Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

([12]). Let 0<α<n,1/q=1/p−α/n,1≤s′<p<n/α,ws′∈A(p/s,′q/s′) and let Ω be homogeneous of degree zero with Ω ∈ L_s(S^n-1). Moreover,for 1 ≤ j ≤ k, |γ_j| = m_j - 1, m_j ≥ 2 and DγjAj∈BMO(Rn). Then there exists a constant C, independent of A_j,1 ≤ j ≤ k and f, such that

Here and in the sequel, we always denote by p' the conjugate index of any p > 1, that is 1=p + 1=p' = 1, and by C a constant which is independent of the main parameters and may vary from line to line. We define the generalized weighed Morrey spaces as follows.

Let 1 ≤ ρ < ∞, ϕ be a positive measurable function on ℝ × (0, ∞) and w be non-negative measurable function on ℝⁿ. We denote by M_p,ϕ(w) the generalized weighted Morrey space, the space of all functions f∈Lp,wlocRn with finite norm

where L_p,w(B(x, r)) denotes the weighted L_p-space of measurable functions f for which

Furthermore, by WM_p,ϕ(w) we denote the weak generalized weighted Morrey space of all functions f∈WLp,wlocRn for which

where WL_p,w(B(x, r)) denotes the weak L_p,w-space of measurable functions f for which

(1) If w ≡ 1, then M_p,ϕ(1) = M_p,ϕ is the generalized Morrey space.

(2) If φ(x,r)≡w(B(x,r))κ−1p,thenMp,φ(w)=Lp,κ(w) is the weighted Morrey space.

(3) If φ(x,r)≡v(B(x,r))κpw(B(x,r))−1p,thenMp,φ(w)=Lp,κ(v,w) is the two weighted Morrey space.

(4) Ifw≡1andφ(x,r)=rλ−npwith0<λ<n,thenMp,φw=Lp,λRnis the classical Morrey space and WM_ρ,ϕ(w) = WL_p,λ(ℝⁿ) is the weak Morrey space.

(5) If φ(x,r)≡w(B(x,r))−1p,thenMp,φ,(w)=Lp,w(Rn) is the weighted Lebesgue space.

The commutators are useful in many nondivergence elliptic equations with discontinuous coefficients, [15-17, 26, 27, 41]. In the recent development of commutators, Pérez and Trujillo-González [42] generalized these multilinear commutators and proved the weighted Lebesgue estimates. Ye and Zhu in [45] obtained the boundedness of the multilinear commutators in weighted Morrey spaces L_p,k(w) for 1 < p < ∞ and 0 < κ < 1, where the symbol b→ belongs to bounded mean oscillation (BMO)ⁿ. Furthermore, the weighted weak type estimate of these operators in weighted Morrey spaces is L_{p, k}(w) for p = 1 and 0 < κ < 1. The following statement was proved by Guliyev in [24].

([24]). Let 0 < α < n, 1 < p < n/α and 1/q = 1/p — α/n, Ω ∈ L_∞(𝕊^n-1), w ∈ A_p,q, A ∈ BMO(ℝⁿ), and (ϕ₁, ϕ₂) satisfies the condition

where C does not depend on χ and r. Then the operator TΩ,αA,k is bounded from Mp,φ1wptoMq,φ2wq.

It has been proved by many authors that most of the operators which are bounded on a weighted (unweighted) Lebesgue space are also bounded in an appropriate weighted (unweighted) Morrey space, see [8, 44]. As far as we know, there is no research regarding boundedness of the fractional multilinear integral operator on Morrey space. In this paper, we are going to prove that these results are valid for the rough fractional multilinear integral operator TΩ,αA1,A2,…,Ak on generalized weighted Morrey space. Our main results can be formulated as follows.

Let 0 < α < n, 1 ≤ s' < p < n/α and 1/q = 1/ p — α/n. Suppose that Ω is homogeneous of degree zero with Ω ∈ L_s(S^n-1) and (ϕ₁,ϕ₂) satisfy the condition (1). Let also, for 1 ≤ j ≤ k, |γ_j| = m_j - 1, m_j ≥ 2 and DγjAj∈BMORn. Suppose ws′∈Aps′,qs′,then the operator TΩ,αA1,A2,…,Ak is bounded from Mp,φ1wptoMq,φ2wq. Moreover, then there is a constant C > 0 independent of f and A₁, A₂, ... , A_k such that

In the case m_j = 1 and A_j = A for j = 1,..., k from the Theorem 1.5 we get the Theorem 1.4. Also, in the case ω ≡ 1 we get the following corollary, which was proved in [1].

([1]). Let 0 < a < n, 1 ≤ s' < p < n/α and 1/q = 1/p — α/n. Suppose that Ω is homogeneous of degree zero with Ω ∈ L_s(S^n-1), and (ϕ₁, ϕ₂) satisfy the condition

where C₀ does not depend on x and r. Let also, for 1 ≤ j ≤ k, |γ_j| = m_j - 1, m_j ≥ 2 and DγjAj∈BMORn. Then the operator TΩ,αA1,A2,…,Ak is bounded from Mp,φ1RntoMq,φ2Rn. Moreover, there is a constant C > 0 independent of f such that

Let φ1(x,t)=(wq(B(x,t)))kp(wp(B(x,t)))−1p,φ2(x,t)=(wq(B(x,t)))kp−1q,0<k<p/q and w^q ∈A_∞(ℝⁿ). Then (ϕ₁, ϕ₂) satisfies the condition (1).

In fact, from (2.8) in Section 2 we have constant δ > 0 such that

Since 0 < κ < p/q, then κp−1q<0. Thus

If ws′∈Aps′,qs′, then by Lemma 2.2 in Section 2 we know wq∈A1+q/p′Rn. Therefore, we have the following corollaries.

Let 0 < α < n, let 1 ≤ s' < p < n/α, and let 1/q = 1/p — α/n. Let also, for 1 ≤ j ≤k, |γ_j| = m_j - 1, m_j ≥ 2 and DγjAj∈BMORn. Suppose ws′∈Aps′,qs′,thenTΩ,αA1,A2,…,Ak is bounded from L_p,k(w^p,w^q)(ℝⁿ) to L_{q,k q/p}(w^q,ℝⁿ) and

where the constant C > 0 is independent of f and A₁, A₂,..., A_k.

Note that, in [2] the Nikolskii-Morrey type spaces were introduced and the authors studied some embedding theorems. In the next paper, we shall introduce the generalized weighted Nikolskii-Morrey spaces and will study some embedding theorems. We will also investigate the boundedness of fractional multilinear integral operators with rough kernels TΩ,αA1,A2,…,Ak on the generalized weighted Nikolskii-Morrey spaces, see for example, [30]. These results may be applicable to some problems of partial differential equations; see for example [6, 7, 19, 20, 26, 28, 30, 43].

([21, 37]). Suppose w ∈A_p and the following statements hold.

(i) For any 1 ≤ p < ∞, there is a positive number C such that

(ii) For any 1 ≤ p < ∞, there is a positive number C and S such that

(iii) For any 1 < p < ∞, one has w1−p′∈Ap′.

We also need another weight class A_p,q introduced by Muckenhoupt and Wheeden in [38] to study weighted boundedness of fractional integral operators.

Given 1 ≤ p ≤ q < ∞. We say that ω ∈ A_p,q if there exists a constant C such that for every ball B ⊂ ℝⁿ, the inequality

holds when 1 < p < ∞, and for every ball B ⊂ ℝⁿ the inequality

holds when p = 1.

By (3), we have

We summarize some properties about weights A_p,q; see [21, 38].

Given 1 ≤ p ≤ q < ∞.

(i) w ∈A_p,q if and only if wq∈A1+q/p′;

(ii) w ∈ A_p,q if and only ifw−p′∈A1+p′/q;

(iii) w ∈A_p,p if and only if w^p ∈A_p;

(iv) If p₁ < p₂ and q₂ > q₁, then Ap1,q1⊂Ap2,q2.

In this paper, we need the following statement on the boundedness of the Hardy type operator

where μ is a non-negative Borei measure on (0, ∞).

The inequality

holds for all non-negative and non-increasing g on (0, ∞) if and only if

and c ≈ A₁.

Note that Theorem 2.3 is proved analogously to Theorem 4.3 in [24, 25].

([39, Theorem 5, p. 236]). Let w ∈ A_∞. Then the norm of BMO(ℝⁿ) is equivalent to the norm of BMO(w), where

and

([24]).

(1) The John-Nirenberg inequality : there are constants C₁,C₂ > 0, such that for all b ∈ BMO(ℝⁿ) and β > 0

(2) For 1 < p < ∞ the John-Nirenberg inequality implies that

and for 1 < ρ < ∞ and w ∈ A_∞

The following lemma was proved by Guliyev in [24].

([24]).

i) Let w ∈ A_∞and b be a function in BMO(ℝⁿ). Let also 1 ≤p < ∞, x ∈ℝⁿ, and r₁,r₂ > 0. Then

where C > 0 is independent of f, x, r₁ and r₂.

ii) Let w ∈A_p and b be a function in BMO(ℝⁿ). Let also 1 < p < ∞, x ∈ ℝⁿ, and r₁,r₂ > 0. Then

where C > 0 is independent of f, x, r₁ and r₂.

Below we present some conclusions about R_m(A;x, y).

([22]). Suppose b is a function on ℝⁿwith the m-th derivatives in L_q(ℝⁿ), q > n. Then

The following property is valid.

Let x ∈ B(x₀, r), y ∈ B(x₀, 2^j+1r)\B(x₀, 2^jr). Assume that A has derivatives of order m - 1 in BMO(ℝⁿ). Then there exists a constant C, independent of A, such that

For fixed x ∈ ℝⁿ, let

Then

From Lemma 2.7 we have,

When x ∈ B(x₀,r), y ∈ B(x₀, 2^j+1r)\B(x₀, 2^jr), then 2^j-1r ≤ |x - y| ≤2^j+2 r. Thus, we have

Then

Hence

Note that

Then

Thus

Combining with (6), (7) and (8), then (5) is proved.

Finally, we present a relationship between essential supremum and essential infimum.

([10]). Let f be a real-valued nonnegative function and measurable on E. Then

Let 1 ≤ s' <p <n/α, and let 1/q = 1/p – α/n. Let also, for 1 ≤ j ≤ k, |γ_j | = m_j – 1, m_j ≥ 2 andDγjAj∈BMO(Rn).A_j ∈ BMO. (ℝⁿ) Suppose that Ω is homogeneous of degree zero with Ω ∈ L_s(S^n-1), ws′∈Aps′,qs′,then for any r > 0, there is a constant C independent of fsuch that

We write f as f = f₁ + f₂, where f1(y)=f(y)χB(x0,2r)(y),χB(x0,2r) denotes the characteristic function of B(x₀,2r) Then

Since f1∈Lp,wp(Rn),, by the boundedness of TΩ,αA1,A2,…,Ak from Lp,wp(Rn)toLq,wq(Rn) (Theorem 1.1) we get

Note that q > p > 1 and s′pp′(p−s′)≥1, then by Hölder’s inequality,

This means

Then