The boundedness of a class of fractional type rough higher order commutators on vanishing generalized weighted Morrey spaces

TL;DR

This paper establishes new bounds for a class of fractional type rough higher order commutators on vanishing generalized weighted Morrey spaces, advancing the understanding of their boundedness properties.

Contribution

It introduces improved bounds for these commutators specifically on vanishing generalized weighted Morrey spaces, a novel contribution in this area.

Findings

New bounds for fractional type rough higher order commutators

Enhanced understanding of boundedness on vanishing generalized weighted Morrey spaces

Extension of existing theories to more general weighted spaces

Abstract

This paper includes new bounds concepting the vanishing generalized weighted Morrey space. In this sense, it is outlined improved bounds about the a class of fractional type rough higher order commutators on vanishing generalized weighted Morrey spaces.

Equations

$$\Omega (\lambda x)=\Omega (x)\text{ for any }\lambda >0\text{, }x\in {\mathbb{R}}^n\setminus \{0\}$$

$$\underset{S^{n-1}}{\int }\Omega (x^{\prime })d\sigma (x^{\prime })=0,$$

$$T_{\Omega ,\alpha }^{A,k}f(x)$$

$$M_{\Omega ,\alpha }^{A,k}f(x)$$

$$T_{\Omega ,\alpha }f(x)=\underset{{\mathbb{R}}^n}{\int }\frac{\Omega (x-y)}{|x-y{|}^{n-\alpha }}f(y)dy0<\alpha <n$$

$$M_{\Omega ,\alpha }f(x)=\underset{r>0}{\sup }\frac{1}{r^{n-\alpha }}\underset{|x-y|<r}{\int }\left|\Omega (x-y)\right|\left|f(y)\right|dy0<\alpha <n.$$

$$\left[A,T_{\Omega ,\alpha }\right]f\left(x\right)$$

$$\left[A,M_{\Omega ,\alpha }\right]f\left(x\right)$$

$$f_B=M\left(f,B\right)=M\left(f,x,r\right)=\frac{1}{|B|}\underset{B}{\int }f(y)dy,$$

$$MO\left(f,B\right)=MO\left(f,x,r\right)=\frac{1}{|B|}\underset{B}{\int }|f(y)-f_B|dy.$$

$$M{O}_{\varphi }\left(f,B\right)=M{O}_{\varphi }\left(f,x,r\right)=\frac{1}{\varphi \left(|B|\right)|B|}\underset{B}{\int }|f(y)-f_B|dy.$$

$$BM{O}_{\varphi }=\left\{f\in L_1^{loc}({\mathbb{R}}^n):\underset{B}{\sup }M{O}_{\varphi }\left(f,B\right)<\infty \right\}$$

$${\Vert f\Vert }_{BM{O}_{\varphi }}=\underset{B}{\sup }M{O}_{\varphi }\left(f,B\right).$$

$$L_p(w)$$

$$L_{\infty ,w}$$

$$[w{]}_{A\left(p,q\right)}:=\underset{B}{\sup }{\left(|B{|}^{-1}\underset{B}{\int }w(x{)}^{q}dx\right)}^{\frac{1}{q}}{\left(|B{|}^{-1}\underset{B}{\int }w(x{)}^{-p^{\prime }}dx\right)}^{\frac{1}{p^{\prime }}}<\infty ,$$

$$[w{]}_{A\left(p,q\right)}\ge [w{]}_{A\left(p,q\right)(B)}=|B{|}^{\frac{1}{p}-\frac{1}{q}-1}\Vert w{\Vert }_{L_q(B)}\Vert w^{-1}{\Vert }_{L_{p^{\prime }}(B)}\ge 1.$$

$${\left(\underset{B}{\int }w(x{)}^{q}dx\right)}^{\frac{1}{q}}{\left(\underset{B}{\int }w(x{)}^{-p^{\prime }}dx\right)}^{\frac{1}{p^{\prime }}}\lesssim {\left|B\right|}^{\frac{1}{q}+\frac{1}{p^{\prime }}}.$$

$$\Vert f{\Vert }_{M_{p,\phi }(w)}=\underset{x\in {\mathbb{R}}^n,r>0}{\sup }\frac{1}{\phi (x,r)}\Vert f{\Vert }_{L_p(B(x,r),w)}<\infty .$$

$$\underset{r\to 0}{\lim }\underset{x\in {\mathbb{R}}^n}{\sup }\frac{1}{\phi (x,r)}\Vert f{\Vert }_{L_p(B(x,r),w)}=0.$$

$$\underset{t\to 0}{\lim }\underset{x\in {\mathbb{R}}^n}{\sup }\frac{{\left(w(B(x,t))\right)}^{{}^{\frac{1}{p}}}}{\phi (x,t)}=0,$$

$$\underset{t>1}{\inf }\underset{x\in {\mathbb{R}}^n}{\sup }\frac{{\left(w(B(x,t))\right)}^{{}^{\frac{1}{p}}}}{\phi (x,t)}>0.$$

$$\Vert f{\Vert }_{V{M}_{p,\phi }(w)}=\underset{x\in {\mathbb{R}}^n,r>0}{\sup }\frac{1}{\phi (x,r)}\Vert f{\Vert }_{L_p(B(x,r),w)},$$

$$\underset{r\to 0}{\lim }\underset{x\in {\mathbb{R}}^n}{\sup }\frac{1}{\phi (x,r)}\Vert f{\Vert }_{L_p(B(x,r),w)}=0,$$

$$V{M}_{p,\phi }\left(w\right)\subset M_{p,\phi }\left(w\right),\Vert f{\Vert }_{M_{p,\phi }\left(w\right)}\le \Vert f{\Vert }_{V{M}_{p,\phi }\left(w\right)}.$$

$$c_{\delta }:=\underset{\delta }{\overset{\infty }{\int }}{\left(1+\ln \frac{t}{r}\right)}^k\underset{x\in {\mathbb{R}}^n}{\sup }\frac{{\phi }_1\left(x,t\right)}{{\left(w^q\left(B\left(x,t\right)\right)\right)}^{\frac{1}{q}}}\frac{1}{t}dt<\infty$$

$$\underset{r}{\overset{\infty }{\int }}{\left(1+\ln \frac{t}{r}\right)}^k\frac{{\phi }_1\left(x,t\right)}{{\left(w^q\left(B\left(x,t\right)\right)\right)}^{\frac{1}{q}}}\frac{1}{t}dt\lesssim \frac{{\phi }_2(x,r)}{{\left(w^q\left(B\left(x,t\right)\right)\right)}^{\frac{1}{q}}},$$

$${\Vert T_{\Omega ,\alpha }^{A,k}f\Vert }_{V{M}_{q,{\phi }_2}\left(w^q,{\mathbb{R}}^n\right)}\lesssim {\Vert A\Vert }_{\ast }^k{\Vert f\Vert }_{V{M}_{p,{\phi }_1}\left(w^p,{\mathbb{R}}^n\right)},$$

$${\Vert M_{\Omega ,\alpha }^{A,k}f\Vert }_{V{M}_{q,{\phi }_2}\left(w^q,{\mathbb{R}}^n\right)}\lesssim {\Vert A\Vert }_{\ast }^k{\Vert f\Vert }_{V{M}_{p,{\phi }_1}\left(w^p,{\mathbb{R}}^n\right)}.$$

$$T_{\Omega }^{A,k}f(x)$$

$$M_{\Omega }^{A,k}f(x)$$

Full text

1. Introduction

Let $\Omega\in L_{s}(S^{n-1})$, $1<s\leq\infty$. $\Omega$ is the function defined on ${\mathbb{R}^{n}}\setminus\{0\}$ satisfying the homogeneous of degree zero condition, that is,

[TABLE]

and the integral zero property (=the vanishing moment condition) over the unit sphere $S^{n-1}$, that is,

[TABLE]

where $x^{\prime}=\frac{x}{|x|}$ for any $x\neq 0$.

In this paper we consider the following higher order ($=k$-th order) commutator operators of rough fractional integral and maximal operators,

[TABLE]

and

[TABLE]

as long as the integrals above make sense, where rough fractional integral operator $T_{\Omega,\alpha}$ and rough fractional maximal operator $M_{\Omega,\alpha}$ are defined by

[TABLE]

and

[TABLE]

For $k=1$ above, $T_{\Omega,\alpha}^{A,k}$ and $M_{\Omega,\alpha}^{A,k}$ are obviously reduced to the rough commutator operators of $T_{\Omega,\alpha}$ and $M_{\Omega,\alpha}$, respectively:

[TABLE]

and

[TABLE]

Moreover, $T_{\Omega,\alpha}^{A,k}$ and $M_{\Omega,\alpha}^{A,k}$ are trivial generalizations of the above commutators, respectively.

Here and henceforth, $F\approx G$ means $F\gtrsim G\gtrsim F$; while $F\gtrsim G$ means $F\geq CG$ for a constant $C>0$; and also $C$ stands for a positive constant that can change its value in each statement without explicit mention.

Now, let us list some definitions that we need in the proof of following Theorem 1:

Definition 1**.**

$\left(\text{{Bounded Mean Oscillation (BMO)}}\right)$* We denote the mean value of $f$ on $B=B(x,r)\subset{\mathbb{R}^{n}}$ by*

[TABLE]

and the mean oscillation of $f$ on $B=B(x,r)$ by

[TABLE]

We also define for a non-negative function $\phi$ on ${\mathbb{R}^{n}}$

[TABLE]

Now we define

[TABLE]

and

[TABLE]

The most important of these spaces occurs when $\phi=1$, in which case $BMO_{\phi}=BMO$.

Definition 2**.**

[1, 3]$\left(\text{{Weighted Lebesgue space}}\right)$* Let $1\leq p\leq\infty$ and given a weight $w\left(x\right)\in A_{p}\left({{\mathbb{R}^{n}}}\right)$, we shall define weighted Lebesgue spaces as*

[TABLE]

Here and later, we refer to $A_{p}$ as the the Muckenhoupt classes. That is, $w\left(x\right)\in A_{p}\left({{\mathbb{R}^{n}}}\right)$ for some $1<p<\infty$ if $\left(\frac{1}{|B|}\mathop{\displaystyle\int}\limits_{B}w(y)dy\right)\left(\frac{1}{|B|}\mathop{\displaystyle\int}\limits_{B}w(y)^{-\frac{1}{p-1}}dy\right)^{p-1}\leq C$ for all balls $B$ (see [1] for more details).

Now, let us consider the Muckenhoupt-Wheeden class $A\left(p,q\right)$ in [5]. One says that $w\left(x\right)\in A\left(p,q\right)$ for $1<p<q<\infty$ if and only if

[TABLE]

where the supremum is taken over all the balls $B$. Note that, by Hölder’s inequality, for all balls $B$ we have

[TABLE]

By (1.5), we have

[TABLE]

On the other hand, let $\mu\left(x\right)=w\left(x\right)^{s^{\prime}}$, $\tilde{p}=\frac{p}{s^{\prime}}$ and $\tilde{q}=\frac{q}{s^{\prime}}$. If $w\left(x\right)^{s^{\prime}}\in A\left(\frac{p}{s^{\prime}},\frac{q}{s^{\prime}}\right)$, then we get $\mu\left(x\right)\in A\left(\tilde{p},\tilde{q}\right)$.

Now, we introduce some spaces which play important roles in PDE. Except the weighted Lebesgue space $L_{p}(w)$, the weighted Morrey space $L_{p,\kappa}(w)$, which is a natural generalization of $L_{p}(w)$ is another important function space. Then, the definition of generalized weighted Morrey spaces $M_{p,\varphi}\left(w\right)$ which could be viewed as extension of $L_{p,\kappa}(w)$ has been given as follows:

For $1\leq p<\infty$, positive measurable function $\varphi(x,r)$ on ${\mathbb{R}^{n}}\times(0,\infty)$ and nonnegative measurable function $w$ on ${\mathbb{R}^{n}}$, $f\in M_{p,\varphi}(w)\equiv M_{p,\varphi}({\mathbb{R}^{n}},w)$ if $f\in L_{p,w}^{loc}({\mathbb{R}^{n}})$ and

[TABLE]

is finite. Note that for $\varphi(x,r)\equiv w(B(x,r))^{\frac{\kappa}{p}}$, $0<\kappa<1$ and $\varphi(x,r)\equiv 1$, we have $M_{p,\varphi}(w)=L_{p,\kappa}(w)$ and $M_{p,\varphi}(w)=L_{p}(w)$, respectively. Moreover, Gürbüz [2] proved that the operators $T_{\Omega,\alpha}^{A,k}$ and $M_{\Omega,\alpha}^{A,k}$ are bounded from one generalized weighted Morrey space $M_{p,\varphi_{1}}\left(w^{p},{\mathbb{R}^{n}}\right)$ to another $M_{q,\varphi_{2}}\left(w^{q},{\mathbb{R}^{n}}\right)$.

The following definition was introduced by Gürbüz [4].

Definition 3**.**

(Vanishing generalized weighted Morrey spaces) For $1\leq p<\infty$, $\varphi(x,r)$ is a positive measurable function on ${\mathbb{R}^{n}}\times(0,\infty)$ and nonnegative measurable function $w$ on ${\mathbb{R}^{n}}$, $f\in VM_{p,\varphi}\left(w\right)\equiv VM_{p,\varphi}({\mathbb{R}^{n},w})$ if $f\in L_{p,w}^{loc}({\mathbb{R}^{n}})$ and

[TABLE]

Inherently, it is appropriate to impose on $\varphi(x,t)$ with the following circumstances:

[TABLE]

and

[TABLE]

From (1.7) and (1.8), we easily know that the bounded functions with compact support belong to $VM_{p,\varphi}\left(w\right)$. On the other hand, the space $VM_{p,\varphi}(w)$ is Banach space with respect to the following finite quasi-norm

[TABLE]

such that

[TABLE]

we omit the details. Moreover, we have the following embeddings:

[TABLE]

Henceforth, we denote by $\varphi\in\mathcal{B}\left(w\right)$ if $\varphi(x,r)$ is a positive measurable function on ${\mathbb{R}^{n}}\times(0,\infty)$ and positive for all $(x,r)\in{\mathbb{R}^{n}}\times(0,\infty)$ and satisfies (1.7) and (1.8).

Inspired of [2], the aim of the present paper is to study the boundedness of the operators $T_{\Omega,\alpha}^{A,k}$ and $M_{\Omega,\alpha}^{A,k}$ generated by $T_{\Omega,\alpha}$ and $M_{\Omega,\alpha}$ with a $BMO$ functions on vanishing generalized weighted Morrey spaces, respectively. That is, in this paper we will consider this problem.

2. Main results

Let us state our main result as follows.

Theorem 1**.**

Suppose that $0<\alpha<n$, $1\leq s^{\prime}<p<\frac{n}{\alpha}$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$, $1<q<\infty$, $\Omega\in L_{s}(S^{n-1})\left(s>1\right)$ satisfies (1.1) such that $k\in\mathbb{N}$, $w\left(x\right)^{s^{\prime}}\in A\left(\frac{p}{s^{\prime}},\frac{q}{s^{\prime}}\right)$, $A\in BMO\left({\mathbb{R}^{n}}\right)$, $T_{\Omega,\alpha}^{A,k}$, $M_{\Omega,\alpha}^{A,k}$ are defined as (1.3), (1.4) and $T_{\Omega,\alpha}^{A,k}$ satisfies (13) in [2]. If $\varphi_{1}\in\mathcal{B}\left(w^{p}\right)$, $\varphi_{2}\in\mathcal{B}\left(w^{q}\right)$ and the pair $(\varphi_{1},\varphi_{2})$ satisfies the conditions

[TABLE]

for every $\delta>0$, and

[TABLE]

then the operator $T_{\Omega,\alpha}^{A,k}$ is bounded from $VM_{p,\varphi_{1}}\left(w^{p}\right)$ to $VM_{q,\varphi_{2}}\left(w^{q}\right)$. Moreover,

[TABLE]

[TABLE]

For $\alpha=0$, from Theorem 1, we get the following:

Corollary 1**.**

Suppose that $1<p<\infty$, $s^{\prime}<p$, $\Omega\in L_{s}(S^{n-1})\left(s>1\right)$ satisfies (1.1) and (1.2) such that $k\in\mathbb{N}$, $w\left(x\right)^{s^{\prime}}\in A_{\frac{p}{s^{\prime}}}$, $A\in BMO\left({\mathbb{R}^{n}}\right)$, $T_{\Omega}^{A,k}$, $M_{\Omega}^{A,k}$ are defined as

[TABLE]

and the corresponding higher order ($=k$-th order) commutator operator of $M_{\Omega}$:

[TABLE]

and $T_{\Omega}^{A,k}$ satisfies (11) in [3]. If $\varphi\in\mathcal{B}\left(w\right)$ and the pair $(\varphi_{1},\varphi_{2})$ satisfies the conditions

[TABLE]

for every $\delta^{\prime}>0$, and

[TABLE]

Then,

[TABLE]

[TABLE]

3. Proof of the main result

**Proof of Theorem 1. **

Proof.

By Definition 3, (13) in [2] and (2.2) we get

[TABLE]

At last, we need to prove that

[TABLE]

Indeed, for any $\epsilon>0$, let $0<r<\psi$. By (13) in [2], we have

[TABLE]

where

[TABLE]

and

[TABLE]

For $\sup\limits_{x\in{\mathbb{R}^{n}}}\sup\limits_{0<r<t}\frac{\left\|f\right\|_{L_{q}\left(w^{q},B(x,r)\right)}}{\varphi_{2}(x,r)}<\frac{\epsilon}{2}$, we can select any constant $\psi>0$. This allows to guess the first term properly from the type $r\in\left(0,\psi\right)$ such that

[TABLE]

For the second term, in view of (2.1), we obtain

[TABLE]

Since $\varphi_{2}\in\mathcal{B}\left(w^{q}\right)$, it gets along to select $r$ minor sufficient such that

[TABLE]

Hence,

[TABLE]

Thus,

[TABLE]

Therefore,

[TABLE]

As a result, (2.3) holds. On the other hand, since $M_{\Omega,\alpha}^{A,k}f(x)\leq\widetilde{T}_{\left|\Omega\right|,\alpha}^{A,k}\left(\left|f\right|\right)(x)$, $x\in{\mathbb{R}^{n}}$ (see Lemma 6 in [2]) we can also use the same method for $M_{\Omega,\alpha}^{A,k}$, so we omit the details. As a result, we complete the proof of Theorem 1.