Boundedness of some operators on grand Herz spaces with variable exponent

1.   Introduction

Function spaces and operator theory are most important tools in harmonic analysis. There is a vast literature dealing with variable exponent spaces, some instances of these works are in ^{[1,2,3,4,5,6,7]}. In recent times, variable exponent function spaces has witnessed tremendous progress. In fact, it is widely recognized that variable exponent function spaces play an important role in partial differential equations and applied mathematics.

The problem on the boundedness of an intrinsic square function on Lebesgue spaces is considered by ^[8]. The first generalization of Herz spaces with variable exponent is given by Izuki ^[9]. He proved boundedness of sublinear operators in these spaces. Herz-Morrey spaces is the generalization of Herz spaces with variable exponent. This class of function spaces is initially defined by the author ^[10]. In ^[11], variable parameters were used to define continual Herz spaces, and demonstrated the boundedness of sublinear operators in these spaces.

The idea of grand Morrey spaces introduced in ^[12] and took considerable amount of attention of researchers, author also proved boundedness of class of integral operators in newly defined grand Morrey spaces. Grand Herz spaces with variable exponent was introduced in ^[13]. Inspired by the concept, in this article we demonstrated the boundedness of an intrinsic square function and higher order commutators of fractional integral operator in grand Herz spaces with variable exponent.

We divided this article into different sections. Apart from introduction, a section is dedicated to basic lemmas and definitions. One section is for boundedness of intrinsic square function on grand Herz spaces with variable exponent. Last section contains the boundedness of higher order commutators of fractional integral operator in grand Herz spaces with variable exponent.

2.   Preliminaries

For this section we refer to ^{[14,15,16,17,18]}.

Definition 2.1. If $H$ is a measurable set in $\mathbb{R}^n$ and $p(\cdot) : H \rightarrow [1, \infty)$ is a measurable function.

$\rm (a)$ Lebesgue space with variable exponent $L^{p(\cdot)}(H)$ can be defined as

Norm in $L^{p(\cdot)}(H)$ can be defined as,

$\rm (b)$ The space $L_{\mathrm{loc}}^{p(\cdot)}(H)$ can be defined as

We use these notations in this paper:

$\rm(i)$ The Hardy-Littlewood maximal operator $\mathcal{M}$ for $\boldsymbol{f} \in L_{\mathrm{loc}}^1(H)$ is defined as

where $\boldsymbol{B}(y, s): = \{x \in H : |y-x| < s\}$.

$\rm(ii)$ The set $\mathfrak{P}(H)$ is consists of all measuable functions $p(\cdot)$ satisfying

$\rm(iii)$ $\mathfrak{P}^{\log} = \mathfrak{P}^{\log}(H)$ is the class of functions $p \in \mathfrak{P}(H)$ satisfying (2.1) and log-condition defined as,

$\rm(iv)$ Let $H$ is unbounded, $\mathfrak{P}_\infty (H)$ and $\mathfrak{P}_{0, \infty} (H)$ are the subsets of $\mathfrak{P}(H)$ and values are in $[1, \infty)$ satisfying following conditions respectively

where $\Omega_\infty \in (1, \infty)$.

in the case of homogenous Herz spaces.

$\rm (v)$ Let $H$ is bounded, then $\mathfrak{P}_\infty (H)$ and $\mathfrak{P}_{0, \infty} (H)$ are the subsets of $\mathfrak{P}(H)$.

$\rm (vi)$ Let $H$ is unbounded, then $\mathfrak{P}_\infty(H)$ are the subsets of exponents in $L^\infty(H)$ and its values are in $[1, \infty]$ satisfying both conditions (2.2) and (2.3), respectively and $\mathfrak{P}_\infty^{\log}(H)$ is the set of exponent $p\in \mathfrak{P}_\infty(H)$ satisfying condition (2.1).

$\rm (vii)$ $\mathcal{B}(H)$ is the collection of $p(\cdot)\in {H}$ satisfying the condition that $M$ is bounded on $L^{p(\cdot)}(H)$.

$\rm (viii)$ $\chi _ l = \chi_{R_l}$, $R_l = D_l \setminus D_{l-1}$, $D_l = D(0, 2^l) = \{z_1 \in \mathbb{R}^n : |z_1| < 2^l\}$ for all $l \in \mathbb{Z}.$

$C$ is a constant, its value varies from line to line and independent of main parameters involved.

Lemma 2.1. ^[11] Let $D > 1$ and $\omega \in \mathfrak{P}_{0, \infty}(\mathbb{R}^n)$. Then

and

respectively, where $t_0 \geq 1$ and $t_\infty \geq 1$ is depending on $D$ but not depending on $s$.

Lemma 2.2. ^[15] [Generalised Hölder's inequality] Assume that $H$ is a measurable subset of $\mathbb{R}^n$, and $1 \leq p_-(H) \leq p_+(H)\leq \infty$. Then

holds, where $\boldsymbol{f} \in {L^{p(\cdot)}(H)}$, $\boldsymbol{g} \in {L^{q(\cdot)}(H)}$ and $\frac{1}{r(z)} = \frac{1}{p(z)}+\frac{1}{q(z)}$ for every $z \in H$.

Definition 2.2. [BMO space] A BMO function is a locally integrable function $u$ whose mean oscillation given by ${1\over |Q|} \int_Q|u(y)-u_Q|dy$ is bounded. Mathematically,

Lemma 2.3. ^[19] Let $k$ is a positive integers. Then for all $b \in BMO(\mathbb{R}^n)$ and all $j, i \in \mathbb{Z}$ for $j > i$,

Lemma 2.4. ^[19] Let $r(\cdot) \in \mathfrak{B}(\mathbb{R}^n)$; then for all balls $D$ in $\mathbb{R}^n$,

2.1.   Variable exponent Herz spaces

In this section we will define variable exponent Herz spaces.

Definition 2.3. Let $u, v \in [1, \infty)$, $\zeta \in \mathbb{R}$, the classical versions of homogenous and non-homogenous Herz spaces, can be defined by the norms,

respectively, where $F_{t, \tau}$ stands for the annulus $F_{t, \tau}: = D(0, \tau)\backslash D(0, t)$.

Definition 2.4. Let $u \in [1, \infty)$, $\zeta \in \mathbb{R}$ and $v(\cdot) \in \mathfrak{P}(\mathbb{R}^n)$. The homogenous Herz space $\dot{K}^{\zeta, u}_{v(\cdot)}(\mathbb{R}^n)$ can be defined as

where

Definition 2.5. Let $u\in [1, \infty)$, $\zeta \in \mathbb{R}$ and $v(\cdot)\in \mathfrak{P}(\mathbb{R}^n)$. The non-homogenous Herz space ${K}^{\zeta, u}_{v(\cdot)}(\mathbb{R}^n)$ can be defined as

where

3.   Grand Herz spaces with variable exponent

Next we define Grand Herz spaces with variable exponent.

Definition 3.1. ^[20] Let $a(\cdot) \in L^\infty(\mathbb{R}^n)$, $u \in [1, \infty)$, $v :\mathbb{R}^n \rightarrow [1, \infty)$, $\theta > 0$. A grand Herz spaces with variable exponent ${\dot{K} ^{a(\cdot), u), \theta}_{ v(\cdot)}}$ is defined by

where

4.   Boundedness of an intrinsic square function

In this section, we show that an intrinsic square function is bounded on ${\dot{K} ^{a(\cdot), u), \theta}_{ s(\cdot)}(\mathbb{R}^n)}$. First we will define intrinsic square function $S_\zeta f(z_1)$.

Definition 4.1. Let $z_1 \in \mathbb{R}^n$, the set $\Gamma (z_1)$ is defined as,

where $\mathbb{R}^{n+1}_+ = \mathbb{R}^n \times (0, \infty)$. Let $0 < \zeta \leq 1$ is a constant. The set $\mathcal{C}_\zeta$ consists of all functions $\phi$ defined on $\mathbb{R}^n$ such that

$\rm(i)$ supp $\phi \subset \{ |z_1| \leq 1 \}$,

$\rm (ii)$ $\int \limits _{\mathbb{R}^n} \phi (z_1) dz_1 = 0$,

$\rm(iii)$ $|\phi (z_1) -\phi(z_1')| \leq |z_1-z_1'| ^\zeta$ for $z_1, z_1' \in \mathbb{R}^n$.

For every $(z_2, t) \in \mathbb{R}^{n+1}_+$ we write $\phi _t(z_2) = t^{-n} \phi (z_2/t)$. Then we define a maximal function for $f \in L^1_{\mathrm{loc}}(\mathbb{R}^n)$.

$A_\zeta f(z_2, t) : = \sup \limits _{\phi \in \mathcal{C}_\zeta} |f * \phi _t(z_2)|$, where $(z_2, t) \in \mathbb{R}^{n+1}_+$. Using above, we define the intrinsic square function with order $\zeta$ by

An intrinsic square function $S_\zeta$ is bounded in variable Lebesgue spaces $L^{p(\cdot)}$, for more detail see ^[8].

Theorem 4.1. Let $1 < u < \infty$, $a(\cdot), s(\cdot) \in \mathfrak{P}_{0, \infty}(\mathbb{R}^n)$, and $\zeta$ be such that

(i) $\frac{-n}{s(0)} < a(0) < \frac{n}{s'(0)}$,

(ii) $\frac{-n}{s_\infty} < a_\infty < \frac{n}{s'_\infty }$.

Suppose that an intrinsic square function $S_\zeta$ bounded on Lebesgue spaces will be bounded on ${\dot{K} ^{a(\cdot), u), \theta}_{ s(\cdot)}(\mathbb{R}^n)}$.

Proof. Let $f \in {\dot{K} ^{a(\cdot), u), \theta}_{ s(\cdot)}(\mathbb{R}^n)}$, we have

As operator $S_\zeta$ is bounded on Lebesgue space $L^{s(\cdot)}(\mathbb{R}^n)$ so for $E_2$,

By using the fact $2^{k a(z_1)} = 2^{k a(0)}$, $k < 0$, $z_1 \in R_k$ implies that

For $E_{22}$, we use the fact $2^{k a(z_1)} = 2^{k a_\infty}$, $k \geq 0$, $z_1\in R_k$, we get,

For $E_1$, for $k \in \mathbb{Z}$, $l \leq k-2$, $z_1 \in R_k$ and $(z_2, t) \in \Gamma (z_1)$. For $\phi \in \mathcal{C}_\zeta$ we have

$x \in R_l$ with $|z_2-x| < t$ we obtain

As a result we get

Note that $z_1\in R_k$ and $|z_1| > 2^{k-1}$, implies that $|z_1|^{-n} \leq 2^{-kn}$, by using Hölder's inequality,

By using Lemma (2.1) we have

Splitting $E_1$ by means of Minkowski's inequality we have

Applying above results to $E_{11}$ we can get

Let $b = \frac{n}{s'(0)}-a(0)$,

by using Hölder's inequality, Fubini's theorem for series and the inequality $2^{-u(1+\psi)} < 2^{-u}$ we get,

Now for $E_{12}$ using Minkowski's inequality we have

The estimate for $A_2$ can be obtained by similar way to $E_{11}$ by replacing $s'(0)$ with $s'_\infty$ and using the fact $\frac{n}{s'_\infty}-a_\infty > 0$. For $A_1$ using Lemma (2.1) we have

As $a_\infty -\frac{n}{s'_\infty} < 0$ we have

Now by applying Hölder's inequality and using the fact that $\frac{n}{s'(0)}-a(0) > 0$ we have

Now we estimate $E_3$, we take $k \in \mathbb{Z}$, $l \geq k-2$, $z_1 \in R_k$ and $(z_2, t) \in \Gamma (z_1)$. For $x \in R_l$, $\phi \in \mathcal{C}_\zeta$ with $|z_2-x| < t$ we obtain

As a result we get

Splitting $E_3$ by using Minkowski's inequality we have

For $E_{32}$ Lemma (2.1) yields

we get

where $d = \frac{n}{s_\infty} + \alpha _\infty > 0$. Then we use Hölder's theorem for series and $2^{-u(1+\psi)} < 2^{-u}$ to obtain

Now for $E_{31}$ using Monkowski's inequality we have

The estimate for $B_1$ can be obtained by similar way to $E_{32}$ by replacing $s_\infty$ with $s(0)$ and using the fact that $\frac{n}{s(0)}+a(0) > 0$. For $B_2$ using Lemma (2.1) we have

Now by applying Hölder's inequality and using the fact that $\frac{n}{s_\infty}+a_\infty > 0$ we have

Combining the estimates for $E_1$, $E_2$ and $E_3$ yields

which ends the proof.

5.   Boundedness of higher order commutators of fractional integral operator

5.1.   Higher order commutators of fractional integral operator

Let $b$ be a locally integrable function, $0 < \zeta < n$, and $m \in \mathbb{N}$; the higher order commutators of fractional integrable operator $I ^m _{\zeta, b}$ are defined by

When $m = 0$ then $I^0_{\zeta, b} = I_\zeta$ and for, $m = 1$ $I^1_{\zeta, b} = [b, I_\zeta]$. According to Hardy-Littlewood-Sobolev theorem the fractional integral operator $I_\zeta$ is bounded operator from Lebesgue spaces $L^{p_1}(\mathbb{R}^n)$ to $L^{p_2}(\mathbb{R}^n)$ when $0 < p_1 < p_2 < \infty$ and $1/p_1- 1/p_2 = \zeta /n$.

Lemma 5.1. ^[19] Suppose that $q_1(\cdot) \in \mathfrak{P}(\mathbb{R}^n)$ satisfies (2.2), (2.3), $0 < \zeta < n/(q_1)_+$ and $1/q_1(x)-1/q_2(x) = \zeta/n$, $b \in BMO(\mathbb{R}^n)$ then

Theorem 5.1. Let $1 < p < \infty$, $b \in BMO (\mathbb{R}^n)$, $\alpha, q_2 \in \mathfrak{P}_{0, \infty}(\mathbb{R}^n)$, $1/q_1(x)-1/q_2(x) = \zeta /n$. If $0 < r < \min \{1/(q_1)_+, 1(q'_2)_+\}$, $0 < \zeta < nr$, and

Suppose that $I ^m _{\zeta, b}$ is higher order commutators of fractional integral operator bounded on Lebesgue spaces will be bounded from $\dot{K} ^{a(\cdot), p), \theta}_{ q_1(\cdot)}(\mathbb{R}^n)$ to $\dot{K} ^{a(\cdot), p), \theta}_{ q_2(\cdot)}(\mathbb{R}^n)$.

Proof. Let $f \in\dot{K} ^{a(\cdot), p), \theta}_{ q_2(\cdot)}(\mathbb{R}^n)$, and $f(x) = \sum_{l = -\infty}^\infty f(x) \chi _l(x) = \sum _{l = -\infty}^\infty f_l(x)$, we have

As operator $I ^m _{\zeta, b}$ is bounded on Lebesgue space $q_2(\cdot)$ so for $E_2$,

Here we use the fact $2^{k \alpha(x)} = 2^{k \alpha(0)}$, $k < 0$, $x \in R_k$ equivalent to say that

For $E_{22}$, we use the fact $2^{k \alpha(x)} = 2^{k a_\infty}$, $k < 0$, $x \in R_k$, we have,

For $E_1$, we use the facts that, if $x \in R_k$, $y \in R _l$ and $l \leq k-2$, then $|x-y| \sim |x| \sim 2^k$, we get

by using size condition and Hölder's inequality, we get

Thus, from Lemma (2.3), and the fact that $\| \chi_l \| _{L^{s(\cdot)}} \leq \| \chi _{B_l} \| _{L^{s(\cdot)}}$, we get

It is known, see e.g., ^[19] that $\chi_{B_k}(x) \leq C 2^{-k \zeta} I_ \zeta (\chi _{B_k})(x)$, which yields

As a result we get,

Consequently, we have

splitting $E_1$ by means of Minkowski's inequality we have

applying above results to $E_{11}$,

let $b: = \frac{n}{q_1'(0)}-\alpha(0)$, applying Hölder's inequality, Fubini's theorem for series and $2^{-p(1+\epsilon)} < 2^{-p}$ we get,

Now for $E_{12}$ using Minkowski's inequality we have

The estimate for $A_2$ follows in a similar manner to $E_{11}$ with $q_1'(0)$ replaced by $q'_{1 _\infty}$ and using the fact $b: = \frac{n}{q_{1'_\infty}}-a_\infty > 0$. For $A_1$ using Lemma (2.1) we have

as $a_\infty -\frac{n}{q'_{1_\infty}} < 0$ we have

Now we estimate $E_3$, for each $k \in \mathbb{Z}$ and $l \geq k+2$ and a.e. $x\in R_k$; splitting $E_3$ by using Minkowski's inequality and using similar method of $E_1, E_2$ we have

For $E_{31}$ by using Monkowski's inequality

The estimate for $B_1$ follows in a similar manner to $E_{32}$ with $q_{1_\infty}$ replaced by $q_1(0)$ and using the facts that $\frac{n}{q_1(0)}+\alpha(0) > 0$, $\frac{n}{q_{1_\infty}}+a_\infty > 0$, For $B_2$ using Lemma (2.1) and Hölder's inequality we have,

Finally we will estimate $E_{32}$,

Thus, from Lemma (2.3), and the fact that $\| \chi_l \| _{L^{s(\cdot)}} \leq \| \chi _{B_l} \| _{L^{s(\cdot)}}$, we get

It is known, see e.g., ^[19], that $\chi_{B_l}(x) \leq C 2^{-l \zeta} I_ \zeta (\chi _{B_l})(x)$, which yields

As a result we get,

For $E_{32}$ Lemma (2.2) yields

we get

where $d = \frac{n}{q_{1_\infty}} + a_\infty > 0$. Then we use Hölder's theorem for series and $2^{-p(1+\epsilon)} < 2^{-p}$ to obtain

Combining the estimates for $E_1$, $E_2$ and $E_3$ yields

which ends the proof.

Acknowledgements

The authors A. ALoqaily and N. Mlaiki would like to thank Prince Sultan University for paying the APC for this work through TAS LAB.

Conflicts of interest

The authors declare no conflict of interest.