Boundedness of some operators on grand generalized Morrey spaces over non-homogeneous spaces

1.   Introduction

Let $G$ be a bounded domain in $\mathbb{R}^{n}$. Recall that a Radon measure $\mu$ on the domain $G$ is said to satisfy the polynomial growth condition, if there exists a positive constant $C_{0}$ such that, for all $x\in G$ and $r\in(0, \infty)$,

where $d$ is a fixed number in $(0, n]$ and $B(x, r): = \{y\in G: |x-y| < r\}$. The bounded domain $G$ with a such Radon measure is also called a non-homogeneous space. Moreover, Tolsa ^[24] showed that the analysis associated with the non-homogeneous space over Euclidean space $\mathbb{R}^{n}$ plays a key role in solving the long-standing open Painlevé's problem and Vitushkin's conjecture. On the development and research of the operators and function spaces over non-homogeneous spaces, we refer readers to see ^{[5,7,17,19,21,22,23,25]}.

On the other hand, Iwaniec and Sbordone ^[9] introduced the theory of grand Lebesgue space $L^{p)}$, which is one of the intensively developing directions in Modern analysis. What's more, the grand Lebesgue spaces have important applications in geometric function theory, Sobolev spaces theory and PDEs; for example, see ^[1,2,3,6,10], respectively. Since then, many papers focus on the grand spaces and the boundedness of operators on these spaces. For example, Kokilashvili ^[11] obtained the boundedness of several well-known operators on weighted grand Lebesgue spaces. In 2019, Kokilashvili et al. established the weighted extrapolation results in grand Morrey spaces and obtained some applications in PDE (see ^[15]). In 2021, Kokilashvili and Meskhi ^[12] obtain the boundedness of maximal operators, fractional integral operators and singular integral operators on generalized weighted grand Lebesgue spaces over non-doubling measures. More researches on the boundedness of integral operators in grand spaces can be seen ^{[13,14,16,20]} and the references therein. The interpolation result in grand spaces can be seen in ^[4,8].

In this paper, we will consider the boundedness of maximal operators, fractional integral operators and $\theta$-type Calderón-Zygmund operators in grand generalized Morrey spaces $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$ over non-homogeneous spaces. For the study of maximal operators, fractional integral operators and $\theta$-type Calderón-Zygmund operators in generalized Morrey spaces defined on non-homogeneous spaces, we rely on the results of references ^[5,18,21].

Now let us begin to recall some necessary notions. The following definitions of the coefficient $K_{B, S}$ and $(\alpha, \beta)$-doubling ball are from ^[23], also see ^[5].

Definition 1.1. For any two balls $B\subset S$, define

where $r_{B}$ and $r_{S}$ respectively denote the radii of the balls $B$ and $S$, and $N_{B, S}$ the smallest integer satisfying $2^{N_{B, S}}r_{B}\geq r_{S}$.

Definition 1.2. Let $\alpha, \beta\in(1, \infty)$. A ball $B\subset G$ is said to be $(\alpha, \beta)$-doubling if $\mu(\alpha B)\leq\beta\mu(B)$.

In ^[23], Tolsa showed that there exists a lot of "big" doubling balls. To be precise, given any point $x\in\mathrm{supp}(\mu)$ and $c > 0$, there exists some $(\alpha, \beta)$-doubling ball $B$ centered at $x$ with radius $r_{B}\geq c$ due to the growth condition (1.1).

Let $1 < p < \infty$ and $\varphi$ be a function on $(0, p-1]$ which is a positive bounded and satisfies $\mathop{\lim}\limits_{x\rightarrow0}\varphi(x) = 0$. The class of such functions will be simply denoted by $\Phi_{p}$. Then the norm of functions $f$ in grand Lebesgue space $L^{p), \varphi}_{\mu}(G)$ is defined by

where $L^{r}_{\mu}(G)$ is the classical Lebesgue space with respect to a measure $\mu$, and defined by the norm:

On the base of grand Lebesgue space $L^{p), \varphi}_{\mu}(G)$, we recall the definition of grand generalized Morrey spaces as follows.

Definition 1.3. Let $1 < p < \infty$ and $\varphi\in\Phi_{p}$. Suppose that $\phi: (0, \infty)\rightarrow (0, \infty)$ is an increasing function. Then grand generalized Morrey space $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$ is defined by

where

Especially, if we take $\varphi(\varepsilon) = \varepsilon^{\theta}$ with $\theta > 0$ in (1.4), then we can denote

Remark 1.4. $\mathrm{(1)}$ If we take the bounded domain $G = \mathbb{R}^{n}$ in Definition 1.1, then generalized Morrey space $\mathcal{L}^{r, \phi}_{\mu}(G)$ is just the generalized Morrey space $\mathcal{L}^{r, \phi}_{\mu}(\mathbb{R}^{n})$ (see ^[21]), namely,

$\mathrm{(2)}$ If we take function $\phi(t) = t^{\frac{p}{q}-1}$ for $t > 0$ and $1 < p\leq q < \infty$, then grand generalized Morrey space $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$ defined as in $\mathrm{(1.4)}$ is just the grand Morrey space $L^{p), q}_{\mu}(G)$ which is sightly modified in ^[15], that is,

Throughout the whole paper, $C$ represents a positive constant which is independent of the main parameters. Given any $q\in(1, \infty)$, let $q': = q/(q-1)$ denote its conjugate index. Furthermore, $m_{B}(f)$ denotes the mean value of function $f$ over ball $B$, that is, $m_{B}(f) = \frac{1}{\mu(B)}\int_{B}f(y)\mathrm{d}\mu(y)$.

2.   Hardy-Littlewood maximal operator on $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$

Let $G$ be a bounded set in $\mathbb{R}^{n}$ with non-negative Radon measure $\mu$ without mass-point and $\mu(G) < \infty$. Throughout the paper, for a function $f: G\rightarrow \mathbb{R}$, we denote

For a $\mu$-integral function $f: G\rightarrow \mathbb{R}$, the Hardy-Littlewood center maximal function in ^[12] is defined by

By (2.1), if the bounded set $G$ tends to space $\mathbb{R}^{n}$, then Hardy-Littlewood center maximal function $M_{G}$ defined as in (2.2) is denoted by

The main result of this section is stated as follows.

Theorem 2.1. Let $1 < p < \infty$, $\varphi\in\Phi_{p}$ and $\phi: (0, \infty)\rightarrow (0, \infty)$ be an increasing function. Assume that the mapping $t\mapsto\frac{\phi(t)}{t}$ is almost decreasing: there exists a positive constant $C$ such that

for $s\geq t$. Then $M_{G}$ defined as in $\mathrm{(2.2)}$ is bounded on $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$.

Proof. Choosing a number $\delta$ such that $0 < \varepsilon\leq \delta < p-1$, then, by applying Definition 1.3, write

For $\mathrm{D}_{1}$, by applying the $\mathcal{L}^{p, \phi}_{\mu}(\mathbb{R}^{n})$-boundedness of $M$ (see ^[21]) and (1.4), we can deduce that

Now let us estimate $\mathrm{D}_{2}$. Since $\delta < \varepsilon < p-1$, then we have $\frac{p-\delta}{p-\varepsilon} > 1$. Further, by applying Hölder's inequality and $\mathcal{L}^{p, \phi}_{\mu}(\mathbb{R}^{n})$-boundedness of $M$, we have

Which, together with the estimate for $\mathrm{D}_{1}$, the proof of Theorem 2.1 is finished.

With an argument similar to that used in the proof of Theorem 2.1, it is easy to obtain the following result on the maximal operator $\widetilde{M}_{r, G}$.

Corollary 2.2. Let $1 < p < \infty$, $\varphi\in\Phi_{p}$ and $\phi: (0, \infty)\rightarrow (0, \infty)$ be an increasing function. Assume that the mapping $t\mapsto\frac{\phi(t)}{t}$ is almost decreasing function satisfying $\mathrm{(2.4)}$. Then non-centered maximal operator $\widetilde{M}_{r, G}$ is bounded on $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$, where $\widetilde{M}_{r, G}$ is defined by

3.   Fractional integral operator on $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$

Let $G$ be a bounded domain in $\mathbb{R}^{n}$, then fractional integral operator $I^{G}_{\alpha}$ being associated with $G$ is defined by

By applying (2.1), it is easy to see that (3.1) is equivalent to the following form

Theorem 3.1. Let $G$ be a bounded domain in $\mathbb{R}^{n}$, $0 < \alpha < n$, $1 < p < q < \infty$ and $\frac{1}{q} = \frac{1}{p}-\frac{\alpha}{n}$. Suppose that measure $\mu$ satisfies $\mathrm{(1.1)}$, and $\phi$ satisfies $\mathrm{(2.4)}$ and the following inequality

We set

where $\varphi\in\Phi_{p}$. Then $I^{G}_{\alpha}$ is bounded from $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$ to $\mathcal{L}^{q), \psi, \phi^{\frac{p}{q}}}_{\mu}(G)$.

To prove the above theorem, we need the following lemma in ^[21].

Lemma 3.2. Let $0 < \alpha < n$, $1 < p < q < \infty$ and $\frac{1}{q} = \frac{1}{p}-\frac{\alpha}{n}$. Suppose that measure $\phi$ satisfies $\mathrm{(2.4)}$ and $\mathrm{(3.3)}$. Then $I_{\alpha}$ is bounded from $\mathcal{L}^{p, \varphi, \phi}_{\mu}(G)$ to $\mathcal{L}^{q, \psi, \phi^{\frac{p}{q}}}_{\mu}(G)$.

Proof of Theorem $\mathrm{3.1}$. Via the definition of $\psi$ and $\varphi\in\Phi_{p}$, it is easy to see that $\psi\in\Phi_{q}$. Let us fix $\sigma$ with $\sigma\in(0, q-1)$, then write

For $\mathrm{E}_{2}$. By applying Hölder's inequality and Definition 1.3, we obtain that

Thus, for small constant $\sigma > 0$,

By applying Lemma 3.2 and (1.4), we find that there exist $0 < \varepsilon\leq\sigma$ and $0 < \eta\leq\delta$ such that

holds. So we have

Further, taking the supremum on $\varepsilon$, we can obtain that

Thus, the proof of Theorem 3.1 is completed.

4.   $\theta$-Type Calderón-Zygmund operators on $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$

Definition 4.1. Let $\theta$ be a non-negative and non-decreasing function on $(0, \infty)$ with satisfying

A kernel $K_{\theta}\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n}\times\mathbb{R}^{n}\setminus\{(x, x): x\in\mathbb{R}^{n}\})$ is called a $\theta$-type kernel if there exists a constant $C > 0$ such that

for all $x, y\in\mathbb{R}^{n}$ with $x\neq y$, and for all $x, x', y\in\mathbb{R}^{n}$ with $|x-y|\geq 2|x-x'|$,

Remark 4.2. If we take the function $\theta(t) = t^{\epsilon}$ with $t > 0$ and $\epsilon > 0$, then $\theta$-type kernel $K_{\theta}$ defined as in Definition 4.1 is just the standard kernel $K$ on non-doubling measure space (see ^[21]).

Definition 4.3. Let $\rho\in(1, \infty)$ and $G$ be a bounded domain in $\mathbb{R}^{n}$. A locally integrable function $f$ is said to be in the space $\mathrm{RBMO}(\mu)$ if there exist a positive constant $C$ and, for any ball $B\subset G$, a number $f_{B}$ such that

and, for any two balls $B$ and $S$ such that $B\subset S$,

where $f_{B}$ represents the mean value of function $f$ over ball $B$, that is,

The infimum of the positive constants $C$ satisfying both $\mathrm{(4.4)}$ and $\mathrm{(4.5)}$ is defined to be the $\mathrm{RBMO}(\mu)$ norm of $f$, and it will be denoted by $\|f\|_{\mathrm{RBMO}(\mu)}$ (or $\|f\|_{\ast}$).

Let $L^{\infty}_{b}(\mu)$ be the space of all $L^{\infty}(\mu)$ functions with bounded support. A linear operator $T^{G}_{\theta}$ is called a $\theta$-type Calderón-Zygmund operator $T^{G}_{\theta}$ with kernel $K_{\theta}$ satisfying (4.2) and (4.3) if, for all $f\in L^{\infty}_{b}(\mu)$ and $x\notin\mathrm{supp}(f)$,

Thus, we also denote $T^{G}_{\theta}(f)(x) = T_{\theta}(\overline{f})(x)$.

Given a function $b\in\mathrm{RBMO}(\mu)$, the commutator $[b, T^{G}_{\theta}]$ which is generated by $b$ and $T^{G}_{\theta}$ is defined by

The main theorems of this section is stated as follows.

Theorem 4.4. Let $p\in(1, \infty)$, $\varphi\in\Phi_{p}$ and $\mu$ satisfy condition $\mathrm{(1.1)}$. Suppose that $\phi$ is a function satisfying $\mathrm{(2.4)}$, the doubling condition

and

Then $T^{G}_{\theta}$ defined as in $\mathrm{(4.6)}$ is bounded on $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$, that is, there exists a constant $C > 0$ such that, for all $f\in\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$,

Theorem 4.5. Let $\theta$ be a non-negative and non-decreasing function on $(0, \infty)$ with satisfying $\mathrm{(4.1)}$, $p\in(1, \infty)$ and $K$ satisfy $\mathrm{(4.2)}$ and $\mathrm{(4.3)}$. Suppose that $\mu$ satisfies condition $\mathrm{(1.1)}$ and $G$ is a bounded domain in $\mathbb{R}^{n}$. Then $T^{G}_{\theta, \varepsilon}$ is bounded on $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$, that is, there exists a constant $C > 0$ such that, for any $f\in\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$,

where the truncated operator $T^{G}_{\theta, \varepsilon}$ is defined by

Moreover, we also denote $T^{G}_{\theta, \varepsilon}f(x) = T_{\theta, \varepsilon}\overline{f}(x)$.

Remark 4.6. Once we prove that $\{T^{G}_{\theta, \varepsilon}\}_{\varepsilon > 0}$ is bounded in grand generalized Morrey space $\mathcal{L}^{p, \varphi, \phi}_{\mu}(G)$ uniformly on $\varepsilon > 0$, then it is east to obtain that $T^{G}_{\theta}$ is bounded in grand generalized Morrey space $\mathcal{L}^{p, \varphi, \phi}_{\mu}(G)$. Thus, we only need to prove the Theorem 4.5.

Theorem 4.7. Let $p\in(1, \infty)$, $b\in\mathrm{RBMO}(\mu)$, $\varphi\in\Phi_{p}$ and $\mu$ satisfy condition $\mathrm{(1.1)}$. Suppose that $T^{G}_{\theta}$ is bounded on $L^{2}_{\mu}(G)$, and $\phi$ is a function satisfying $\mathrm{(2.4)}$, $\mathrm{(4.8)}$ and $\mathrm{(4.9)}$. Then, the commutator $[b, T^{G}_{\theta}]$ defined as in $\mathrm{(4.7)}$ is bounded on $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$, that is, there exists a constant $C > 0$ such that, for all $f\in\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$,

To prove the above theorems, we should recall the following lemma which is slightly modified in ^[5].

Lemma 4.8. Let $\theta$ be a non-negative and non-decreasing function on $(0, \infty)$ with satisfying $\mathrm{(4.1)}$, $p\in(1, \infty)$ and $K$ satisfy $\mathrm{(4.2)}$ and $\mathrm{(4.3)}$. Suppose that $\mu$ satisfies condition $\mathrm{(1.1)}$. Then $T_{\theta, \varepsilon}$ is bounded on $L^{p}_{\mu}(\mathbb{R}^{n})$, that is, there exists a constant $C > 0$ such that, for all $f\in L^{p}_{\mu}(\mathbb{R}^{n})$,

where the truncated operator $T_{\theta, \varepsilon}$ is defined by

Also, we should establish the following lemmas about $T_{\theta, \varepsilon}$ and commutator $[b, T_{\theta}]$.

Lemma 4.9. Let $\theta$ be a non-negative and non-decreasing function on $(0, \infty)$ with satisfying $\mathrm{(4.1)}$, $p\in(1, \infty)$, and $K$ satisfy $\mathrm{(4.2)}$ and $\mathrm{(4.3)}$. Suppose that $\phi$ is a function satisfying $\mathrm{(2.4)}$, $\mathrm{(4.8)}$ and $\mathrm{(4.9)}$. Then $T_{\theta, \varepsilon}$ defined as in $\mathrm{(4.11)}$ is bounded on $\mathcal{L}^{p, \phi}_{\mu}(\mathbb{R}^{n})$.

Proof.. Let $B: = B(c_{B}, r_{B})$ be a fixed ball with center at $c_{B}$ and radius $r_{B}$, and set $\varepsilon < r_{B}$. Decompose function $f$ as

Then write

With an argument similar to that used in the estimate of Theorem 1.1 in ^[21], it is easy to see that

For $\mathrm{F}_{1}$. By applying (1.3), (2.4) and Lemma 4.8, we can deduce that

Which, combing the estimate of $\mathrm{F}_{1}$, the proof of Lemma 4.9 is finished.

Moreover, we say that $T_{\theta}$ is bounded in Lebesgue space $L^{p}_{\mu}(\mathbb{R}^{n})$ if the family of truncate operators $\{T_{\theta, \varepsilon}\}_{\varepsilon > 0}$ is bounded in $L^{p}_{\mu}(\mathbb{R}^{n})$ uniformly on $\varepsilon > 0$, and $T_{\theta}$ is bounded in generalized Morrey space $\mathcal{L}^{p, \phi}_{\mu}(\mathbb{R}^{n})$ if $\{T_{\theta, \varepsilon}\}_{\varepsilon > 0}$ is bounded in $\mathcal{L}^{p, \phi}_{\mu}(\mathbb{R}^{n})$ uniformly on $\varepsilon > 0$, where $T_{\theta}$ is defined by

Respectively, given a function $b\in\mathrm{RBMO}(\mu)$, commutator $[b, T_{\theta}]$ which is generated by $T_{\theta}$ and $b$ is defined by

Lemma 4.10. Let $b\in\mathrm{RBMO}(\mu)$, $\theta$ be a non-negative and non-decreasing function on $(0, \infty)$ with satisfying $\mathrm{(4.1)}$, $p\in(1, \infty)$, and $K$ satisfy $\mathrm{(4.2)}$ and $\mathrm{(4.3)}$. Suppose that $\phi$ is a function satisfying $\mathrm{(2.4)}$, $\mathrm{(4.8)}$ and $\mathrm{(4.9)}$. Then $[b, T_{\theta}]$ defined as in $\mathrm{(4.13)}$ is bounded on $\mathcal{L}^{p, \phi}_{\mu}(\mathbb{R}^{n})$.

To prove Lemma 4.10, we should recall the sharp maximal function $M^{\sharp}$

and the non-centered doubling maximal operator

By applying Lebesgue differential theorem, it is easy to see that for any $f\in L^{1}_{\mathrm{loc}}(\mu)$,

for $\mu$-a.e. $x\in\mathbb{R}^{n}$; see ^[22].

Proof of Lemma $4.10$. With a slightly modified argument similar to that used in the estimate of (9.4) in ^[22], we also obtain the following pointwise inequality on the commutator $[b, T_{\theta}]f$, that is,

where, for any $\rho > 1$, $\widetilde{M}_{r, (\rho)}$ is the non-centered maximal operator defined by

From (4.14), the $\mathcal{L}^{p, \phi}_{\mu}(\mathbb{R}^{n})$-boundedness of $\widetilde{M}_{r, (\rho)}$ (see ^[21]) and Lemma 4.9, it follows that

which is our desired result.

Now we state the proofs of Theorems 4.5 and 4.7 as follows.

Proof of Theorem $\mathrm{4.5}$. Let $\delta$ be a fixed constant satisfying $0 < \varepsilon < \delta < p-1$. By applying Definition 1.3, write

The estimates for $\mathrm{G}_{1}$ goes as follows. From Definition 1.3 and Lemma 4.9, it follows that

Since $0 < \delta < \varepsilon < p-1$, we notice that $\frac{p-\delta}{p-\varepsilon} > 1$. Applying Hölder inequality and the boundedness of $T^{G}_{\theta, \varepsilon}$ in $L^{p}_{\mu}(\mathbb{R}^{n})$ (see Lemma 4.8), we can deduce that

For the above result, we divide into the following cases.

Case Ⅰ If $r_{B} > 1$, then, by applying (2.4), we obtain that

Case Ⅱ If $r_{B}\leq1$, then, by applying the monotonicity of $\phi$, we can deduce that

Combing the cases Ⅰ and Ⅱ, we further obtain that

Which, together with estimate of $\mathrm{G}_{1}$, we obtain the desired result.

Proof of Theorem $\mathrm{4.7}$. Let $\delta$ be a fixed constant satisfying $0 < \varepsilon < \delta < p-1$. By applying Definition 1.3, write

The estimates for $\mathrm{F}_{1}$ is given as follows. By applying Lemma 4.10 and Definition 1.3, we obtain that

where $\overline{f}$ is defined as in (2.1).

Now let us estimate $\mathrm{F}_{2}$. By virtue of Hölder's inequality, the $L^{p}_{\mu}(\mathbb{R}^{n})$-boundedness of $[b, T^{G}_{\theta}]$ (see Lemma 4.10) and Cases Ⅰ and Ⅱ, it then follows that

Which, together with the estimate of $\mathrm{F}_{1}$, we complete the proof of Theorem 4.7

5.   Conclusions

In this paper, we mainly obtain the boundedness of Hardy-Littlewood maximal operator, fractional integral operators and $\theta$-type Calderón-Zygmund operators on the non-homogeneous grand generalized Morrey space $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$. In addition, the boundedness of commutator $[b, T^{G}_{\theta}]$ which is generated by $\theta$-type Calderón-Zygmund operator $T_{\theta}$ and $b$ on spaces $\mathcal{L}^{p), \varphi, \phi}_{\mu}(G)$ is also established.

Acknowledgments

This research was supported by National Natural Science Foundation of China(Grant No. 11561062).

Conflict of interest

The authors declare that they have no conflict of interest.