Boundedness of (k + 1)-linear fractional integral with a multiple variable kernel

Abstract

In this article, we discuss the boundedness of (k + 1)-linear fractional integrals with variable kernels on product L^pspaces. Our results improved some known results.

2000 Mathematics Subject Classification: 42B20; 42B25.

1. Introduction

It is well known that multilinear theory plays an important role in harmonic analysis and mathematicians pay much attention to it, see [1, 2] for more details. In 1992, Grafakos [1] first proved that the multilinear fractional operator $I_{\beta }^m\left(\overrightarrow{f}\right)\left(x\right)$ is bounded from $L^{p_1}\times \cdots \times L^{p_m}$ spaces to L^rspace with 1/r + α/n = 1/s, where 1/s = 1/p_l + ··· + 1/p _m and s satisfies n/(n + α) ≤ s < n/α with multilinear fractional $I_{\beta }^m\left(\overrightarrow{f}\right)\left(x\right)$ defined as following:

$$I_{\beta }^m\left(\overrightarrow{f}\right)\left(x\right)=\underset{{ℝ}^n}{\int }\frac{1}{{\left|y\right|}^{n-\beta }}\prod _{i=1}^m{f}_i\left(x-{\theta }_{i}y\right)dy,$$

(1.1)

for fixed nonzero real numbers θ _i (i = 1, ..., m) and 0 < β < n.

Later, Ding and Lu [3] improved Grafakos's results to the case when $I_{\beta }^m\left(\overrightarrow{f}\right)\left(x\right)$ has a rough kernel Ω₀(x) with Ω₀(x) ∈ L^r(S^n-1) and

$$I_{\beta ,{\Omega }_0}^m\left(\overrightarrow{f}\right)\left(x\right)=\underset{{ℝ}^n}{\int }\frac{{\Omega }_0\left(y\right)}{{\left|y\right|}^{n-\beta }}\prod _{i=1}^m{f}_i\left(x-{\theta }_{i}y\right)dy,$$

(1.2)

Ding and Lu proved that $I_{\beta ,{\Omega }_0}^m\left(\overrightarrow{f}\right)\left(x\right)$ is bounded from $L^{p_1}\times \cdots \times L^{p_k}$ spaces to L^qspaces with 1/p₁ + ...... + 1/p _k - 1/q = β/n. Obviously, Ding and Lu's results improved the main results in [1].

In 1999, Kenig and Stein [2] studied a new kind of multilinear fractional integral associated with the bilinear fractional integrals operators, they defined the (k + 1)-linear fractional integrals as following,

$$\begin{array}{c}{I}_{\alpha ,A}\left(f_1,...,f_{k+1}\right)\left(x\right)=\underset{{\left(R^n\right)}^k}{\int }{f}_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\\ \cdots f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)\frac{d{y}_1,...,d{y}_k}{{\left|\left(y_1,...,y_k\right)\right|}^{nk-\alpha }},0<\alpha <kn,\end{array}$$

where for a fixed k ∈ N and 1 ≤ i, j ≤ k + 1, a linear mapping ℓ _j : R^n(k+1)→ Rⁿ, 1 ≤ j ≤ k + 1 is defined by

$${\ell }_j\left(x_1...,x_k,x\right)=A_{1j}{x}_1+\cdots +A_{kj}{x}_k+A_{k+1,j}x.$$

(1.3)

Here, A _ij is an n × n matrix and a (k + 1)n × (k + 1)n matrix A = (A _ij ) (i = 1, ..., k + 1, j = 1, ..., k + 1,) satisfies the following assumptions:

• (I) For each 1 ≤ j ≤ k + 1, A _k+1,iis an invertible n × n matrix.

• (II) A is an invertible (k + 1)n × (k + 1)n matrix.

• (III) For each j ₀, 1 ≤ j ₀ ≤ k + 1, consider the kn × kn matrix $A_{j_0}={\left(A_{j_0}\right)}_{\ell m}$, where

  $${\left(A_{j_0}\right)}_{\ell m}=\left\{\begin{array}{cc}{A}_{\ell ,m}\hfill & 1\le \ell \le k,1\le m\le k,m<j_0\hfill \\ A_{\ell ,m+1}\hfill & 1\le \ell \le k,1\le m\le k,m<j_{0.}\hfill \end{array}.\right.$$

Obviously, when k = 1 and A₁₁ = I, A₂₁ = I, A₁₂ = -I, A₂₂ = I, I _α,A (f₁, f₂)(x) becomes the classical bilinear fractional integral, that is

$$I_{\alpha ,A}\left(f_1,f_2\right)\left(x\right)=B_{\alpha }\left(f_1,f_2\right)\left(x\right)=\underset{R^n}{\int }{f}_1\left(x+t\right)f_2\left(x-t\right)\frac{dt}{{\left|t\right|}^{n-\alpha }}.$$

(1.4)

In [2], Kenig and Stein proved that B _α (f₁, f₂)(x) is bounded from $L^{p_1}\times L^{p_2}$ to L^qwith 1/p₁ + 1/p₂-1/q = α/n for 1 ≤ p₁, p₂ ≤ ∞. Later, Ding and Lin [4] considered the following bilinear fractional integral with a rough kernel,

$$B_{\alpha ,{\Omega }_0}\left(f_1,f_2\right)\left(x\right)=\underset{R^n}{\int }{f}_1\left(x+t\right)f_2\left(x-t\right){\Omega }_0\left(t\right)\frac{dt}{{\left|t\right|}^{n-\alpha }},$$

where Ω₀(y') is a rough kernel belongs to L^s(S^n-1)(s > 1) without any smoothness on the unit sphere.

Ding and Lin proved the following theorem,

Theorem A ([4])

Assume that $0<\alpha <n,1<s^{\prime }<\frac{n}{\alpha }$, 1/p₁ + 1/p₂ - α/n, 1/q = 1/p₁ + 1/p₂ - α/n, and that s < min{p₁, p₂}, then for 1 ≤ p₁, p₂ ≤ ∞, we have

$${∥B_{\alpha ,{\Omega }_0}\left(f_1,f_2\right)∥}_{L^q}\le C{∥f_1∥}_{L^{p_1}}{∥f_2∥}_{L^{p_2}}.$$

For the research of partial differential equation, mathematicians pay much attention to the singular integral (or fractional integral) with a variable kernel Ω(x, y), see [5, 6] for more details. A function Ω(x, y) is said to be belonged to L^∞(Rⁿ) × L^q(S^n-1) if the function Ω(x, y) satisfies the following conditions:

• (i) Ω(x, λz) = Ω(x, z) for any x, z ∈R ⁿand λ > 0.

• (ii) ${{∥\Omega ∥}_{L^{\infty }}}_{\left(R^n\right)\times L^q\left(S^{n-1}\right)}=\underset{x\in R^n}{\text{sup}}{\left({\int }_{S^{n-1}}{\left|\Omega \left(x,z^{\prime }\right)\right|}^{q}d\sigma \left(z^{\prime }\right)\right)}^{1/q}<\infty$.

Recently, Chen and Fan [7] considered the following bilinear fractional integral with a variable kernel,

$$B_{\alpha ,\Omega }\left(f_1,f_2\right)\left(x\right)=\underset{R^n}{\int }{f}_1\left(x+t\right)f_2\left(x-t\right)\frac{\Omega \left(x,t\right)}{{\left|t\right|}^{n-\alpha }}dt,$$

(1.5)

they proved the following result,

Theorem B([7])

Let 1/p = 1/p₁ + 1/p₂ - α/n and Ω(x, y) ∈ L^∞(Rⁿ) × L^s(S^n-1) with s' < min{p₁, p₂} and $s>\frac{n}{n-\alpha }$, then

$${∥B_{\alpha ,\Omega }\left(f_1,f_2\right)∥}_{L^p}\le C{∥f_1∥}_{L^{p_1}}{∥f_2∥}_{L^{p_2}}.$$

Obviously, Chen and Fan's result improved the main results in [4] and the method they used is different from [4].

In this article, we will consider the (k + 1)-linear fractional integral with a multiple variable kernel $\Omega \left(x,\overrightarrow{y}\right)$. Before state the main results in this article, we first introduce a multiple variable function $\Omega \left(x,\overrightarrow{y}\right)\in L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)$ satisfying the following conditions:

• (i) $\Omega \left(x,\lambda \overrightarrow{y}\right)=\Omega \left(x,\overrightarrow{y}\right)$ for any λ > 0.

• (ii) ${{∥\Omega ∥}_{L^{\infty }}}_{\left(R^n\right)\times L^r\left(S^{nk-1}\right)}=\underset{x\in R^n}{\text{sup}}\left({\int }_{S^{nk-1}}{\left|\Omega \left(x,{\overrightarrow{y}}^{\prime }\right)\right|}^{r}d\sigma \left({\overrightarrow{y}}^{\prime }\right)\right)<\infty .$.

Now, we define the (k + 1)-linear fractional integral with a multiple variable kernel $\Omega \left(x,\overrightarrow{y}\right)\in L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)$ as following:

$$\begin{array}{c}{I}_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)\left(x\right)=\underset{{\left(R^n\right)}^k}{\int }{f}_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\\ \cdots f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)\frac{\Omega \left(x,\overrightarrow{y}\right)}{{\left|\left(y_1,...,y_k\right)\right|}^{nk-\alpha }},d{y}_1,...,d{y}_k,\end{array}$$

where the linear mapping ℓ _j is defined as in (1.3) and the corresponding matrix A satisfies the assumptions (I), (II) and (III). What's more, we assume that for each 1 ≤ j₀ ≤ k + 1, $A_{j_0}$ is an invertible kn × kn matrix.

Our main results are as following,

Theorem 1.1.

Assume that (I), (II) and (III) hold, if $\Omega \left(x,\overrightarrow{y}\right)\in L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)$ for $r>\frac{nk}{nk-\alpha }$ and 0 < α < kn, then

$${∥I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{p.\infty }}\le C\prod _{i=1}^{k+1}{∥f_i∥}_{L^{r^{\prime }}}$$

with 1/p = (k + 1)/r' - α/n.

Theorem 1.2.

Assume that (I), (II) and (III) hold, if $\Omega \left(x,\overrightarrow{y}\right)\in L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)$ for r' < min{p₁, ..., p_k+1}, $r>\frac{nk}{nk-\alpha }$ and 0 < α < kn, then

$${∥I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^q}\le C\prod _{i=1}^{k+1}{∥f_i∥}_{L^{p_i}}$$

with 1/q = 1/p₁ + ...... + 1/p_k+1- α/n.

Remark 1.3.

As far as we know, our results are also new even in the case that if we replace $\Omega \left(x,\overrightarrow{y}\right)$ by ${\Omega }_0\left(\overrightarrow{y}\right)\in L^r\left(S^{nk-1}\right)$.

Remark 1.4.

Obviously, our results improved the main results in [2, 4, 7].

2. Proof of Theorem 1.1

In this section, we will give the proof of Theorem 1.1. First we introduce some definitions and lemmas that will be used throughout this article.

Denote

$$\begin{array}{c}{M}_{A,s}\left(f_1,...,f_{k+1}\right)\left(x\right)=\underset{2^{-s}\le \left|\left(y_1,...,y_k\right)\right|\le 2^{-s+1}}{\int }\Omega \left(x,\overrightarrow{y}\right)f_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)...\\ f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)d\overrightarrow{y},\end{array}$$

thus we have the following conclusion.

Lemma 2.1.

Let $\Omega \left(x,\overrightarrow{y}\right)$ be as in Theorem 1.1 and assume (I), (II) and (III) hold, then

$${∥M_{A,s}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{\frac{r^{\prime }}{k+1}}}\le C{∥\Omega ∥}_{L\infty \left(R^n\right)\times L^r\left(S^{nk-1}\right)}{2}^{-nks}\prod _{i=1}^{k+1}{∥f_i∥}_{L^{r^{\prime }}}.$$

Proof. By Hölder's inequality, we have

$$\begin{array}{l}{M}_{A,s}^{\Omega }\left(f_1,...,f_{k+1}\right)\left(x\right)\le C{\left(\underset{2^{-s}\le \left|\left(y_{1,...,y_k}\right)\right|\le 2^{-s+1}}{\int }{\left|\Omega \left(x,\overrightarrow{y}\right)\right|}^{r}d\overrightarrow{y}\right)}^{1/r}\\ \times {\left(\underset{2^{-s}\le \left|\left(y_{1,...,y_k}\right)\right|\le 2^{-s+1}}{\int }{\left|f_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\cdots f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)\right|}^{r^{\prime }}d\overrightarrow{y}\right)}^{1/r^{\prime }}\\ \le C{2}^{\frac{-nks}{r}}{∥\Omega ∥}_{L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)}\\ \times {\left(\underset{2^{-s}\le \left|\left(y_{1,...,y_k}\right)\right|\le 2^{-s+1}}{\int }{\left|f_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\cdots f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)\right|}^{r^{\prime }}d\overrightarrow{y}\right)}^{1/r^{\prime }}\end{array}$$

Then by the estimate in page 8 of [2], we have

$$\begin{array}{l}{∥M_{A,s}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{\frac{r^{\prime }}{k+1}}}\le C{2}^{\frac{-nks}{r}}{∥\Omega ∥}_{L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)}\\ \times {\left({\underset{R^n}{\int }\left|{\underset{2^{-s}\le \left|\left(y_1,...,y_k\right)\right|\le 2^{-s+1}}{\int }\left|f_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\cdots f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)\right|}^{r^{\prime }}dy\right|}^{\frac{1}{k+1}}dx\right)}^{\frac{k+1}{r^{\prime }}}\\ \le C{∥\Omega ∥}_{L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)}{2}^{\frac{-nks}{r^{\prime }}}\prod _{i=1}^{k+1}{∥f_i∥}_{L^{r^{\prime }}}\\ \le C{∥\Omega ∥}_{L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)}{2}^{-kns}\prod _{i=1}^{k+1}{∥f_i∥}_{L^{r^{\prime }}}.\end{array}$$

So far, the proof of Lemma 2.1 has been finished.

Lemma 2.2.

Under the same conditions as in Theorem 1.1, for

$$I_{\alpha ,\Omega }\left(\overrightarrow{f}\right)\left(x\right)=\underset{R^{nk}}{\int }\frac{\Omega \left(x,\overrightarrow{y}\right)}{{\left|\left(y_1,...,y_k\right)\right|}^{kn-\alpha }}{f}_1\left(x-y_1\right)\cdots f_k\left(x-y_k\right)d\overrightarrow{y},$$

with $\Omega \left(x,\overrightarrow{y}\right)\in L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)$ for $1<r^{\prime }<\frac{kn}{\alpha }$ and 0 < α < kn.

Let 1/s = 1/r₁ + ...... 1/r _k -α/n > 0 with 1 ≤ r _i ≤ ∞, then,

• (i) if each r _i > r', then there exists a constant C such that

  $${∥I_{\alpha ,\Omega }\left(\overrightarrow{f}\right)∥}_{L^s}\le C\prod _{i=1}^k{∥f_i∥}_{L^{r_i}},$$

• (ii) if r _i = r' for some i, then there exists a constant C such that

  $${∥I_{\alpha ,\Omega }\left(\overrightarrow{f}\right)∥}_{L^{s,\infty }}\le C\prod _{i=1}^k{∥f_i∥}_{L^{r_i}}.$$

Proof. In [8], Lemma 2.2 was proved in the case $\Omega \left(x,\overrightarrow{y}\right)={\Omega }_0\left(\overrightarrow{y}\right)\in L^r\left(S^{nk-1}\right)$. When consider the case if the multiple kernel function is a multiple variable kernel, by the similar argument as in [3, 8], we can prove Lemma 2.2. Here we state the main steps to prove Lemma 2.2 for the completeness of this article.

First, we introduce the multilinear fractional maximal function ${ℳ}_{\alpha }\left(\overrightarrow{f}\right)\left(x\right)$ and multilinear fractional maximal function with a multiple variable kernel ${ℳ}_{\Omega ,\alpha }\left(\overrightarrow{f}\right)\left(x\right)$, respectively.

$${ℳ}_{\alpha }\left(\overrightarrow{f}\right)\left(x\right)=\underset{r>0}{\text{sup}}\frac{1}{r^{kn-\alpha }}\underset{\left|\overrightarrow{y}\right|<r}{\int }\prod _{i=1}^k\left|f_i\left(x-y_i\right)\right|d\overrightarrow{y}.$$

$${ℳ}_{\Omega ,\alpha }\left(\overrightarrow{f}\right)\left(x\right)=\underset{r>0}{\text{sup}}\frac{1}{r^{kn-\alpha }}\underset{\left|\overrightarrow{y}\right|<r}{\int }\Omega \left(x,\overrightarrow{y}\right)\prod _{i=1}^k\left|f_i\left(x-y_i\right)\right|d\overrightarrow{y}.$$

By Hölder's inequality, we can easily get the boundedness of $M_{\alpha }\left(\overrightarrow{f}\right)\left(x\right)$ on product L^pspaces and the following fact is also obvious by a simple computation,

$$M_{\Omega ,\alpha }\left(\overrightarrow{f}\right)\left(x\right)\le C{∥\Omega ∥}_{L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)}{\left[M_{\alpha s^{\prime }}\left({\left|f_1\right|}^{s^{\prime }},{\left|f_2\right|}^{s^{\prime }},...,{\left|f_k\right|}^{s^{\prime }}\right)\left(x\right)\right]}^{1/s^{\prime }}$$

which implies the boundedness of $M_{\Omega ,\alpha }\left(\overrightarrow{f}\right)\left(x\right)$ on product L^pspaces.

Then by a classical augment as in [3, 8], we have the following point estimate for $I_{\alpha ,\Omega }\left(\overrightarrow{f}\right)\left(x\right)$,

$$\left|I_{\alpha ,\Omega }\left(\overrightarrow{f}\right)\left(x\right)\right|\le C{\left[M_{\Omega ,\alpha +\epsilon }\left(\overrightarrow{f}\right)\left(x\right)\right]}^{\frac{1}{2}}{\left[M_{\Omega ,\alpha -\epsilon }\left(\overrightarrow{f}\right)\left(x\right)\right]}^{\frac{1}{2}},$$

(2.1)

So, by inequality (2.1) and the boundedness of $M_{\Omega ,\alpha }\left(\overrightarrow{f}\right)\left(x\right)$ on product L^pspaces, we get Lemma 2.2 easily.

To finish the proof of Theorem 1.1, we define

$$\begin{array}{c}{F}_{A,s}^{\Omega }\left(f_1,...,f_{k+1}\right)\left(x\right)=\underset{2^{-s}\le \left|\left(y_1,...,y_k\right)\right|\le 2^{-s+1}}{\int }\frac{\Omega \left(x,\overrightarrow{y}\right)}{{\left|\left(y_1,...,y_k\right)\right|}^{nk}}{f}_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\cdots \\ f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)d\overrightarrow{y}.\end{array}$$

Then, we have

$$I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)\left(x\right)\le H\left(x\right)+G\left(x\right)$$

with $H\left(x\right)=\sum _{s\ge s_0}{2}^{-s\alpha }{F}_{A,s}^{\Omega }\left(x\right)$ and

$$\begin{array}{c}G\left(x\right)=\underset{\left|\left(y_1,...,y_k\right)\right|\ge 2^{-s_0}}{\int }\frac{\Omega \left(x,\overrightarrow{y}\right)}{{\left|\left(y_1,...,y_k\right)\right|}^{nk-\alpha }}\\ f_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\cdots f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)d\overrightarrow{y}.\end{array}$$

For $r>\frac{kn}{kn-\alpha }$, we have

$$\begin{array}{ll}\hfill G\left(x\right)& =\underset{\left|\left(y_1,...,y_k\right)\right|\ge 2^{-s_0}}{\int }\frac{\Omega \left(x,\overrightarrow{y}\right)}{{\left|\left(y_1,...,y_k\right)\right|}^{nk-\alpha }}{f}_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\cdots f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)d\overrightarrow{y}\\ \le C{\left(\underset{\left|\left(y_1,...,y_k\right)\right|\ge 2^{-s_0}}{\int }\frac{{\left|\Omega \left(x,\overrightarrow{y}\right)\right|}^r}{{\left|\left(y_1,...,y_k\right)\right|}^{\left(nk-\alpha \right)r}}d\overrightarrow{y}\right)}^{1/r}\\ \times {\left(\underset{R^{nk}}{\int }{\left|f_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\cdots f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)\right|}^{r^{\prime }}d\overrightarrow{y}\right)}^{1/r^{\prime }}\\ \le 2^{s_0\left[\left(kn-\alpha \right)-\frac{kn}{r}\right]}{\left(\underset{R^{nk}}{\int }{\left|f_1\left({\ell }_1\left(y_1,...,y_k,x\right)\right)\cdots f_{k+1}\left({\ell }_{k+1}\left(y_1,...,y_k,x\right)\right)\right|}^{r^{\prime }}d\overrightarrow{y}\right)}^{1/r^{\prime }}\end{array}$$

Now using the linear change of variables as in page 14 of [2], that is for each 1 ≤ j ≤ k + 1, we define $f_j\left({\ell }_j\left(y_1,...,y_k,x\right)\right)=f_j^{\prime }\left(A_{k+1,j}^{-1}{\ell }_j\left(x\right)\right)=f_j^{\prime }\left(x-\sum _{i=1}^k{A^{\prime }}_{ij}{x}_i\right)=f_j^{\prime }\left(x-y_j\right)$ with $A_{ij}^{\prime }=-A_{k+1,j}^{-1}{A}_{ij}$ and $y_j=\sum _{i=1}^k{A^{\prime }}_{ij}{x}_i$ we have

$${∥G∥}_{L^{r^{\prime }}}\le 2^{s_0\left[\left(kn-\alpha \right)-\frac{kn}{r}\right]}\prod _{i=1}^{k+1}{∥f_i∥}_{L^{r^{\prime }}}$$

For the estimate of H(x), first by Lemma 2.1, we have

$${∥F_{A,s}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{\frac{r^{\prime }}{k+1}}}\le C{∥\Omega ∥}_{L^{\infty }\left(R^n\right)\times L^r\left(S^{nk-1}\right)}\prod _{i=1}^{k+1}{∥f_i∥}_{L^{r^{\prime }}}.$$

So when $\frac{r^{\prime }}{k+1}\le 1$, we get

$$\begin{array}{ll}\hfill {∥H∥}_{L^{\frac{r^{\prime }}{k+1}}}^{\frac{r^{\prime }}{k+1}}& \le C\sum _{s\ge s_0}{2}^{-\frac{r^{\prime }}{k+1}s\alpha }{∥F_{A,s}^{\Omega }∥}_{L^{\frac{r^{\prime }}{k+1}}}^{\frac{r^{\prime }}{k+1}}\\ \le C{2}^{-\frac{r^{\prime }}{k+1}{s}_0\alpha }\prod _{i=1}^m{∥f_i∥}_{L^{r^{\prime }}}^{\frac{r^{\prime }}{k+1}}\end{array}$$

When $\frac{r^{\prime }}{k+1}>1$, we can easily get

$$\begin{array}{ll}\hfill {∥H∥}_{L^{\frac{r^{\prime }}{k+1}}}& \le C\sum _{s\ge s_0}{2}^{-s\alpha }{∥F_{A,s}^{\Omega }∥}_{L^{\frac{r^{\prime }}{k+1}}}\\ \le C{2}^{-s_0\alpha }\prod _{i=1}^m{∥f_i∥}_{L^{r^{\prime }}}.\end{array}$$

Combine the estimate above can we easily get

$${∥H∥}_{L^{\frac{r^{\prime }}{k+1}}}\le 2^{-s_0\alpha }\prod _{i=1}^{k+1}{∥f_i∥}_{L^{r^{\prime }}}.$$

By the above estimates, we have

$$\begin{array}{l}\left|\left\{I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)\left(x\right)>\lambda \right\}\right|\\ \le \left|\left\{x\in R^n:H\left(x\right)>\frac{\lambda }{2}\right\}\right|+\left|\left\{x\in R^n:G\left(x\right)>\frac{\lambda }{2}\right\}\right|\\ \le \frac{{∥G∥}_{L^{r^{\prime }}}^{r^{\prime }}}{{\lambda }^{r^{\prime }}}+\frac{{∥H∥}_{L^{\frac{r^{\prime }}{k+1}}}^{\frac{r^{\prime }}{k+1}}}{{\lambda }^{\frac{r^{\prime }}{k+1}}}\\ \le \frac{2^{s_0\left(kn-\alpha -\frac{kn}{r}\right)r^{\prime }}}{{\lambda }^{r^{\prime }}}\prod _{i=1}^{k+1}{∥f_i∥}_{L^{r^{\prime }}}^{r^{\prime }}+\frac{2^{-s_0\alpha \frac{r^{\prime }}{k+1}}}{{\lambda }^{\frac{r^{\prime }}{k+1}}}\prod _{i=1}^{k+1}{∥f_i∥}_{L^{r^{\prime }}}^{\frac{r^{\prime }}{k+1}}.\end{array}$$

Now we may assume that ${∥f_i∥}_{L^{r^{\prime }}}=1$ for i = 1,..., k +1, and choose $s_0=\frac{\frac{k}{k+1}lo{g}_2\lambda }{\frac{kn}{r^{\prime }}-\frac{\alpha k}{k+1}},$, we get

$$\left|\left\{x\in R^n:I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)\left(x\right)>\lambda \right\}\right|\le \frac{C}{{\lambda }^p},$$

with $1/p=\frac{k+1}{r^{\prime }}-\frac{\alpha }{n}$.

So far, the proof of Theorem 1.1 has been finished.

3. Proof of Theorem 1.2

For any p₁ that is larger than and sufficiently close to r', by the proof of Theorem 1.1, we get

$${∥I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{q_{1,\infty }}}\le C{∥f_1∥}_{L^{p_1}}\cdots {∥f_k∥}_{L^{p_1}}{∥f_{k+1}∥}_{L^{p_1}}$$

with 1/q₁ = (k + 1)/p₁ - α/n. On the other hand, by Lemma 2.2 and the same linear change of variables as in Section 2, we have

$${∥I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{q_2}}\le C{∥f_1∥}_{L^{p_1}}\cdots {∥f_k∥}_{L^{p_1}}{∥f_{k+1}∥}_{L^{\infty }}$$

with 1/q₂ = k/p₁ - α/n.

Then by interpolation, we have

$${∥I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{q_{3,}\infty }}\le C{∥f_1∥}_{L^{p_1}}\cdots {∥f_k∥}_{L^{p_1}}{∥f_{k+1}∥}_{L^{p_k+1}}$$

with 1/q₃ = k/p₁ + 1/p_k+1- α/n and p₁ ≤ p_k+1.

Again, by Lemma 2.2 and the same linear change of variables as in Section 2, we have

$${∥I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{q_4}}\le C{∥f_1∥}_{L^{p_1}}\cdots {∥f_{k-1}∥}_{L^{p_1}}{∥f_k∥}_{L^{\infty }}{∥f_{k+1}∥}_{L^{p_k+1}}$$

with 1/q₄ = (k - 1)/p₁ + 1/p_k+1- α/n

Then by interpolation, we have,

$${∥I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{q_{5,}\infty }}\le C{∥f_1∥}_{L^{p_1}}\cdots {∥f_{k-1}∥}_{L^{p_1}}{∥f_k∥}_{L^{p_k}}{∥f_{k+1}∥}_{L^{p_k+1}}$$

with 1/q₅ = (k- 1)/p₁ + 1/p _k + 1/p_k+1- α/n and p₁ ≤ min{p _k , p_k+1}.

Again using the above methods can we easily get

$${∥I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{q,\infty }}\le C\prod _{i=1}^{k+1}{∥f_i∥}_{L^{p_i}}$$

for any p₁ ≤ min{p₂,... ,p_k+1} with 1/q = 1/p₁ + · · · 1/p_k+1- α/n.

Similarly, for any p _i (1 ≤ i ≤ k + 1) that is larger than and sufficiently close to r', we can also get

$${∥I_{\alpha ,A}^{\Omega }\left(f_1,...,f_{k+1}\right)∥}_{L^{q\infty }}\le C\prod _{i=1}^{k+1}{∥f_i∥}_{L^{p_i}},$$

for any p _i ≤ min{p₁,..., p_i-1, p_i+1,...p_k+1} with 1/q = 1/p₁+· · · 1/p_k+1-α/n.

Now, we obtain Theorem 1.2 by multilinear interpolation from [2, 9].