By Hölder’s inequality, the boundedness of $$ from $$ to $$ and the fact σ₂ = σ₁ − n(1/q₁+⋯+1/q_m + ν₁+⋯+ν_m), we get

For $$, we denote by

and 1/ω = 1/z + 1/q − Re(α)/n, then 1/r = 1/h + 1/ω. Using Hölder’s inequality and the L^ω-boundedness of $$, we have

Then, similarly to the method estimating for $$, it is not difficult for us to get

Next, we will estimate $$ and $$. Let

Therefore, by Minkowski’s inequality, Hölder’s inequality, and Lemma 13, it follows that

Thus, similarly to the method estimating for $$ and $$, we get

So far, by a combination for the estimates of $$, $$, and $$, we will finish the proof for Theorem 3 of the case m ≥ 2. This completes the proof of Theorem 17.