Boundedness of rough fractional multilinear integral operators on generalized Morrey spaces

Abstract

We consider the boundedness of fractional multilinear integral operators with rough kernels $T_{\Omega,\alpha}^{A,m}$ on the generalized Morrey spaces $M_{p,\varphi}$. We find the sufficient conditions on the pair $(\varphi_{1},\varphi _{2})$, which ensures the boundedness of the operators $T_{\Omega,\alpha}^{A,m}$ from $M_{p,\varphi_{1}}$ to $M_{p,\varphi_{2}}$ for $1< p<\infty$. In all cases the conditions for the boundedness of the operator $T_{\Omega,\alpha}^{A,m}$ is given in terms of Zygmund-type integral inequalities on $(\varphi_{1},\varphi_{2})$, which do not make any assumption on the monotonicity of $\varphi_{1}(x,r)$, $\varphi_{2}(x,r)$ in r.

1 Introduction and results

The classical Morrey spaces were originally introduced by Morrey in [1] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [1–9]. Mizuhara [10] introduced generalized Morrey spaces. Later, Guliyev [5] defined the generalized Morrey spaces $M_{p,\varphi}$ with normalized norm.

Suppose that $\Omega\in L_{s}({\mathbb{S}}^{n-1})$ ($s>1$) is homogeneous of degree zero on ${\mathbb{R}^{n}}$ with zero means value on ${\mathbb{S}}^{n-1}$, A is a function defined on ${\mathbb {R}^{n}}$. Following [11], the rough fractional multilinear integral operator $T_{\Omega,\alpha}^{A,m}$ is defined by

$$ T_{\Omega,\alpha}^{A,m}(f) (x)=\int_{{\mathbb{R}^{n}}} \frac {R_{m}(A;x,y)}{|x-y|^{n-\alpha+m-1}}\Omega(x-y)f(y)\, dy, $$

(1.1)

where $0<\alpha<n$, and $R_{m}(A;x,y)$ is the mth remainder of Taylor series of A at x about y. More precisely,

$$ R_{m}(A;x,y)=A(x)-\sum_{|\gamma|< m} \frac{1}{\gamma!}D^{\gamma}A(y) (x-y)^{\gamma}. $$

(1.2)

When $m=1$, then $T_{\Omega,\alpha}^{A} \equiv T_{\Omega,\alpha }^{A,1}$ is just the commutator of the fractional integral $T_{\Omega ,\alpha}f(x)$ with function A,

$$ T_{\Omega,\alpha}^{A}(f) (x)=\int_{{\mathbb{R}^{n}}} \frac{\Omega (x-y)}{|x-y|^{n-\alpha}} \bigl(A(x)-A(y)\bigr)f(y)\, dy. $$

The weighted $(L_{p},L_{q})$-boundedness of such a commutator is given by Ding and Lu in [12]. When $m\ge2$, $T_{\Omega,\alpha}^{A}$ is a non-trivial generalization of the above commutator. In [13], Wu and Yang proved the following result.

Theorem A

Suppose that $\Omega\in L_{s}({\mathbb{S}}^{n-1})$ and assume that A has derivatives of order $m-1$ in $\mathit{BMO}({\mathbb{R}^{n}})$. Let $m \ge2$, $0<\alpha <n$, $1\le s'< p< n/\alpha$, and $1/q=1/p-\alpha/n$. Then there exists a constant C, independent of A and f, such that

$$ \bigl\Vert T_{\Omega,\alpha}^{A,m} f\bigr\Vert _{L_{q}({\mathbb{R}^{n}})}\le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}\|f\|_{L_{p}({\mathbb{R}^{n}})}. $$

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.

The commutators are useful in many nondivergence elliptic equations with discontinuous coefficients [3, 14–16]. In the recent development of commutators, Pérez and Trujillo-González [17] generalized these multilinear commutators and proved the weighted Lebesgue estimates.

In [18], Guliyev proved the following result.

Theorem B

Let $0 < \alpha< n$, $1 < p < n/\alpha$, and $1/q=1/p-\alpha /n$, $\Omega\in L_{s}({\mathbb{S}}^{n-1})$, $1< s \le\infty$, $A \in \mathit{BMO}$, and $(\varphi_{1},\varphi_{2})$ satisfies the condition

$$ \int_{r}^{\infty} \ln \biggl(e+\frac{t}{r} \biggr) \frac{ {\operatorname{ess\, sup}} _{t< s< \infty} \varphi_{1}(x,s) s^{\frac{n}{p}}}{t^{\frac{n}{q}}} \frac{dt}{t} \le C \varphi_{2}(x,r), $$

where C does not depend on x and r. Then the operator $T_{\Omega ,\alpha}^{A}$ is bounded from $M_{p,\varphi_{1}}$ to $M_{q,\varphi_{2}}$.

It has been proved by many authors that most of the operators which are bounded on a Lebesgue space are also bounded in an appropriate Morrey space; see [19]. As far as we know, there is no research regarding the 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 $T_{\Omega,\alpha }^{A,m}$ on generalized Morrey spaces. Our main results can be formulated as follows.

Theorem 1.1

Let $0 < \alpha< n$, $1\le s' < p < n/\alpha$, and $1/q=1/p-\alpha/n$. Suppose that $\Omega\in L_{s}({\mathbb{S}}^{n-1})$ and $(\varphi _{1},\varphi_{2})$ satisfy the condition

$$ \int_{r}^{\infty} \biggl(1+\ln \frac{t}{r} \biggr) \frac{ {\operatorname{ess\, inf}} _{t< \tau< \infty } \varphi_{1}(x,\tau) \tau^{\frac{n}{p}}}{t^{\frac{n}{q}}} \frac{dt}{t} \le C_{0} \varphi_{2}(x,r), $$

(1.3)

where $C_{0}$ does not depend on x and r. If A has derivatives of order $m-1$ in $\mathit{BMO}({\mathbb{R}^{n}})$, then the operator $T_{\Omega,\alpha}^{A,m}$ is bounded from $M_{p,\varphi_{1}}({\mathbb{R}^{n}})$ to $M_{q,\varphi_{2}}({\mathbb {R}^{n}})$. Moreover, then there is a constant $C>0$ independent of f such that

$$ \bigl\Vert T_{\Omega,\alpha}^{A,m} f\bigr\Vert _{M_{q,\varphi_{2}}}\le C \sum_{|\gamma|=m-1} \bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}\|f\|_{M_{p,\varphi_{1}}}. $$

Remark 1.1

Note that in the case $m=1$ from Theorem 1.1 we get Theorem B.

2 Generalized Morrey spaces $M_{p,\varphi}$

The classical Morrey spaces $M_{p,\lambda}$ were originally introduced by Morrey in [1] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [20, 21].

We denote by $M_{p,\lambda} \equiv M_{p,\lambda}({\mathbb{R}^{n}})$ the Morrey space, the space of all functions $f\in L_{p}^{\mathrm{loc}}({\mathbb{R}^{n}})$ with finite quasinorm,

$$\Vert f\Vert _{M_{p,\lambda}} = \sup_{x\in{\mathbb{R}^{n}}, r>0 } r^{-\frac{\lambda}{p}} \|f\|_{L_{p}(B(x,r))}, $$

where $1\le p < \infty$ and $0 \le\lambda\le n$.

Note that $M_{p,0}=L_{p}({\mathbb{R}^{n}})$ and $M_{p,n}=L_{\infty }({\mathbb{R}^{n}})$. If $\lambda<0$ or $\lambda>n$, then $M_{p,\lambda}={\Theta}$, where Θ is the set of all functions equivalent to 0 on ${\mathbb{R}^{n}}$.

In [10], Mizuhara introduced the generalized Morrey spaces $\mathcal{M}_{p,\varphi}({\mathbb{R}^{n}})$ in the following form and discussed the boundedness of the Calderón-Zygmund singular integral operators.

Definition 2.1

Let $\varphi(x,r)$ be a positive measurable function on ${\mathbb {R}^{n}}\times (0,\infty)$ and $1 \le p < \infty$. We denote by $\mathcal{M}_{p,\varphi} \equiv\mathcal{M}_{p,\varphi }({\mathbb{R}^{n}})$ the generalized Morrey space, the space of all functions $f\in L_{p}^{\mathrm{loc}}({\mathbb{R}^{n}})$ with finite quasinorm

$$\|f\|_{\mathcal{M}_{p,\varphi}} = \sup_{x\in{\mathbb {R}^{n}}, r>0} \varphi(x,r)^{-1} \|f\|_{L_{p}(B(x,r))}. $$

Note that the generalized Morrey spaces $M_{p,\varphi} \equiv M_{p,\varphi}({\mathbb{R}^{n}})$ with normalized norm,

$$\|f\|_{M_{p,\varphi}} = \sup_{x\in{\mathbb{R}^{n}}, r>0} \varphi (x,r)^{-1} \bigl\vert B(x,r)\bigr\vert ^{-\frac{1}{p}} \|f \|_{L_{p}(B(x,r))}, $$

were first defined by Guliyev in [5].

Also, in [5], by $\mathit{WM}_{p,\varphi} \equiv \mathit{WM}_{p,\varphi }({\mathbb{R}^{n}} )$ we denote the weak generalized Morrey space of all functions $f\in WL_{p}^{\mathrm{loc}}({\mathbb{R}^{n}})$ for which

$$\|f\|_{\mathit{WM}_{p,\varphi}} = \sup_{x\in{\mathbb{R}^{n}}, r>0} \varphi (x,r)^{-1} \bigl\vert B(x,r)\bigr\vert ^{-\frac{1}{p}} \|f \|_{WL_{p}(B(x,r))} < \infty. $$

By the definition, we recover the Morrey space $M_{p,\lambda}$ and weak Morrey space $\mathit{WM}_{p,\lambda}$ under the choice $\varphi(x,r)=r^{\frac{\lambda-n}{p}}$:

$$M_{p,\lambda}= {\bigl.M_{p,\varphi}\bigr|_{\varphi(x,r)=r^{\frac{\lambda-n}{p}}}},\qquad \mathit{WM}_{p,\lambda}={\bigl. \mathit{WM}_{p,\varphi}\bigr|_{\varphi(x,r)=r^{\frac{\lambda-n}{p}}}}. $$

There are many papers discussing the conditions on $\varphi(x,r)$ to obtain the boundedness of operators on the generalized Morrey spaces. For example, in [10], the function φ is supposed to be a positive growth function and satisfy the double condition: for all $r>0$, $\varphi(2r) \le D \varphi(r)$, where $D \ge1$ is a constant independent of r. This type of conditions on φ is studied by many authors; see, for example, [22, 23]. In [24], the following statement was proved by Nakai for the Riesz potential $I_{\alpha }$:

$$I_{\alpha }f(x) = \int_{{\mathbb{R}^{n}}} \frac {f(y)\,dy}{|x-y|^{n-\alpha }}, \quad 0 < \alpha < n. $$

Theorem C

Let $1\le p<\infty$, $0 < \alpha < \frac {n}{p}$, $\frac {1}{q}=\frac{1}{p}-\frac{\alpha }{n}$, and let $\varphi(x,r)$ satisfy the conditions

$$ c^{-1} \varphi(x,r) \le\varphi(x,t) \le c \varphi(x,r), $$

(2.1)

whenever $r \le t \le2r$, where c ($c \ge1$) does not depend on t, r, x, and

$$ \int_{r}^{\infty } t^{\alpha } \varphi^{q}(x,t)\frac{dt}{t} \le C \varphi^{p}(x,r), $$

(2.2)

where C does not depend on x and r. Then the operator $I_{\alpha }$ is bounded from $M_{p,\varphi}$ to $M_{q,\varphi}$ for $p > 1$ and from $M_{1,\varphi}$ to $\mathit{WM}_{1,\varphi}$ for $p=1$.

The following statements, containing the Mizuhara and Nakai results obtained in [10, 24], were proved by Guliyev in [5, 25] (see also [26]).

Theorem D

Let $1\le p<\infty$, $0 < \alpha < \frac {n}{p}$, $\frac {1}{q}=\frac{1}{p}-\frac{\alpha }{n}$, and $(\varphi_{1},\varphi_{2})$ satisfy the condition

$$ \int_{r}^{\infty } t^{\alpha } \varphi_{1}(x,t)\frac{dt}{t} \le C \varphi_{2}(x,r), $$

(2.3)

where C does not depend on x and r. Then the operator $I_{\alpha }$ is bounded from $M_{p,\varphi_{1}}$ to $M_{q,\varphi_{2}}$ for $p > 1$ and from $M_{1,\varphi_{1}}$ to $\mathit{WM}_{q,\varphi_{2}}$ for $p=1$.

Recently, in [27] and [28], Guliyev et al. introduced a weaker condition for the boundedness of Riesz potential from $M_{p,\varphi_{1}}$ to $M_{q,\varphi_{2}}$.

Theorem E

Let $1 \le p < \infty$, $0<\alpha <\frac{n}{p}$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$, and $(\varphi_{1},\varphi_{2})$ satisfy the condition

$$ \int_{t}^{\infty} \frac{ {\operatorname{ess\, sup}}_{r< s< \infty} \varphi_{1}(x,s) s^{\frac {n}{p}}}{r^{\frac{n}{q}+1}}\, dr \le C \varphi_{2}(x,t), $$

(2.4)

where C does not depend on x and t. Then the operator $I_{\alpha }$ is bounded from $M_{p,\varphi_{1}}$ to $M_{q,\varphi_{2}}$ for $1< p< q<\infty$ and from $M_{1,\varphi_{1}}$ to $\mathit{WM}_{q,\varphi_{2}}$ for $1< q<\infty$.

By an easy computation, we can check that if the pair $(\varphi _{1},\varphi_{2})$ satisfies the double condition, then it will satisfy condition (2.3). Moreover, if $(\varphi _{1},\varphi_{2})$ satisfies condition (2.3), it will also satisfy condition (2.4). But the opposite is not true. We refer to [24] and Remark 4.7 in [28] for details.

3 Some preliminaries

Let $B=B(x_{0},r_{B})$ denote the ball with the center $x_{0}$ and radius $r_{B}$. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by $|E|$. For any given $\Omega\subseteq{\mathbb{R}^{n}}$ and $0 < p < \infty$, denote by $L_{p}(\Omega)$ the space of all function f satisfying

$$ \|f\|_{L_{p}(\Omega)}= \biggl(\int_{\Omega} \bigl\vert f(x) \bigr\vert ^{p} \, dx \biggr)^{\frac {1}{p}}< \infty. $$

First we recall the definition of the space $\mathit{BMO}({\mathbb{R}^{n}})$.

Definition 3.1

Suppose that $f\in L_{1}^{\mathrm{loc}}({\mathbb{R}^{n}})$, let

$$ \|f\|_{\ast}=\sup_{x\in{\mathbb{R}^{n}}, r>0}\frac{1}{|B(x,r)|} \int _{B(x,r)}\bigl\vert f(y)-f_{B(x,r)}\bigr\vert \,dy< \infty, $$

where

$$f_{B(x,r)}=\frac{1}{|B(x,r)|} \int_{B(x,r)} f(y)\,dy. $$

Define

$$\mathit{BMO}\bigl({\mathbb{R}^{n}}\bigr)=\bigl\{ f \in L_{1}^{\mathrm{loc}}\bigl({\mathbb{R}^{n}}\bigr) : \| f \|_{\ast} < \infty \bigr\} . $$

If one regards two functions whose difference is a constant as one, then space $\mathit{BMO}({\mathbb{R}^{n}})$ is a Banach space with respect to norm $\| \cdot \|_{\ast}$.

Remark 3.1

[29]

• (1) The John-Nirenberg inequality: there are constants $C_{1}, C_{2}>0$, such that, for all $f \in \mathit{BMO}({\mathbb{R}^{n}})$ and $\beta>0$,

  $$\bigl\vert \bigl\{ x \in B : \bigl\vert f(x)-f_{B}\bigr\vert > \beta \bigr\} \bigr\vert \le C_{1} |B| e^{-C_{2} \beta/\| f \|_{\ast}}, \quad \forall B \subset {\mathbb{R}^{n}}. $$

• (2) The John-Nirenberg inequality implies that

  $$ \|f\|_{\ast}\approx\sup_{x\in{\mathbb{R}^{n}}, r>0} \biggl(\frac {1}{|B(x,r)|} \int_{B(x,r)}\bigl\vert f(y)-f_{B(x,r)}\bigr\vert ^{p} \,dy \biggr)^{\frac{1}{p}} $$

  (3.1)

  for $1< p<\infty$.

• (3) Let $f\in \mathit{BMO}({\mathbb{R}^{n}})$. Then there is a constant $C>0$ such that

  $$ \vert f_{B(x,r)}-f_{B(x,t)}\vert \le C \|f \|_{\ast}\ln\frac{t}{r} \quad \mbox{for } 0< 2r< t, $$

  (3.2)

  where C is independent of f, x, r, and t.

Lemma 3.1

[30]

Let b be a function in $\mathit{BMO}({\mathbb{R}^{n}})$, $1 \le p < \infty$, and $r_{1}, r_{2} > 0$. Then

$$\biggl( \frac{1}{|B(x,r_{1})|} \int_{B(x,r_{1})} \bigl\vert b(y)-b_{B(x,r_{2})}\bigr\vert ^{p} \,dy \biggr)^{\frac{1}{p}} \le C \biggl( 1+ \biggl\vert \ln\frac{r_{1}}{r_{2}} \biggr\vert \biggr) \| b \| _{\ast}, $$

where $C>0$ is independent of b, x, $r_{1}$, and $r_{2}$.

Below we present some conclusions as regards $R_{m}(A;x,y)$.

Lemma 3.2

[21]

Suppose b be a function on ${\mathbb{R}^{n}}$ with the mth derivatives in $L_{q}({\mathbb{R}^{n}})$, $q > n$. Then

$$ \bigl\vert R_{m}(b;x,y)\bigr\vert \le C |x-y|^{m} \sum _{|\gamma|=m} \biggl(\frac {1}{B(x,5\sqrt{n}|x-y|)}\int _{B(x,5\sqrt{n}|x-y|)} \bigl\vert D^{\gamma}b(z)\bigr\vert \, dz \biggr)^{1/q}. $$

Lemma 3.3

Let $x\in B(x_{0},r)$, $y\in B(x_{0},2^{j+1}r)\setminus B(x_{0},2^{j}r)$. Then

$$ \bigl\vert R_{m}(A;x,y)\bigr\vert \le C|x-y|^{m-1} \biggl(j\sum_{|\gamma|=m-1} \bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}+\sum _{|\gamma|=m-1} \bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\vert \biggr). $$

(3.3)

Proof

For fixed $x\in{\mathbb{R}}^{n}$, let

$$ \bar{A}(x)=A(x)-\sum_{|\gamma|=m-1}\frac{1}{\gamma !} \bigl(D^{\gamma}A\bigr)_{B(x,5\sqrt{n}|x-y|)}x^{\gamma}. $$

So

$$\begin{aligned} \bigl\vert R_{m}(A;x,y)\bigr\vert &=\bigl\vert R_{m}(\bar{A};x,y)\bigr\vert \\ & \le\bigl\vert R_{m-1}(\bar{A};x,y)\bigr\vert +\sum _{|\gamma|=m-1}\frac {1}{\gamma!}\bigl\vert \bigl(D^{\gamma}\bar{A}(y)\bigr)\bigr\vert |x-y|^{m-1}. \end{aligned}$$

(3.4)

By Lemma 3.2, we get

$$ \bigl\vert R_{m-1}(\bar{A};x,y)\bigr\vert \le C |x-y|^{m-1}\sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}. $$

(3.5)

Since $x\in B(x_{0},r)$, $y\in B(x_{0},2^{j+1}r)\setminus B(x_{0},2^{j}r)$, it is easy to see that $2^{j-1}r \le|x-y| \le2^{j+2}r$. In this way, we have

$$ B\bigl(x_{0},2^{j-1}r\bigr)\subset B\bigl(x,5\sqrt{n}\vert x-y\vert \bigr)\subset100\sqrt{n}B\bigl(x_{0},2^{j}r \bigr). $$

Then

$$ \frac{|100\sqrt{n}B(x_{0},2^{j}r)|}{|B(x,5\sqrt{n}|x-y|)|}\le\frac {|100\sqrt{n}B(x_{0},2^{j}r)|}{|B(x_{0},2^{j-1}r)|}\le C. $$

Therefore

$$\begin{aligned}& \bigl\vert \bigl(D^{\gamma}A\bigr)_{B(x,5\sqrt{n}|x-y|)}- \bigl(D^{\gamma}A\bigr)_{B(x_{0},2^{j}r)}\bigr\vert \\& \quad \le\frac{1}{|B(x,5\sqrt{n}|x-y|)|}\int_{B(x,5\sqrt {n}|x-y|)}\bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A\bigr)_{B(x_{0},2^{j}r)}\bigr\vert \,dy \\& \quad \le\frac{1}{|100\sqrt{n}B(x_{0},2^{j}r)|}\int_{100\sqrt {n}B(x_{0},2^{j}r)}\bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A\bigr)_{B(x_{0},2^{j}r)}\bigr\vert \,dy \\& \quad \le C \bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}. \end{aligned}$$

Note that

$$\begin{aligned}& \bigl\vert \bigl(D^{\gamma}A\bigr)_{B(x_{0},2^{j}r)}- \bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\vert \\& \quad \le\sum_{k=1}^{j} \bigl\vert \bigl(D^{\gamma}A\bigr)_{B(x_{0},2^{k}r)}-\bigl(D^{\gamma}A \bigr)_{B(x_{0},2^{k-1}r)}\bigr\vert \\& \quad \le2^{n} j \bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}. \end{aligned}$$

Then

$$\begin{aligned}& \bigl\vert \bigl(D^{\gamma}A\bigr)_{B(x,5\sqrt{n}|x-y|)}- \bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\vert \\& \quad \le\bigl\vert \bigl(D^{\gamma}A\bigr)_{B(x,5\sqrt{n}|x-y|)}- \bigl(D^{\gamma}A\bigr)_{B(x_{0},2^{j}r)}\bigr\vert +\bigl\vert \bigl(D^{\gamma}A\bigr)_{B(x_{0},2^{j}r)}-\bigl(D^{\gamma}A \bigr)_{B(x_{0},r)}\bigr\vert \\& \quad \le C_{j} \bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}. \end{aligned}$$

Thus

$$\begin{aligned} \bigl\vert D^{\gamma}\bar{A}(y)\bigr\vert & =\bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A\bigr)_{B(x,5\sqrt {n}|x-y|)}\bigr\vert \\ & \le\bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A \bigr)_{B(x_{0},r)}\bigr\vert +\bigl\vert \bigl(D^{\gamma}A \bigr)_{B(x,5\sqrt {n}|x-y|)}-\bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\vert \\ & \le\bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A \bigr)_{B(x_{0},r)}\bigr\vert + C_{j} \bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}. \end{aligned}$$

(3.6)

Combining with (3.4), (3.5), and (3.6), then (3.3) is proved. □

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

Lemma 3.4

[31]

Let f be a real-valued nonnegative function and measurable on E. Then

$$ \Bigl(\mathop{\operatorname{ess\, inf}} _{x\in E} f(x) \Bigr)^{-1}=\mathop{\operatorname{ess\, sup}} _{x\in E} \frac {1}{f(x)}. $$

4 A local Guliyev-type estimates

In the following theorem we get Guliyev-type local estimate (see, for example, [5, 25]) for the operator $T_{\Omega,\alpha}^{A,m}$.

Theorem 4.1

Let $\Omega\in L_{s}({\mathbb{S}}^{n-1})$, $1\le s' < p < n/\alpha$, and let $1/q = 1/p - \alpha/n$. Let A be a function defined on ${\mathbb{R}^{n}}$. Suppose that it has derivatives of order $m-1$ in $\mathit{BMO}({\mathbb{R}^{n}})$, then the inequality

$$ \bigl\Vert T_{\Omega,\alpha}^{A,m}(f)\bigr\Vert _{L_{q}(B(x_{0},r))} \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast} r^{\frac{n}{q}} \int _{2r}^{\infty } \|f\|_{L_{p}(B(x_{0},t))} t^{-\frac{n}{q}-1} \,dt $$

(4.1)

holds for any ball $B(x_{0},r)$, and for all $f\in L_{p}^{\mathrm {loc}}({\mathbb{R}^{n}})$, where the constant C is independent of f, r, and $x_{0}$.

Proof

We write f as $f=f_{1}+f_{2}$, where $f_{1}(y)=f(y)_{\chi _{B(x_{0},2r)}(y)}$, $\chi_{B(x_{0},2r)}$ denotes the characteristic function of $B(x_{0},2r)$. Then

$$ \bigl\Vert T_{\Omega,\alpha}^{A,m}(f)\bigr\Vert _{L_{q}(B(x_{0},r))}\le \bigl\Vert T_{\Omega,\alpha }^{A,m}(f_{1})\bigr\Vert _{L_{q}(B(x_{0},r))}+ \bigl\Vert T_{\Omega,\alpha}^{A,m}(f_{2}) \bigr\Vert _{L_{q}(B(x_{0},r))}. $$

Since $f_{1}\in L_{p}({\mathbb{R}^{n}})$, by the boundedness of $T^{A}_{\Omega ,\alpha}$ from $L_{p}({\mathbb{R}^{n}})$ to $L_{q}({\mathbb{R}^{n}})$ (Theorem A) we get

$$\begin{aligned} \bigl\Vert T_{\Omega,\alpha}^{A,m}(f_{1})\bigr\Vert _{L_{q}(B(x_{0},r))}&\le\bigl\Vert T_{\Omega ,\alpha}^{A,m}(f_{1}) \bigr\Vert _{L_{q}({\mathbb{R}^{n}})} \\ & \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A \bigr\Vert _{\ast}\|f_{1}\| _{L_{p}({\mathbb{R}^{n}})} \\ & = C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}\|f\|_{L_{p}(B(x_{0},2r))}. \end{aligned}$$

Moreover,

$$\begin{aligned} \|f\|_{L_{p}(B(x_{0},2r))} &\le C r^{\frac{n}{s'}-\alpha}\|f\| _{L_{p}(B(x_{0},2r))}\int _{2r}^{\infty} \frac{dt}{t^{n-\alpha+1}} \\ & \le C r^{\frac{n}{q}}\|1\|_{L_{\frac{s'p}{p-s'}}(B(x_{0},r))} \int_{2r}^{\infty} \|f\|_{L_{p}(B(x_{0},t))} \frac{dt}{t^{n-\alpha+1}} \\ & \le C r^{\frac{n}{q}}\int_{2r}^{\infty}\|f \|_{L_{p}(B(x_{0},t))} \|1\|_{L_{\frac{s'p}{p-s'}}(B(x_{0},t))}\frac{dt}{t^{n-\alpha+1}}. \end{aligned}$$

Thus

$$\begin{aligned}& \bigl\Vert T_{\Omega,\alpha}^{A,m}(f_{1}) \bigr\Vert _{L_{q}(B(x_{0},r))} \\& \quad \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}r^{\frac{n}{q}} \int_{2r}^{\infty} \|f\|_{L_{p}(B(x_{0},t))} t^{-\frac{n}{q}} \frac{dt}{t}. \end{aligned}$$

(4.2)

Let $\triangle_{i}=(B(x_{0},2^{j+1}r))\setminus(B(x_{0},2^{j}r))$, and let $x\in B(x_{0},r)$. By Lemma 3.3 we get

$$\begin{aligned} \bigl\vert T_{\Omega,\alpha}^{A,m} (f_{2}) (x)\bigr\vert \le&\biggl\vert \int_{(B(x_{0},2r))^{c}} \frac{R_{m}(A;x,y)}{|x-y|^{n-\alpha+m-1}}\Omega(x-y)f(y)\,dy\biggr\vert \\ \le& C \sum_{j=1}^{\infty}\int _{\triangle_{i}}\frac{|\Omega (x-y)f(y)|}{|x-y|^{n-\alpha}} \\ &{} \times \biggl(j \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast} +\sum _{|\gamma|=m-1}\bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\vert \biggr)\,dy \\ \le& C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A \bigr\Vert _{\ast}\sum_{j=1}^{\infty}j \int_{\triangle_{i}}\frac{|\Omega(x-y)f(y)|}{|x-y|^{n-\alpha}}\,dy \\ &{} + C \sum_{|\gamma|=m-1} \sum _{j=1}^{\infty} \int_{\triangle_{i}} \frac{|\Omega(x-y)f(y)|}{|x-y|^{n-\alpha}}\bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A \bigr)_{B(x_{0},r)}\bigr\vert \,dy \\ =&I_{1}+I_{2}. \end{aligned}$$

(4.3)

By Hölder’s inequality we have

$$ \int_{\triangle_{i}}\frac{|\Omega(x-y)f(y)|}{|x-y|^{n-\alpha}} \,dy\le \biggl(\int _{\triangle_{i}}\bigl\vert \Omega(x-y)\bigr\vert ^{s}\, dy \biggr)^{\frac{1}{s}} \biggl(\int_{\triangle_{i}} \frac{|f(y)|^{s'}}{|x-y|^{(n-\alpha )s'}} \,dy \biggr)^{\frac{1}{s'}}. $$

When $x \in B(x_{0},s)$ and $y\in\triangle_{i}$, then by a direct calculation, we can see that $2^{j-1}r\le|y-x|<2^{j+1}r$. Hence

$$ \biggl(\int_{\triangle_{i}}\bigl\vert \Omega(x-y)\bigr\vert ^{s}\, dy \biggr)^{\frac{1}{s}}\le C \|\Omega\|_{L^{s}({\mathbb{S}}^{n-1})} \bigl\vert B\bigl(x_{0},2^{j+1}r\bigr)\bigr\vert ^{\frac{1}{s}}. $$

(4.4)

We also note that if $x\in B(x_{0},r)$, $y\in B(x_{0},2r)^{c}$, then $\frac {1}{2} |x_{0}-y| \le|x-y| \le\frac{3}{2} |x_{0}-y|$. Consequently

$$ \biggl(\int_{\triangle_{i}} \frac{|f(y)|^{s'}}{|x-y|^{(n-\alpha )s'}}\,dy \biggr)^{\frac{1}{s'}}\le \frac{1}{|B(x_{0},2^{j+1}r)|^{1-\alpha/n}} \biggl(\int_{B(x_{0},2^{j+1}r)} \bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac{1}{s'}}. $$

(4.5)

Then

$$ I_{1}\le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}\sum _{j=1}^{\infty} j \bigl(2^{j+1} r \bigr)^{\alpha-n} \biggl(\int_{B(x_{0},2^{j+1}r)}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac{1}{s'}}. $$

(4.6)

Since $s' < p$, it follows from Hölder’s inequality that

$$ \biggl(\int_{B(x_{0},2^{j+1}r)}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac{1}{s'}}\le C \| f\|_{L_{p}(B(x_{0},2^{j+1}r))} \|1 \|_{L_{\frac{s'p}{p-s'}}(B(x_{0},2^{j+1}r))}. $$

(4.7)

Then

$$\begin{aligned} I_{1} & \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}\sum_{j=1}^{\infty} j \bigl(2^{j+1} r\bigr)^{\alpha-n} \biggl(\int_{B(x_{0},2^{j+1}s)} \bigl\vert f(y)\bigr\vert ^{q'}\,dy \biggr)^{\frac{1}{q'}} \\ & \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A \bigr\Vert _{\ast}\sum_{j=1}^{\infty} \biggl(1+\ln{\frac{2^{j+1}r}{r}} \biggr) \bigl(2^{j+1}r \bigr)^{\alpha-n}\|f\|_{L_{p}(B(x_{0},2^{j+1}r))} \|1\|_{L_{\frac{s'p}{p-s'}}(B(x_{0},2^{j+1}r))} \\ & \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A \bigr\Vert _{\ast}\sum_{j=1}^{\infty} \int_{2^{j+1}r}^{2^{j+2}l} \biggl(1+\ln\frac{t}{r} \biggr)\|f\|_{L_{p}(B(x_{0},t))} \|1\|_{L_{\frac{s'p}{p-s'}}(B(x_{0},t))}\frac{dt}{r^{1-\alpha+n/s'}} \\ & \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A \bigr\Vert _{\ast}\int_{2r}^{\infty} \biggl(1+\ln\frac{t}{r} \biggr)\|f\|_{L_{p}(B(x_{0},t))} \|1\|_{L_{\frac{s'p}{p-s'}}(B(x_{0},t))} \frac{dt}{t^{1-\alpha+n/s'}}. \end{aligned}$$

Then

$$ I_{1}\le C \int_{2r}^{\infty} \biggl(1+\ln\frac{t}{r} \biggr)\|f\| _{L_{p}(B(x_{0},t))} t^{-\frac{n}{q}-1} \,dt. $$

(4.8)

On the other hand, by Hölder’s inequality and (4.4), (4.5), we have

$$\begin{aligned}& \int_{\triangle_{i}}\frac{|\Omega(x-y)f(y)|}{|x-y|^{n-\alpha }}\bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\vert \,dy \\& \quad \le \biggl(\int_{\triangle_{i}}\bigl\vert \Omega(x-y)\bigr\vert ^{s}\, dy \biggr)^{\frac {1}{s}} \biggl(\int _{\triangle_{i}}\frac{|D^{\gamma}A(y)- (D^{\gamma}A)_{B(x_{0},r)}f(y)|^{s'}}{|x-y|^{(n-\alpha)s'}}\,dy \biggr)^{\frac{1}{s'}} \\& \quad \le C \sum_{|\gamma|=m-1}\sum _{j=1}^{\infty }\bigl(2^{j+1}r \bigr)^{\alpha-n} \biggl(\int_{B(x_{0},2^{j+1}r)} \bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\vert ^{s'}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac{1}{s'}}. \end{aligned}$$

Applying Hölder’s inequality we get

$$\begin{aligned}& \biggl(\int_{B(x_{0},2^{j+1}r)}\bigl\vert D^{\gamma}A(y)- \bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\vert ^{s'}\bigl\vert f(y)\bigr\vert ^{s'}\,dy \biggr)^{\frac{1}{s'}} \\& \quad \le C \|f\|_{L_{p}(B(x_{0},2^{j+1}r))}\bigl\Vert D^{\gamma}A(\cdot)- \bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\Vert _{L^{\frac{ps'}{p-s'}}(B(x_{0},2^{j+1}r))}. \end{aligned}$$

Consequently,

$$\begin{aligned} I_{2} \le& C \sum_{|\gamma|=m-1}\sum _{j=1}^{\infty} \int_{2^{j+1}r}^{2^{j+2}r} \bigl(2^{j+1}r\bigr)^{\alpha-n} \|f\|_{L_{p}(B(x_{0},t))} \\ &{} \times\bigl\Vert D^{\gamma}A(\cdot)-\bigl(D^{\gamma}A \bigr)_{B(x_{0},r)}\bigr\Vert _{L^{\frac{ps'}{p-s'}}(B(x_{0},t))}\,dt \\ \le& C \sum_{|\gamma|=m-1}\int_{2r}^{\infty}\|f\|_{L_{p}(B(x_{0},t))} \\ &{} \times\bigl\Vert D^{\gamma}A(\cdot)-\bigl(D^{\gamma}A \bigr)_{B(x_{0},r)}\bigr\Vert _{L^{\frac {ps'}{p-s'}}(B(x_{0},t))} \frac{dt}{t^{1-\alpha+n/s'}}. \end{aligned}$$

Then it follows from Lemma 3.1 that

$$\begin{aligned}& \bigl\Vert D^{\gamma}A(\cdot)-\bigl(D^{\gamma}A \bigr)_{B(x_{0},r)}\bigr\Vert _{L^{\frac {ps'}{p-s'}}(B(x_{0},t))} \\& \quad \le \biggl(\int_{B(x_{0},t)}\bigl\vert D^{\gamma}A(y)-\bigl(D^{\gamma}A\bigr)_{B(x_{0},r)}\bigr\vert ^{\frac{p s'}{p-s'}} \,dy \biggr)^{\frac{p-s'}{p s'}} \\& \quad \le C \bigl\Vert D^{\gamma}A\bigr\Vert _{\ast} \biggl(1+\ln\frac{t}{r} \biggr)r^{\frac {n(p-s')}{p s'}} \\& \quad \le C \bigl\Vert D^{\gamma}A\bigr\Vert _{\ast} \biggl(1+\ln\frac{t}{r} \biggr)r^{\frac {n}{s'}-\alpha}r^{-\frac{n}{q}}. \end{aligned}$$

Then

$$ I_{2} \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}\int_{2r}^{\infty} \biggl(1+\ln\frac{t}{r} \biggr)\|f\|_{L_{p}(B(x_{0},t))} t^{-\frac{n}{q}-1} \,dt. $$

(4.9)

Combining the estimates of $I_{1}$ and $I_{2}$, we have

$$\begin{aligned}& \sup_{x\in B(x_{0},r)}\bigl\vert T_{\Omega,\alpha}^{A,m}(f_{2}) (x)\bigr\vert \\& \quad \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}\int_{2r}^{\infty} \biggl(1+\ln\frac{t}{r} \biggr)\|f\|_{L_{p}(B(x_{0},t))} t^{-\frac {n}{q}-1} \,dt. \end{aligned}$$

Then we get

$$\begin{aligned}& \bigl\Vert T_{\Omega,\alpha}^{A,m}(f_{2}) \bigr\Vert _{L_{q}(B(x_{0},r))} \\& \quad \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}r^{\frac {n}{q}}\int_{2r}^{\infty} \|f\|_{L_{p}(B(x_{0},t))}t^{-\frac{n}{q}-1} \,dt. \end{aligned}$$

(4.10)

This completes the proof of Theorem 4.1. □

5 Proof of Theorem 1.1

Since $f\in M_{p,\varphi_{1}}({\mathbb{R}^{n}})$, by Lemma 3.4 and the non-decreasing, with respect to t, of the norm $\|f\| _{L_{p}(B(x_{0},t))}$, we get

$$\begin{aligned}& \frac{\|f\|_{L_{p}(B(x_{0},t))}}{{\operatorname{ess\, inf}}_{0< t< \tau< \infty } \varphi _{1}(x_{0},\tau) \tau^{\frac{n}{p}}} \\& \quad \le\mathop{\operatorname{ess\, sup}}_{0< t< \tau< \infty } \frac{\| f\|_{L_{p}(B(x_{0},t))}}{\varphi _{1}(x_{0},\tau) \tau^{\frac{n}{p}}} \\& \quad \le\sup_{\tau>0} \frac{\|f\|_{L_{p}(B(x_{0},\tau))}}{\varphi _{1}(x_{0},\tau) \tau^{\frac{n}{p}}} \\& \quad \le\|f\|_{M_{p,\varphi_{1}}}. \end{aligned}$$

Since $(\varphi_{1},\varphi_{2})$ satisfies (1.3), we have

$$\begin{aligned}& \int_{r}^{\infty } \biggl(1+\ln\frac{t}{r} \biggr)\|f\|_{L_{p}(B(x_{0},t))} t^{-\frac{n}{q}-1} \,dt \\& \quad = \int_{r}^{\infty }\frac{\|f\|_{L_{p}(B(x_{0},t))}}{ {\operatorname{ess\, inf}}_{t< \tau< \infty } \varphi_{1}(x_{0},\tau) \tau^{\frac{n}{p}}} \biggl(1+ \ln\frac{t}{r} \biggr) \frac{ {\operatorname{ess\, inf}}_{t< \tau< \infty } \varphi _{1}(x_{0},\tau)\tau^{\frac {n}{p}}}{t^{\frac{n}{q}}}\frac{dt}{t} \\& \quad \le\|f\|_{M_{p,\varphi_{1}}}\int_{r}^{\infty } \biggl(1+\ln\frac {t}{r} \biggr) \frac{ {\operatorname{ess\, inf}}_{t< \tau< \infty } \varphi _{1}(x_{0},\tau)\tau^{\frac {n}{p}}}{t^{\frac{n}{q}}}\frac{dt}{t} \\& \quad \le C \|f\|_{M_{p,\varphi_{1}}}\varphi_{2}(x_{0},t). \end{aligned}$$

Then by (4.1) we get

$$\begin{aligned}& \bigl\Vert T_{\Omega,\alpha}^{A,m}(f)\bigr\Vert _{M_{q,\varphi_{2}}} \\& \quad = \sup_{x_{0}\in{\mathbb{R}^{n}},t>0}\frac{1}{\varphi _{2}(x_{0},t)} \biggl( \frac{1}{|B(x_{0},t)|}\int_{B(x_{0},t)} \bigl\vert T_{\Omega,\alpha}^{A,m}(f) (y)\bigr\vert ^{q} \,dy \biggr)^{1/q} \\& \quad \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast} \sup_{x_{0}\in{\mathbb{R}^{n}},t>0}\frac{1}{\varphi_{2}(x_{0},t)} \int_{r}^{\infty } \biggl(1+\ln\frac{t}{r} \biggr) \|f\|_{L_{p}(B(x_{0},t))} t^{-\frac{n}{q}-1} \,dt \\& \quad \le C \sum_{|\gamma|=m-1}\bigl\Vert D^{\gamma}A\bigr\Vert _{\ast}\|f\| _{M_{p,\varphi_{1}}}. \end{aligned}$$