Weighted boundedness of multilinear integral operators for the endpoint cases

1.   Introduction and preliminaries

As the development of singular integral operators, their commutators and multilinear operators have been well studied (see ^[1,3,4,5,6]. Let $T$ be the Calderón-Zygmund singular integral operator and $b\in BMO(R^n)$, a classical result of Coifman, Rochberg and Weiss stated that the commutator $[b, T]f = T(bf)-bTf$ is bounded on $L^p(R^n)$ for $1 < p < \infty$. In ^[10], authors obtain the boundedness properties of the commutators for the extreme values of $p$. And note that $[b, T]$ is not bounded for the end point cases (that is $p = 1$ and $p = \infty$). In recent years, the theory of Herz space and Herz type Hardy space, as a local version of Lebesgue space and Hardy space, have been developed (see ^[8,9,10,11]). The purpose of this paper is to introduce some multilinear operators associated to certain non-convolution type integral operators and prove the weighted boundedness properties of the multilinear operators for the endpoint cases ($p = 1$ and $p = \infty$). In fact, we prove the endpoint boundedness of the multilinear operators only under the boundedness of the operators on Lebesgue spaces, that is the boundedness of the operators on $L^p(R^n)$ for $1 < p\leq\infty$ implies the boundedness of the multilinear operators from $L^{\infty}(R^n)$ to $BMO(R^n)$ for $p = \infty$ and from $H^1(R^n)$ to $L^1(R^n)$ for $p = 1$. Then, we apply the boundedness of the integral operator to some concrete including Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.

First, let us introduce some preliminaries (see ^{[7,8,14,15,16]}). Throughout this paper, we denote by $A_p$ the class of the Muckenhoupt weights for $1\le p < \infty$ (see ^[9]). In this paper $Q$ always stand for a cube of $R^n$ with sides parallel to the axes. For a cube $Q$ and a locally integrable function $b$ and a weight function $w$, let $w(Q) = \int_Qw(x)dx$, $b_Q = |Q|^{-1}\int_Qb(x)dx$ and $b^{\#}_w(x) = \sup\limits_{Q\ni x}w(Q)^{-1}\int_Q|b(y)-b_Q|w(y)dy$. $b$ is said to belong to $BMO(w)$ if $b^{\#}_w \in L^\infty(w)$ and define $||b||_{BMO(w)} = ||b^{\#}||_{L^\infty(w)}$, if $w = 1$, we denote $BMO(w) = BMO(R^n)$. We also define the weighted central $BMO$ space by $CMO(w)$, which is the space of those functions $b\in L_{loc}(R^n)$ such that

It is well-known that (see ^[8,9]

Also, we give the concepts of the atom and weighted Hardy spaces $H^1(w)$. A function $a$ is called an $H^1$ atom if there exists a cube $Q$ such that $a$ is supported on $Q$, $||a||_{L^2(w)}\le w(Q)^{-1/2}$ and $\int_{R^n}a(x)dx = 0$. It is well known that the weighted Hardy space $H^1(w)$ has the atomic decomposition characterization (see ^[2,9,16,17]).

For $k\in Z$, define $B_k = \{x\in R^n: |x|\le 2^k\}$ and $C_k = B_k\setminus B_{k-1}$. Denote by $\chi_k$ the characteristic function of $C_k$ and $\tilde\chi_k$ the characteristic function of $C_k$ for $k\ge1$ and $\tilde\chi_0$ the characteristic function of $B_0$.

Definition 1. Let $1 < p < \infty$ and $w_1$, $w_2$ be two non-negative weight functions on $R^n$.

(1) The homogeneous weighted Herz space is defined by

where

(2) The nonhomogeneous weighted Herz space is defined by

where

(3) The homogeneous weighted Herz type Hardy space is defined by

where

(4) The nonhomogeneous weighted Herz type Hardy space is defined by

where

where $G(f)$ is the grand maximal function of $f$, that is

where $K_m = \{\varphi\in S(R^n): \sup_{x\in R^n, |\alpha|\leq m}(1+|u|)^{m+n}|D^\alpha \varphi(u)|\leq 1 \}$, $\varphi_t(x) = t^{-n}\varphi(x/t)$ for $t > 0$, $m$ is a positive integer and $S(R^n)$ is the Schwartz class (see ^[18], p.88).

The Herz type Hardy spaces have the atomic decomposition characterization.

Definition 2. Let $1 < p < \infty$ and $w_1, w_2 \in A_1$. A function $a(x)$ on $R^n$ is called a central $(n(1-1/p), p; w_1, w_2)$-atom (or a central $(n(1-1/p), p; w_1, w_2)$-atom of restrict type), if

1) supp$a\subset B(0, r)$ for some $r > 0$ (or for some $r\ge 1$);

2) $||a||_{L^p(w_2)}\le [w_1(B(0, r))]^{1/p-1}$;

3) $\int_{R^n}a(x)dx = 0$.

Lemma 1. (see ^[7,16]). Let $w_1, w_2 \in A_1$ and $1 < p < \infty$. A tempered distribution $f$ belongs to $H\dot K_p(w_1, w_2; R^n)$(or $HK_p(w_1, w_2; R^n)$) if and only if there exist central $(n(1-1/p), p; w_1, w_2)$-atoms (or central $(n(1-1/p), p; w_1, w_2)$-atoms of restrict type) $a_j$ supported on $B_j = B(0, 2^j)$ and constants $\lambda_j$, $\sum_j|\lambda_j| < \infty$ such that $f = \sum_{j = -\infty}^\infty \lambda_ja_j$(or $f = \sum_{j = 0}^\infty \lambda_j a_j$) in the $S'(R^n)$ sense, and

Definition 3. Let $1 < p < \infty$ and $w$ be a non-negative weight functions on $R^n$. We shall call $B_p(w)$ the space of those functions $f$ on $R^n$ such that

2.   Theorems

In this paper, we will consider a class of multilinear operators associated to some non-convolution type integral operators as following.

Let $m_j$ be the positive integers($j = 1, \cdot\cdot\cdot, l$), $m_1+\cdot\cdot\cdot+m_l = m$ and $b_j$ be the locally integrable functions on $R^n$ ($j = 1, \cdot\cdot\cdot, l$). Set

and

Let $F_t(x, y)$ define on $R^n\times R^n\times [0, +\infty)$. Set

and

for every bounded and compactly supported function $f$. Let $H$ be the Banach space $H = \{h:||h|| < \infty\}$ such that, for each fixed $x\in R^n$, $F_t(f)(x)$ and $F_t^b(f)(x)$ may be viewed as a mapping from $[0, +\infty)$ to $H$. Then, the multilinear operator associated to $F_t$ is defined by

where $F_t$ satisfies: for fixed $0 < \varepsilon\leq 1$,

and

if $2|y-z|\le |x-z|$. Let $T(f)(x) = ||F_t(f)(x)||$. We also consider the variant of $T^b$, which is defined by

where

Note that when $m = 0$, $T^b$ is just the higher order commutators of $T$ and $b$ (see ^[1,12,13,20]). The operator $\tilde T^b$ is a variant of $T^b$, and it has not any form in the commutator, thus, it is a non-trivial variant of the commutator. It is well-known that multilinear operator, as a non-trivial extension of commutator, is of great interest in harmonic analysis and has been widely studied by many authors (see ^[3,4,5,6]). In ^[3], the weak ($H^1$, $L^1$)-boundedness of the multilinear operator related to some singular integral operator are obtained. In this paper, we will study the weighted boundedness properties of the multilinear operators $T^A$ and $\tilde T^A$ for the extreme cases. In Section 4, some examples of theorems in this paper are given.

We shall prove the following theorems in Section 3.

Theorem 1. Let $w \in A_1$ and $D^\alpha b_j\in BMO(R^n)$ for all $\alpha$ with $|\alpha| = m_j$ and $j = 1, \cdot\cdot\cdot, l$. Suppose that $T$ is bounded on $L^q(u)$ for any $1 < q\le\infty$ and $u\in A_1$. Then $T^b$ is bounded from $L^\infty(w)$ to $BMO(w)$.

Theorem 2. Let $w \in A_1$ and $D^\alpha b_j\in BMO(R^n)$ for all $\alpha$ with $|\alpha| = m_j$ and $j = 1, \cdot\cdot\cdot, l$. Suppose that $\tilde T^b$ is bounded on $L^q(u)$ for any $1 < q\le\infty$ and $u\in A_1$. Then $\tilde T^b$ is bounded from $H^1(w)$ to $L^1(w)$.

Theorem 3. Let $1 < p < \infty$, $w\in A_1$ and $D^\alpha b_j\in BMO(R^n)$ for all $\alpha$ with $|\alpha| = m_j$ and $j = 1, \cdot\cdot\cdot, l$. Suppose that $T$ is bounded on $L^q(u)$ for any $1 < q\le\infty$ and $u\in A_1$. Then $T^b$ is bounded from $B_p(w)$ to $CMO(w)$.

Theorem 4. Let $1 < p < \infty$, $w_1, w_2 \in A_1$ and $D^\alpha b_j\in BMO(R^n)$ for all $\alpha$ with $|\alpha| = m_j$ and $j = 1, \cdot\cdot\cdot, l$. Suppose that $\tilde T^b$ is bounded on $L^q(u)$ for any $1 < q\le\infty$ and $u\in A_1$. Then $\tilde T^b$ is bounded from $\dot HK_p(w_1, w_2; R^n)$ (or $HK_p(w_1, w_2; R^n)$) to $\dot K_p(w_1, w_2; R^n)$ (or $HK_p(w_1, w_2; R^n)$).

3.   Proof of theorems

To prove the theorem, we need the following lemma.

Lemma 2. (see ^[5]). Let $b$ be a function on $R^n$ and $D^\alpha b\in L^q(R^n)$ for $|\alpha| = m$ and some $q > n$. Then

where $\tilde Q(x, y)$ is the cube centered at $x$ and having side length $5\sqrt{n}|x-y|$.

Proof of Theorem 1. It is only to prove that there exists a constant $C_Q$ such that

holds for any cube $Q$. Without loss of generality, we may assume $l = 2$. Fix a cube $Q = Q(x_0, d)$. Let $\tilde Q = 5\sqrt{n}Q$ and $\tilde b_j(x) = b_j(x)-\sum\limits_{|\alpha| = m}\frac{1}{\alpha!}(D^\alpha b_j)_{\tilde Q}x^\alpha$, then $R_m(b_j; x, y) = R_m(\tilde b_j; x, y)$ and $D^\alpha\tilde b_j = D^\alpha b_j-(D^\alpha b_j)_{\tilde Q}$ for $|\alpha| = m_j$. We write, for $f_1 = f\chi_{\tilde Q}$ and $f_2 = f\chi_{R^n\setminus\tilde Q}$,

then

thus,

Now, let us estimate $I_1$, $I_2$, $I_3$, $I_4$ and $I_5$, respectively. First, for $x\in Q$ and $y\in \tilde Q$, by Lemma 2, we get

thus, by the $L^\infty(w)$-boundedness of $T$, we get

For $I_2$, since $w\in A_1$, $w$ satisfies the reverse of Hölder's inequality:

for all cube $Q$ and some $1 < q < \infty$ (see ^[9]). Thus, by the $L^p(w)$-boundedness of $T$ for $p > 1$ and Hölder'inequality, we get

For $I_3$, similar to the proof of $I_2$, we get

Similarly, for $I_4$, choose $1 < r_1, r_2 < \infty$ such that $1/r_1+1/r_2+1/q = 1$, we obtain, by Hölder'inequality,

For $I_5$, we write

By Lemma 2 and the following inequality (see ^[18])

we know that, for $x\in Q$ and $y\in 2^{k+1}\tilde Q\setminus 2^k\tilde Q$,

Note that $|x-y|\sim |x_0-y|$ for $x\in Q$ and $y\in R^n\setminus\tilde Q$, we obtain, by the condition on $F_t$,

For $I_5^{(2)}$, by the formula (see ^[5]):

and Lemma 2, we have

thus

Similarly,

For $I_5^{(4)}$, similar to the proof of $I_5^{(1)}$ and $I_5^{(2)}$, we get

Similarly,

For $I_5^{(6)}$, taking $1 < r_1, r_2 < \infty$ such that $1/r_1+1/r_2 = 1$, then

Thus

This completes the proof of Theorem 1.

Proof of Theorem 2. First, we prove that there exists a constant $C > 0$ such that for every $H^1(w)$-atom $a$ (that is that $a$ satisfy: supp $a\subset Q = Q(x_0, r)$, $||a||_{L^2(w)}\le w(Q)^{-1/2}$ and $\int_{R^n} a(y)dy = 0$ (see ^[7])), the following inequality holds:

Without loss of generality, we may assume $l = 2$. Write

For $J_1$, by the $L^2(w)$-boundedness of $\tilde T^b$, we get

To obtain the estimate of $J_2$, we denote that $\tilde b_j(x) = b_j(x)-\sum_{|\alpha_j| = m_j}\frac{1}{\alpha!} (D^\alpha b_j)_{2Q}x^\alpha$. Then $Q_{m_j}(b_j; x, y) = Q_{m_j}(\tilde b_j; x, y)$. We write, by the vanishing moment of $a$,

notice that if $w\in A_1$, then $\frac{w(Q_2)}{|Q_2|}\frac{|Q_1|}{w(Q_1)}\le C$ for all cubes $Q_1, Q_2$ with $Q_1\subset Q_2$. Thus, by Hölder's inequality and the reverse of Hölder's inequality for $w\in A_1$ and $1 < q < \infty$, we obtain, similar to the proof of Theorem 1,

Now, for $f\in H^1(w)$ with $f = \sum_{j = 1}^\infty\lambda_ja_j$, where $a_j's$ are the $H^1(w)$-atom and $\sum_j|\lambda_j|\leq C||f||_{H^1(w)}$. From above, we get

that is $\sum_{j = 1}^\infty|\lambda_j|||\tilde T^b(a_j)|\in L^1(w)$, and

This completes the proof of Theorem 2.

Proof of Theorem 3. It is only to prove that there exists a constant $C_Q$ such that

holds for any cube $Q = Q(0, d)$ with $d > 1$. Without loss of generality, we may assume $l = 2$. Fix a cube $Q = Q(0, d)$ with $d > 1$. Let $\tilde Q = 5\sqrt{n}Q$ and $\tilde b_j(x) = b_j(x)-\sum\limits_{|\alpha| = m_j}\frac{1}{\alpha!}(D^\alpha b_j)_{\tilde Q}x^\alpha$, then $R_{m_j}(b_j; x, y) = R_{m_j}(\tilde b_j; x, y)$ and $D^\alpha\tilde b_j = D^\alpha b_j-(D^\alpha b_j)_{\tilde Q}$ for $|\alpha| = m_j$. Similar to the proof of Theorem 1, we write, for $f_1 = f\chi_{\tilde Q}$ and $f_2 = f\chi_{R^n\setminus\tilde Q}$,

Similar to the proof of Theorem 1, we get

For $L_2$, taking $r, s > 1$ such that $rs < p$ and $q = (ps-rs)/(p-rs)$, then, by the reverse of Hölder's inequality,

For $L_4$, taking $r, t_1, t_2, t_3 > 1$ such that $1/t_1+1/t_2+1/t_3 = 1$, $rt_3 < p$ and $q = (pt_3-rt_3)/(p-rt_3)$, then, by the reverse of Hölder's inequality,

For $L_5$, we write, for $x\in Q$,

Similar to the proof of Theorem 1 and notice that $w\in A_1\subset A_p$, we get

For $L_5^{(4)}$, choose $1 < r < p$, notice that $w\in A_1\subset A_{p/r}$, we get

For $L_5^{(6)}$, choose $1 < r_1, r_2, r_3 < \infty$ such that $r_3 < p$ and $1/r_1+1/r_2+1/r_3 = 1$, notice that $w\in A_1\subset A_{p/r_3}$, we get

Thus

This finishes the proof of Theorem 3.

Proof of Theorem 4. We only give the proof of homogeneous Herz type Hardy spaces. Without loss of generality, we may assume $l = 2$. Let $f\in H\dot K_p(w_1, w_2; R^n)$, by Lemma 1, $f = \sum_{j = -\infty}^\infty\lambda_ja_j$, where $a_j's$ are the central $(n(1-1/p), p; w_1, w_2)$-atom with supp$a_j \subset B_j = B(0, 2^j)$ and $||f||_{H\dot K_p(w_1, w_2)}\sim\sum_j|\lambda_j|$. Write

For $M_2$, by the $L^p(w)$-boundedness of $\tilde T^b$ for $1 < p < \infty$ and $w\in A_1$, we get

To estimate $M_1$, we denote that $\tilde b_j(x) = b_j(x)-\sum_{|\alpha_j| = m_j}\frac{1}{\alpha!}(D^\alpha b_j)_{2Q}x^\alpha$. Then $Q_{m_j}(b_j; x, y) = Q_{m_j}(\tilde b_j; x, y)$. We write, by the moment condition of $a_j$,

Going through a similar argument to Theorem 2, we obtain

thus

This completes the proof of Theorem 4.

4.   Applications

Now we apply the theorems of this paper to some concrete including Littlewood-Paley operators, Marcinkiewicz operators and the Bochner-Riesz operator.

Example 1. Littlewood-Paley operators.

Fixed $\varepsilon > 0$ and $\mu > (3n+2)/n$. Let $\psi$ be a fixed function which satisfies:

(1) $\int_{R^n}\psi(x)dx = 0$,

(2) $|\psi(x)|\le C(1+|x|)^{-(n+1)}$,

(3) $|\psi(x+y)-\psi(x)|\le C|y|^\varepsilon(1+|x|)^{-(n+1+\varepsilon)}$ when $2|y| < |x|$.

We denote that $\Gamma(x) = \{(y, t)\in R_+^{n+1}:|x-y| < t\}$ and the characteristic function of $\Gamma(x)$ by $\chi_{\Gamma(x)}$. The Littlewood-Paley multilinear operators are defined by (see ^[13]),

and

where

and $\psi_t(x) = t^{-n}\psi(x/t)$ for $t > 0$. The variants of $g_\psi^A$, $S_\psi^A$ and $g_\mu^A$ are defined by

and

where

and

Set $F_t(f)(y) = f\ast\psi_t(y)$. We also define that

and

which are the Littlewood-Paley operators (see ^[19]). Let $H$ be the space

or

then, for each fixed $x\in R^n$, $F_t^A(f)(x)$ and $F_t^A(f)(x, y)$ may be viewed as the mapping from $[0, +\infty)$ to $H$, and it is clear that

and

It is easy to see that $g_\psi$, $S_\psi$ and $g_\mu$ satisfy the conditions of Theorems 1–4, thus Theorems 1–4 hold for $g_\psi^A$ and $\tilde g_\psi^A$, $S_\psi^A$ and $\tilde S_\psi^A$, $g_\mu^A$ and $\tilde g_\mu^A$.

Example 2. Marcinkiewicz operators.

Fixed $\lambda > \max(1, 2n/(n+2))$ and $0 < \gamma\le 1$. Let $\Omega$ be homogeneous of degree zero on $R^n$ with $\int_{S^{n-1}}\Omega(x')d\sigma(x') = 0$. Assume that $\Omega\in Lip_\gamma(S^{n-1})$. The Marcinkiewicz multilinear operators are defined by

and

where

and

The variants of $\mu_\Omega^A$, $\mu_S^A$ and $\mu_\lambda^A$ are defined by

and

where

and

Set

We also define that

and

which are the Marcinkiewicz operators (see ^[20]). Let $H$ be the space

or

Then, it is clear that

and

It is easy to see that $\mu_\Omega$, $\mu_S$ and $\mu_\lambda$ satisfy the conditions of Theorems 1–4, thus Theorems 1–4 hold for $\mu_\Omega^A$ and $\tilde\mu_\Omega^A$, $\mu_S^A$ and $\tilde\mu_S^A$, $\mu_\lambda^A$ and $\tilde\mu_\lambda^A$.

Example 3. Bochner-Riesz operator.

Let $\delta > (n-1)/2$, $B_t^\delta(f\hat{)(}\xi) = (1-t^2|\xi|^2)_+^\delta\hat f(\xi)$ and $B_t^\delta(z) = t^{-n}B^\delta(z/t)$ for $t > 0$. Set

and

The maximal Bochner-Riesz multilinear operator and its the variants are defined by

We also define that

which is the maximal Bochner-Riesz operator (see ^[14]). Let $H$ be the space $H = \{h:||h|| = \sup\limits_{t > 0}|h(t)| < \infty\}$, then

It is easy to see that $B_{\delta, \ast}$ satisfies the conditions of Theorems 1–4, thus Theorems 1–4 hold for $B_{\delta, \ast}^A$ and $\tilde B_{\delta, \ast}^A$.

Acknowledgments

Project supported by Scientific Research Fund of Hunan Provincial Education Departments (19A347).

Conflict of interest

No conflict of interest.