BOUNDEDNESS OF DIFFERENTIAL TRANSFORMS FOR FRACTIONAL HEAT SEMIGROUPS GENERATED BY SCHRÖDINGER OPERATORS

Abstract

Let $L=-\Delta +V$ be a Schrödinger operator in ${\mathbb R}^n$ with $n\geq 3$, where $\Delta $ is the Laplace operator denoted by $\Delta =\sum ^{n}_{i=1}({\partial ^{2}}/{\partial x_{i}^{2}})$ and the nonnegative potential V belongs to the reverse Hölder class $(RH)_{q}$ with $q>n/2$. For $\alpha \in (0,1)$, we define the operator

$$ \begin{align*} T_N^{L^{\alpha}} f(x) =\sum_{j=N_1}^{N_2} v_j(e^{-a_{j+1}L^\alpha} f(x)-e^{-a_{j}L^\alpha} f(x)) \quad \mbox{for all }x\in \mathbb R^n, \end{align*} $$

where $\{e^{-tL^\alpha } \}_{t>0}$ is the fractional heat semigroup of the operator L, $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequence and $\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence.

We investigate the boundedness of the operator $T_N^{L^{\alpha }}$ and the related maximal operator $T^*_{L^{\alpha }}f(x):=\sup _N \vert T_N^{L^{\alpha }} f(x)\vert $ on the spaces $L^{p}(\mathbb {R}^{n})$ and $BMO_{L}(\mathbb {R}^{n})$, respectively. As extensions of $L^{p}(\mathbb {R}^{n})$, the boundedness of the operators $T_N^{L^{\alpha }}$ and $T^*_{L^{\alpha }}$ on the Morrey space $L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ and the weak Morrey space $WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$ has also been proved.

1 Introduction

Consider the Schrödinger operator ${L}=-\Delta +V$ in ${\mathbb R}^n$ with $n\ge 3$ , where $ {\Delta = \sum _{j=1}^n({\partial ^2}/ {\partial x^2_j})}$ is the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class $(RH)_q$ : that is, there exists $C>0$ such that

$$ \begin{align*} \bigg(\frac{1}{|B|}\int_B V(x)^q\,dx\bigg)^{{1}/{q}}\leq \frac{C}{|B|}\int_B V(x)\,dx \end{align*} $$

holds for every ball B in ${\mathbb R}^{n}$ . In this paper, we mainly focus on the fractional Schrödinger operator ${L^\alpha }=(-\displaystyle \Delta +V)^\alpha $ for $\alpha \in (0,1)$ . When $V=0$ , $L^{\alpha }$ reduces to the fractional Laplace operator $(-\Delta )^{\alpha }$ , which is represented as

$$ \begin{align*} \mathcal{F}((-\Delta)^\alpha f)(\xi)=|\xi|^{2\alpha}\mathcal F( f) (\xi), \end{align*} $$

where $\mathcal F$ denotes the Fourier transform defined as

$$ \begin{align*}\mathcal{F}(f)(\xi)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^n}f(x)e^{-ix\cdot\xi}\,dx \end{align*} $$

for any $f\in S'$ , where $S'$ denotes the space of tempered distributions. The corresponding fractional heat semigroup is given by

(1-1) $$ \begin{align} \mathcal F(e^{-t(-\Delta)^\alpha}f) (\xi):=e^{-t|\xi|^{2\alpha}}\mathcal F (f)(\xi) \quad \mbox{for all }\alpha\in (0, 1). \end{align} $$

In [12], by an invariant derivative technique and the Fourier analysis method, Miao, Yuan and Zhang concluded that the convolution kernels of $\{e^{- {t(-\Delta )^\alpha }}\}_{t>0}$ , denoted by $K^{-\Delta }_{\alpha , t}(\cdot )$ , satisfy the following pointwise estimate: there exists a positive constant C such that, for any $(x,t)\in {\mathbb R}_+^{n+1}$ ,

$$ \begin{align*} K^{-\Delta}_{\alpha, t}(x) \le \frac {Ct}{(t^{1/(2\alpha)}+|x|)^{n+2\alpha}}. \end{align*} $$

Then, arguing in the same way as in [6, Remark 2.2], we also have the lower estimate

$$ \begin{align*} \frac {Ct}{(t^{1/(2\alpha)}+|x|)^{n+2\alpha}} \leq K^{-\Delta}_{\alpha, t}(x). \end{align*} $$

Different from the case of $-\displaystyle \Delta $ , for the Schrödinger operator ${L}$ with the positive potential V, the fractional heat semigroup $\{e^{- {tL^\alpha }}\}_{t>0}$ with $\alpha \in (0,1)$ generated from $L^\alpha $ cannot be defined via (1-1). In [10], Li et al. used the subordinative formula to obtain the following pointwise estimate of the integral kernel $K^{L}_{\alpha , t}(\cdot ,\cdot )$ of $e^{- {tL^\alpha }}$ : that is, for every $N>0,$ there exists a positive constant $C_N$ such that, for any $x,y\in \mathbb R^{n}$ ,

(1-2) $$ \begin{align} |K^{L}_{\alpha, t}(x,y)|\leq\frac{C_{N}t}{(t^{1/(2\alpha)}+|x-y|)^{n+2 \alpha }} \bigg(1+\frac{t^{1/(2\alpha)}}{\rho(x)}+\frac{t^{1/(2\alpha)}}{\rho(y)}\bigg)^{-N}. \end{align} $$

Let $(\mathbb {X}, \Sigma , m)$ be a complete nonatomic probability space and let $\tau $ be a measurable, invertible, ergodic measure-preserving point transformation mapping $\mathbb {X}$ to itself. For an increasing sequence of positive integers denoted by $\{n_{k}\}_{k=0}^{\infty }$ , the ergodic average related with $\{n_{k}\}^{\infty }_{k=0}$ is defined as

$$ \begin{align*}A_{k}(f)(x):=\frac{1}{n_{k}}\sum^{n_{k}-1}_{j=0}f(\tau^{j}x).\end{align*} $$

To investigate the convergence property of the difference average

$$ \begin{align*}\sum^{\infty}_{k=1}v_{k}(A_{k}(f)(x)-A_{k-1}(f)(x)),\end{align*} $$

Jones and Rosemblatt in [9] considered the partial sum

$$ \begin{align*}T_{n}(f)(x):=\sum^{n}_{k=1}v_{k}(A_{k}(f)(x)-A_{k-1}(f)(x)),\end{align*} $$

where $\{v_{k}\}$ is some bounded sequence of real (or complex) numbers. As one of the main results of [9], the authors proved that if $\{n_{k}\}$ is a lacunary sequence of positive integers and $v=\{v_{k}\}$ is any sequence such that $\|v\|_{\infty }\leq 1$ , then $T_{n}(f)$ converges almost everywhere for all functions $f\in L^{p}$ , $1\leq p\leq \infty $ , and converges in $L^p$ norm for all functions $f\in L^{p}$ , $1< p< \infty $ . In [3], this method of analyzing the convergence of sequences was applied to differential transforms by Bernardis et al. Based on the idea used in [3, 9], in Zhang et al. [19] and Zhang–Torrea [20], the authors investigated the differential transforms related to the one-sided fractional Poisson-type operator sequence and the heat semigroup, which are generated by the Laplacian and Schrödinger operators, respectively.

The aim of this paper is to analyze the behavior of the sum

(1-3) $$ \begin{align} \sum_{j\in {\mathbb Z}} v_j(e^{-a_{j+1}L^\alpha} f(x)-e^{-a_{j}L^\alpha} f(x)), \end{align} $$

where $\{v_j\}_{j\in {\mathbb Z}}$ is a bounded sequence of real numbers and $\{a_j\}_{j\in {\mathbb Z}}$ is an increasing sequence of positive numbers. It follows from [20] that, in the case $v_j\equiv 1$ , the behavior of the above series coincides with that of $e^{-tL^\alpha } f(x)$ . To investigate the behavior of the sum in (1-3), we only need to analyze the ‘partial sums’ of the series in (1-3), which are defined as follows. For each $N=(N_1,N_2)\in {\mathbb Z}^2$ with $N_1<N_2,$ the corresponding partial sum operators are defined as

(1-4) $$ \begin{align} T^{L^\alpha}_N f(x):=\sum_{j=N_1}^{N_2} v_j(e^{-a_{j+1}L^\alpha} f(x)-e^{-a_{j}L^\alpha} f(x))\quad \mbox{for all }x\in \mathbb R^n, \end{align} $$

and

$$ \begin{align*} T^{(-\Delta)^{\alpha}}_N f(x):=\sum_{j=N_1}^{N_2} v_j(e^{-a_{j+1}(-\Delta)^\alpha} f(x)-e^{-a_{j}(-\Delta)^\alpha} f(x))\quad \mbox{for all }x\in \mathbb R^n. \end{align*} $$

We also consider the maximal operators

(1-5) $$ \begin{align} T^*_{L^{\alpha}}f(x):=\sup_N \vert T_N^{L^{\alpha}} f(x)\vert \quad \mbox{for all }x\in{\mathbb R}^n \end{align} $$

and

$$ \begin{align*} T^*_{(-\Delta)^{\alpha}}f(x):=\sup_N \vert T_N^{(-\Delta)^{\alpha}} f(x)\vert \quad \mbox{for all }x\in{\mathbb R}^n, \end{align*} $$

where the supremum is taken over all $N=(N_1,N_2)\in {\mathbb Z}^2$ with $N_1< N_2$ .

In this paper, we present a detailed study of the convergence of $T^{L^{\alpha }}_N$ in order to extend the results of [19, 20]. The structure of the paper is as follows. In Section 2, we collect some notation and basic ingredients that are used in what follows. In Section 3, we mainly focus on the boundedness of the operator $T^{L^{\alpha }}_N$ . More precisely, we investigate for the first time the integral kernel $K_{N}^{\alpha }(\cdot ,\cdot )$ of $T^{L^{\alpha }}_N$ , and its size description and the smoothness estimates are obtained in Proposition 3.5. Moreover, we investigate the uniform boundedness of $T^{L^{\alpha }}_N$ in the BMO-type space denoted by $BMO_{L}(\mathbb R^{n})$ ; see Theorem 3.6. Then with the aid of the vector-valued Calderón–Zygmund theory, we obtain the $L^{p}$ -boundedness of $T^{L^{\alpha }}_N$ . In addition, we get the boundedness of $T^{L^{\alpha }}_N$ on the Morrey spaces $L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ and the weak Morrey spaces $WL^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ ; see Theorem 3.7.

In Section 4, our attention shifts to the maximal operator $T^*_{L^{\alpha }}$ . On the one hand, using a Cotlar-type inequality, the maximal operators and the uniform $L^{p}$ -boundedness of $T^{L^{\alpha }}_N$ , we obtain the $L^{p}$ -boundedness of $T^*_{L^{\alpha }}$ and $T^*_{(-\Delta )^{\alpha }}$ ; see Theorem 4.4. On the other hand, we not only establish the $BMO_{L}$ -boundedness of $T^*_{L^{\alpha }}$ using the $T1$ theorem (see Theorem 4.6), but we also explore the boundedness of $T^*_{L^{\alpha }}$ in Morrey spaces (see Theorem 4.7). Finally, Theorem 4.10 analyzes the behavior of $T^*_{L^{\alpha }}$ and $T^*_{(-\Delta )^{\alpha }}$ in $L^{\infty }$ more deeply. It should be mentioned that some of our results are valid only when the sequence $\{a_j\}_{j\in \mathbb Z}$ is lacunary, which means that there exists a $\lambda>1$ such that $\displaystyle {a_{j+1}}/{a_j} \ge \lambda , \, j \in \mathbb {Z}$ .

Throughout this paper, we use c and C to denote positive constants, which are independent of main parameters and may be different at each occurrence. By $\mathrm {X}\sim \mathrm {Y}$ , we mean that $\mathrm {Y}\lesssim \mathrm {X}\lesssim \mathrm {Y},$ where the second estimate means that there exists a positive constant C, independent of main parameters, such that $\mathrm {X}\leq C\mathrm {Y}.$ Similarly, we write ${\mathsf V}\gtrsim {\mathsf U}$ for ${\mathsf V}\ge c{\mathsf U}$ .

2 Preliminaries

To investigate estimates of the fundamental solution of Schrödinger operators, Shen in [15] introduced the critical radius function, which is defined as

(2-1) $$ \begin{align} \rho(x):=\sup\bigg\{r>0:\frac{1}{r^{n-2}}\int_{B(x,r)}V(y)\,dy\leq1\bigg\}\quad \mbox{for all}\ x\in{\mathbb R}^n, \end{align} $$

where $B(x,r)$ denotes the ball in $\mathbb {R}^{n}$ centered at x and with radius r. It is well known that the auxiliary function $\rho (x)$ determined by $V\in (RH)_{q}$ satisfies $0<\rho (x)<\infty $ for any given $x\in \mathbb {R}^{n}$ . We need the following known results concerning the critical radius function in (2-1).

Lemma 2.1 [15, Lemma 1.4].

There exist constants $C_{0}\geq 1$ and $N_{0}>0$ such that, for any x and y in $\mathbb {R}^{n}$ ,

(2-2) $$ \begin{align} \frac{1}{C_{0}}\bigg(1+\frac{|x-y|}{\rho(x)}\bigg)^{-N_{0}}\leq \frac{\rho(y)}{\rho(x)}\leq C_{0}\bigg(1+\frac{|x-y|}{\rho(x)}\bigg)^{{N_{0}}/{(1+N_{0})}}. \end{align} $$

As a direct consequence of (2-2), we see that, for each fixed $k\in \mathbb {N}$ , the estimate

(2-3) $$ \begin{align} \bigg(1+\frac{r}{\rho(x)}\bigg)^{-{N_{0}}/{(1+N_{0})}}\bigg(1+\frac{2^{k}r}{\rho(x)}\bigg)\leq C_{0}\bigg(1+\frac{2^{k}r}{\rho(y)}\bigg) \end{align} $$

holds for any $y\in B(x,r)$ with $x\in \mathbb {R}^{n}$ and $r\in (0,\infty )$ , where $C_{0}$ appears in (2-2).

The space $BMO_{L}({\mathbb R}^n)$ , which was introduced in [5], is defined as the set of locally integrable functions f such that, for all balls $B= B(x,R)$ ,

$$ \begin{align*} \begin{cases} \displaystyle \dfrac1{|B|} \int_B\bigg|f(z)- \dfrac1{|B|} \int_B f \bigg| \,dz\le C_1 & \text{if }R \le\rho(x),\\\\[-8pt] \displaystyle \dfrac1{|B|} \int_B |f| \le C_2& \text{if }R>\rho(x). \end{cases} \end{align*} $$

The norm $\|f\|_{BMO_L(\mathbb {R}^n)}$ is defined as $\min \{C_1,C_2\}$ , which are given in the above two inequalities.

The classical Morrey spaces $M^{\lambda }_{p}(\mathbb {R}^{n})$ are defined as follows.

Definition 2.2.

1. (i) Let $1\leq p<\infty $ and $0\leq \lambda \leq n$ . The Morrey space $M^{\lambda }_{p}(\mathbb {R}^{n})$ is defined as the set of all p-locally integrable functions f on $\mathbb {R}^{n}$ such that

   $$ \begin{align*}\|f\|_{M^{\lambda}_{p}(\mathbb{R}^{n})}:=\sup_{x\in\mathbb{R}^{n},r>0}r^{-\lambda/p}\|f\|_{L^{p}(B(x,r))}<\infty.\end{align*} $$

2. (ii) The weak Morrey space $WM^{\lambda }_{1}(\mathbb {R}^{n})$ consists of all measurable functions f on $\mathbb {R}^{n}$ such that

   $$ \begin{align*} \|f\|_{WM^{\lambda}_{1}(\mathbb{R}^{n})} &:= \sup_{x\in\mathbb{R}^{n},r>0}r^{-\lambda}\|f\|_{WL^{1}(B(x,r))} \\ &= \sup_{x\in\mathbb{R}^{n},r>0}r^{-\lambda} \sup_{\sigma>0}\sigma|\{y\in B(x,r):|f(y)|>\sigma\}|<\infty. \end{align*} $$

The spaces $M^{\lambda }_{p}(\mathbb {R}^{n})$ , which were originally introduced by Morrey in [13] as a generalization of Lebesgue spaces, have been studied intensively and widely used in elliptic partial differential equations, calculus of variations and other related fields. For the properties and applications of $M^{\lambda }_{p}(\mathbb {R}^{n})$ , we refer the reader to [1, 2, 14, 17] and the references therein. It should be mentioned that $M^{0}_{p}(\mathbb {R}^{n})=L^{p}(\mathbb {R}^{n})$ and $M^{n}_{p}(\mathbb {R}^{n})=L^{\infty }(\mathbb {R}^{n})$ by the Lebesgue differentiation theorem. If $\lambda <0$ or $\lambda>n$ , then $M^{\lambda }_{p}(\mathbb {R}^{n})=\Theta $ , where $\Theta $ is the set of all functions equivalent to 0 on $\mathbb {R}^{n}$ .

In [16], Tang and Dong introduced the following Morrey-type spaces related to L to investigate the boundedness of operators generated by L; for example, Schrödinger-type Calderón–Zygmund operators, Riesz transforms and fractional integrals related to L.

Definition 2.3. Let $1\leq p<\infty $ and $0\leq \kappa <1$ . For a given $0<\theta <\infty $ , the Morrey space $L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ is defined to be the set of all p-locally integrable functions f on $\mathbb {R}^{n}$ such that

(2-4) $$ \begin{align} \bigg(\frac{1}{|B(x_{0},r)|^{\kappa}}\int_{B(x_{0},r)}|f(x)|^{p}\,dx\bigg)^{1/p}\leq C\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{\theta} \end{align} $$

holds for every ball $B(x_{0},r)$ in $\mathbb {R}^{n}$ . A norm for $f\in L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ , denoted by $\|f\|_{L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})}$ , is given by the infimum of the constants appearing in (2-4), or, equivalently,

$$ \begin{align*}\|f\|_{L^{\rho,\theta}_{p,\kappa}(\mathbb{R}^{n})}:=\sup_{B(x_{0},r)}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{-\theta}\bigg(\frac{1}{|B(x_{0},r)|^{\kappa}} \int_{B(x_{0},r)}|f(x)|^{p}\,dx\bigg)^{1/p}<\infty,\end{align*} $$

where the supremum is taken over all balls $B(x_{0},r)$ in $\mathbb {R}^{n}$ . Define

$$ \begin{align*}L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n}):=\bigcup_{0<\theta<\infty}L^{\rho,\theta}_{p,\kappa}(\mathbb{R}^{n}).\end{align*} $$

Definition 2.4. Let $p=1$ and $0\leq \kappa <1$ . For a given $0<\theta <\infty $ , the weak Morrey space $WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$ is defined to be the set of all measurable functions f on $\mathbb {R}^{n}$ such that

$$ \begin{align*}\frac{1}{|B(x_{0},r)|^{\kappa}}\sup_{\lambda>0}\lambda\cdot|\{x\in B(x_{0},r):|f(x)|>\lambda\}|\leq C\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{\theta}\end{align*} $$

holds for every ball $B(x_{0},r)$ in $\mathbb {R}^{n}$ . A norm for $f\in WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$ is given by

$$ \begin{align*}\|f\|_{WL^{\rho,\theta}_{1,\kappa}(\mathbb{R}^{n})}:=\sup_{B(x_{0},r)}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{-\theta}\frac{1}{|B(x_{0},r)|^{\kappa}} \sup_{\lambda>0}\lambda\, |\{x\in B(x_{0},r): |f(x)|>\lambda\}|<\infty.\end{align*} $$

Correspondingly, we define

$$ \begin{align*}WL^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n}):=\bigcup_{0<\theta<\infty}WL^{\rho,\theta}_{1,\kappa}(\mathbb{R}^{n}).\end{align*} $$

Remark 2.5.

1. (i) It is easy to see that if $\theta =0$ or $V=0$ , then the Morrey space $L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ (or the weak Morrey space $WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$ ) is just the Morrey space $L^{\kappa }_{p}(\mathbb {R}^{n})$ (or the weak Morrey space $WL^{\kappa }_{1}(\mathbb {R}^{n})$ ), which was defined and studied by Guliyev et al.; see [8].

2. (ii) According to the above definitions,

   (2-5) $$ \begin{align} L^{\kappa}_{p}(\mathbb{R}^{n})\subset L^{\rho,\theta_{1}}_{p,\kappa}(\mathbb{R}^{n})\subset L^{\rho,\theta_{2}}_{p,\kappa}(\mathbb{R}^{n}) \end{align} $$
   and
   (2-6) $$ \begin{align} WL^{\kappa}_{1}(\mathbb{R}^{n})\subset WL^{\rho,\theta_{1}}_{1,\kappa}(\mathbb{R}^{n})\subset WL^{\rho,\theta_{2}}_{1,\kappa}(\mathbb{R}^{n}), \end{align} $$
   whenever $0<\theta _{1}<\theta _{2}<\infty .$ Hence,
   $$ \begin{align*} L^{\kappa}_{p}(\mathbb{R}^{n})\subset L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})\quad \mathrm{and}\quad WL^{\kappa}_{1}(\mathbb{R}^{n})\subset WL^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n}) \end{align*} $$
   for all $(p,\kappa )\in [1,\infty )\times [0,1)$ .

3. (iii) Define a norm on the space $L^{\rho ,\infty }_{p,\kappa }(\mathbb {R}^{n})$ . Then this space becomes a Banach space. In view of (2-5), for any given $f\in L^{\rho ,\infty }_{p,\kappa }(\mathbb {R}^{n})$ , let

   $$ \begin{align*}\theta^{\ast}:=\inf\{\theta>0:f\in L^{\rho,\theta}_{p,\kappa}(\mathbb{R}^{n})\}.\end{align*} $$

   Now define the functional $\|\cdot \|_{\star }$ by

   $$ \begin{align*} \|f\|_{\star}:=\|f\|_{L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})}:=\|f\|_{L^{\rho,\theta^{\ast}}_{p,\kappa}(\mathbb{R}^{n})}. \end{align*} $$

   It is easy to check that this functional $\|\cdot \|_{\star }$ satisfies the axioms of a norm.

4. (iv) In view of (2-6), for any given $f\in WL^{\rho ,\infty }_{1,\kappa }(\mathbb {R}^{n})$ , let

   $$ \begin{align*}\theta^{\ast\ast}:=\inf\{\theta>0:f\in WL^{\rho,\theta}_{1,\kappa}(\mathbb{R}^{n})\}.\end{align*} $$

   Similarly, we define the functional $\|\cdot \|_{\star \star }$ by

   $$ \begin{align*}\|f\|_{\star\star}:=\|f\|_{WL^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n})}:=\|f\|_{WL^{\rho,\theta^{\ast\ast}}_{1,\kappa}(\mathbb{R}^{n})}.\end{align*} $$

   By definition, we easily show that this functional $\|\cdot \|_{\star \star }$ satisfies the axioms of a (quasi-)norm and that $WL^{\rho ,\infty }_{1,\kappa }(\mathbb {R}^{n})$ is a (quasi-)normed linear space.

Since the Morrey space $L^{\rho ,\theta }_{p,\kappa }(\mathbb {R}^{n})$ (or the weak Morrey space $WL^{\rho ,\theta }_{1,\kappa }(\mathbb {R}^{n})$ ) could be viewed as an extension of the Lebesgue spaces (or the weak Lebesgue) on $\mathbb {R}^{n}$ , it is natural to investigate the boundedness properties of operators $T^{L^{\alpha }}_{N}$ and $T^*_{L^{\alpha }}$ in the framework of Morrey spaces.

3 Boundedness of the operator $T^{L^{\alpha }}_N$

In this section, as a preparation for Theorem 4.4, we prove the uniform boundedness of the operators $T^{L^{\alpha }}_N$ , where the standard Calderón–Zygmund theory is a fundamental tool. We refer the reader to [4, 7] for some basic knowledge about the Calderón–Zygmund theory. The following $L^{2}$ -boundedness of the operator $T_N^{L^{\alpha }}$ is a direct consequence of the functional calculus and the spectral resolution of L.

Proposition 3.1. Let $\alpha \in (0,1)$ . There exists a constant $C>0$ , depending on n, $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ and $\alpha $ (not on N), such that

$$ \begin{align*} \|T^{L^{\alpha}}_N f \|_{L^2({\mathbb R}^{n})}\leq C \|f \|_{L^2({\mathbb R}^{n})}.\end{align*} $$

Proof. Denote by $dE(\lambda )$ the spectral resolution of the operator L. Since $L^\alpha $ is a nonnegative, densely defined and self-adjoint operator on $\mathbb {R}^n$ , by spectral analysis,

$$ \begin{align*} e^{- {tL^\alpha}}=\int_0^\infty e^{- {t\lambda^\alpha}}\,dE(\lambda), \end{align*} $$

which is expressed as

$$ \begin{align*}\langle e^{-t L^\alpha}f,\ g\rangle_{L^2(\mathbb{R}^n)}=\int_0^\infty e^{-t\lambda^\alpha}\,dE_{f,g}(\lambda) \quad \mbox{for all } f, g\in L^2(\mathbb{R}^n),\end{align*} $$

where $dE_{f,g}(\lambda )$ is a regular Borel complex measure of bounded variation concentrated on the spectrum of ${\mathcal L}$ with $d\vert E_{f,g}\vert (0,\infty )\leq \Vert f\Vert _{L^2({\mathbb R}^n)}\Vert g\Vert _{L^2({\mathbb R}^n)}$ . Therefore, we write

$$ \begin{align*} T^{L^{\alpha}}_N =\sum_{j=N_1}^{N_2} v_j\int_0^\infty (e^{-a_{j+1}\lambda^\alpha} -e^{-a_{j}\lambda^\alpha}) \,dE(\lambda) \end{align*} $$

and we obtain that, for any $f, g\in L^2(\mathbb {R}^n)$ ,

$$ \begin{align*} \vert\langle T^{L^{\alpha}}_N f,g\rangle\vert &\le \bigg| \!\sum_{j=N_1}^{N_2} v_j\int_0^\infty \int_{a_j}^{a_{j+1}} \partial_s e^{-s\lambda^\alpha} \,ds \,dE_{f,g}(\lambda)\bigg| \\ &= \bigg|\! \int_0^\infty\sum_{j=N_1}^{N_2} v_j \int_{a_j}^{a_{j+1}} \partial_s e^{-s\lambda^\alpha} \,ds \,dE_{f,g}(\lambda)\bigg| \\ &\le \|v\|_{\ell^\infty(\mathbb Z)} \int_0^\infty \int_{0}^{\infty} \vert\partial_s e^{-s\lambda^\alpha}\vert \,ds ~\vert \,dE_{f,g}\vert(\lambda)\\ &\lesssim \Vert f\Vert_{L^2({\mathbb R}^{n})} \Vert \Vert g\Vert_{L^2({\mathbb R}^{n})}. \end{align*} $$

This completes the proof of Proposition 3.1.

Lemma 3.2 [10, Proposition 3.11].

Let $\alpha \in (0,1)$ . For every $N>0$ , there exists a constant $C_{N}>0$ such that

$$ \begin{align*}|\partial_t K^{L}_{\alpha,t}(x,y)|\leq C_{N}\min\bigg\{\frac{1}{t^{n/2\alpha+1}},\ \frac{t^{N/\alpha}}{|x-y|^{n+2\alpha+2N}}\bigg\}\bigg(\frac{t^{1/2\alpha}}{\rho(x)}\bigg)^{-N}\bigg(\frac{t^{1/2\alpha}}{\rho(y)}\bigg)^{-N},\end{align*} $$

which further gives

$$ \begin{align*}|\partial_t K^{L}_{\alpha,t}(x,y)|\leq \frac{C_N}{(t^{1/2\alpha}+|x-y|)^{n+2\alpha}}\bigg(1+\frac{t^{1/2\alpha}}{\rho(x)}+\frac{t^{1/2\alpha}}{\rho(y)}\bigg)^{-N}.\end{align*} $$

Lemma 3.3 [10, Proposition 3.12].

Let $\alpha \in (0,1)$ and $\beta>0$ . Let $0<\delta '\leq \delta =\min \{2\alpha ,\delta _{0}\}$ , where $\delta _{0}=\min \{1,2-n/q\}$ . For every $N>0$ , there exists a constant $C_{N}>0$ such that, for all $|h|\leq t^{1/2\alpha }$ ,

$$ \begin{align*} &|\partial_t K^{L}_{\alpha,t}(x+h,y)-\partial_t K^{L}_{\alpha,t}(x,y)|\\&\quad\leq C_{N}\bigg(\frac{|h|}{\sqrt {t^{1/\alpha}}}\bigg)^{\delta'}\min\bigg\{\frac{1}{t(\sqrt {t^{1/\alpha}})^n},\ \frac{1}{|x-y|^{n+2\alpha}}\bigg\}\bigg(1+\frac{\sqrt {t^{1/\alpha}}}{\rho(x)}+\frac{\sqrt {t^{1/\alpha}}} {\rho(y)}\bigg)^{-N},\end{align*} $$

which further gives

$$ \begin{align*}&|\partial_t K^{L}_{\alpha,t}(x+h,y)-\partial_t K^{L}_{\alpha,t}(x,y)|\\ &\quad\leq C_{N}\bigg(\frac{|h|}{\sqrt {t^{1/\alpha}}}\bigg)^{\delta'} \frac{1}{(\!\sqrt {t^{1/\alpha}}+|x-y|)^{n+2\alpha}} \bigg(1+\frac{\sqrt {t^{1/\alpha}}}{\rho(x)}+\frac{\sqrt {t^{1/\alpha}}}{\rho(y)}\bigg)^{-N}. \end{align*} $$

Lemma 3.4. Assume that $\delta _{0}=\min \{1,2-n/q\}$ , as in Lemma 3.3.

1. (i) [18, Proposition 7] There exists a constant $C>0$ such that

   $$ \begin{align*} &|t\partial_{t}K^{L}_{\alpha,t}(x,y)-t\partial_{t}K^{-\Delta}_{\alpha,t}(x-y)|\\ &\quad\leq \left\{ \begin{aligned} &C\bigg(\frac{|x-y|}{\rho(x)}\bigg)^{\delta_{0}}\frac{t}{(|x-y|+\sqrt{t^{1/\alpha}})^{n+2\alpha}},& \sqrt{t^{1/\alpha}}\leq |x-y|,\\ &C\bigg(\frac{\sqrt{t^{1/\alpha}}}{\rho(x)}\bigg)^{\delta_{0}}\frac{t}{(|x-y|+\sqrt{t^{1/\alpha}})^{n+2\alpha}},& \sqrt{t^{1/\alpha}}\geq |x-y|. \end{aligned}\right. \end{align*} $$

2. (ii) [18, Proposition 9] For every $0<\delta <\min \{2\alpha , \delta _{0}\}$ , there exist constants $C,c>0$ such that, for $\displaystyle |y-z|<\min \{\rho (x), {|x-y|}/{4}\}$ ,

   $$ \begin{align*} &|(t\partial_{t}K^{L}_{\alpha,t}(x,y)-t\partial_{t}K^{-\Delta}_{\alpha,t}(x-y)) -(t\partial_{t}K^{L}_{\alpha,t}(x,z)-t\partial_{t}K^{-\Delta}_{\alpha,t}(x-z))|\\ &\quad\leq C\bigg(\frac{|y-z|}{\rho(y)}\bigg)^{\delta}\frac{t}{(\!\sqrt {t^{1/\alpha}}+|x-y|)^{n+2\alpha}}. \end{align*} $$

The kernel of $T^{L^{\alpha }}_N$ can be expressed as

$$ \begin{align*} K^\alpha_N(x,y) = \sum_{j=N_1}^{N_2}v_j (K^{L}_{\alpha,a_{j+1}} (x,y)-K^{L}_{\alpha,a_{j}} (x,y)). \end{align*} $$

The following proposition contains the size descriptions and the smoothness estimates of $T^{L^{\alpha }}_N$ , which are required in the Calderón–Zygmund theory.

Proposition 3.5. Let $\alpha \in (0,1)$ and let $0<\delta '\leq \delta =\min \{2\alpha ,\delta _{0}\}$ with $\delta _{0}=\min \{1, 2-n/q\}$ . For every $N>0$ , there exists a constant $C>0$ , depending on $n, \alpha $ and $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ (not on N), such that, for any $x\neq y:$

1. (i)

   $$ \begin{align*} | K^\alpha_N(x,y)|\leq \frac{C}{|x-y|^{n}}\bigg(1+\frac{|x-y|}{\rho(x)}+\frac{|x-y|}{\rho(y)}\bigg)^{-N}; \mbox{and}\end{align*} $$

2. (ii)

   $$ \begin{align*} | K_N^\alpha(x,y)- K_N^\alpha(x,z)|+|K_N^\alpha(y,x)-K_N^\alpha(z,x)| \leq C\frac{\vert y-z\vert^{\delta'}}{|x-y|^{n+\delta'}}, \end{align*} $$
   whenever $\vert x-y\vert>{2}\vert y-z\vert .$

Proof. (i) For the size condition for $K^{\alpha }_{N}(\cdot ,\cdot )$ , by Lemma 3.2,

$$ \begin{align*} | K^\alpha_N (x,y)| &\leq \sum_{j=N_1}^{N_2}\vert v_j\vert \vert K^{L}_{\alpha,a_{j+1}}(x,y)-K^{L}_{\alpha,a_{j}} (x,y)\vert\\[5pt] &\le \|v\|_{l^\infty(\mathbb{Z})} \sum_{j=-\infty}^{\infty} \bigg|\!\int_{a_j}^{a_{j+1}}\partial_t K^{L}_{\alpha,t}(x,y) \,dt\bigg|\\[5pt] &\le \|v\|_{l^\infty(\mathbb{Z})} \sum_{j=-\infty}^{\infty} \int_{a_j}^{a_{j+1}}|\partial_t K^{L}_{\alpha,t}(x,y)| \,dt\\[5pt] &\lesssim \|v\|_{l^\infty(\mathbb{Z})} \sum_{j=-\infty}^{\infty} \int_{a_j}^{a_{j+1}}\frac 1{(t^{1/{(2\alpha)}}+|x-y|)^{n+2\alpha}}\bigg(1+\frac{t^{1/(2\alpha)}}{\rho(x)}+\frac{t^{1/(2\alpha)}}{\rho(y)}\bigg)^{-N} \,dt\\[5pt] &\lesssim \|v\|_{l^\infty(\mathbb{Z})} \sum_{j=-\infty}^{\infty} \int_{a_j}^{a_{j+1}}\frac 1{(t^{1/{(2\alpha)}}+|x-y|)^{n+2\alpha}}\,dt. \end{align*} $$

Since $\{a_{j}\}_{j\in \mathbb {Z}}$ is an increasing sequence of positive numbers, we see that

$$ \begin{align*} | K^\alpha_N (x,y)| &\lesssim \|v\|_{l^\infty(\mathbb{Z})}\int_{0}^{\infty}\frac 1{(t^{1/{(2\alpha)}}+|x-y|)^{n+2\alpha}} \,dt \\[5pt] & \lesssim {1\over {|x-y|^{n}}}. \end{align*} $$

Furthermore, we also have

$$ \begin{align*} &| K^\alpha_N (x, y)|& \\[5pt] &\quad\lesssim \int_0^{|x-y|^{2\alpha}}\frac{t^{N/\alpha}}{|x-y|^{n+2\alpha+2N}}\bigg(\frac{t^{1/2\alpha}}{\rho(x)}\bigg)^{-N}\,dt +\int_{|x-y|^{2\alpha}}^{+\infty}\frac{1}{t^{n/2\alpha+1}}\bigg(\frac{t^{1/2\alpha}}{\rho(x)}\bigg)^{-N}\,dt\\[5pt] &\quad \lesssim \frac{1}{|x-y|^{n}}\bigg({|x-y|\over \rho(x)}\bigg)^{-N}. \end{align*} $$

Then, using the symmetry of $K^{L}_{\alpha ,t}(\cdot , \cdot )$ , we get

$$ \begin{align*}|{K^\alpha_N(x, y)}|\lesssim \frac{1}{|x-y|^{n}}\bigg(1+\frac{|x-y|}{\rho(x)}+\frac{|x-y|}{\rho(y)}\bigg)^{-N}.\end{align*} $$

(ii) Note that if $|x-y|>2|y-z|$ , then $\vert x-y\vert \sim \vert x-z\vert .$ For any $0<\delta '\leq \delta =\min \{2\alpha ,\delta _{0}\}$ , if $\vert y-z\vert \le {t}^{1/2\alpha }$ , using a similar argument as in (i) and by Lemma 3.3, we get

$$ \begin{align*} &\vert K^\alpha_N(x,y)-K^\alpha_N(x,z)\vert\\ \nonumber &\quad\lesssim\Vert v\Vert_{\ell^\infty(\mathbb Z)}\int_0^{+\infty}\vert\partial_t K^{L}_{t}(x,y)-\partial_t K^{L}_{t}(x,z)\vert \,dt\\ \nonumber &\quad\lesssim\Vert v\Vert_{\ell^\infty(\mathbb Z)}\int_0^{|x-y|^{2\alpha}}\frac{1}{|x-y|^{n+2\alpha}}\bigg(\frac{|y-z|}{t^{1/2\alpha}}\bigg)^{\delta'}\,dt +\int_{|x-y|^{2\alpha}}^{+\infty}\bigg(\frac{|y-z|}{t^{1/2\alpha}}\bigg)^{\delta'}\frac{1}{t^{n/2\alpha+1}}\,dt\\ &\quad\lesssim\Vert v\Vert_{\ell^\infty(\mathbb Z)}\frac{\vert y-z\vert^{\delta'}}{\vert x-y\vert^{n+\delta'}}. \end{align*} $$

If $\vert y-z\vert>{t}^{1/2\alpha }$ , Lemma 3.2 gives

$$ \begin{align*} \vert K^\alpha_N(x,y)\vert &\lesssim \int_{0}^{+\infty} |\partial_t K^{L}_{t}(x,y)| \,dt \\ & \lesssim\int_{0}^{+\infty} \bigg(\frac{\vert y-z\vert}{{t}^{1/2\alpha}}\bigg)^{\delta'} \frac{1}{(t^{1/2\alpha}+|x-y|)^{n+2\alpha}}\,dt\\ &\lesssim\frac{\vert y-z\vert^{\delta'}}{\vert x-y\vert^{n+\delta'}}. \end{align*} $$

The same bound is valid for $K^\alpha _N(x,z)$ because $\vert x-z\vert \sim \vert x-y\vert $ . Then the estimate can be obtained directly owing to the inequality

$$ \begin{align*}\vert K^\alpha_N(x,y)-K^\alpha_N (x,z)\vert\le\vert K^\alpha_N(x,y)\vert+\vert K^\alpha_N(x,z)\vert.\end{align*} $$

The symmetry of the kernel $K^\alpha _N(x,y)=K^\alpha _N(y,x)$ gives the conclusion of (ii). This completes the proof of Proposition 3.5.

The uniform boundedness of $T^{L^{\alpha }}_N$ in the BMO-type space $BMO_{L}(\mathbb R^{n})$ is as follows.

Theorem 3.6. Let $\{a_j\}_{j\in \mathbb Z}$ be a positive increasing sequence and let $T^{L^{\alpha }}_N$ be the operator defined in (1-4) with $N=(N_1,N_2)$ .

1. (i) There exists a positive constant C, depending on n, $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ and $\alpha $ , such that, for all functions $f\in L^\infty ({\mathbb R}^{n})$ ,

   $$ \begin{align*}\Vert T^{L^{\alpha}}_N f\Vert_{BMO_L(\mathbb R^{n})}\leq C\Vert f\Vert_{L^\infty(\mathbb R^{n})}.\end{align*} $$

2. (ii) There exists a positive constant C, depending on n, $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ and $\alpha $ , such that, for all functions $f\in BMO_L({\mathbb R}^n)$ ,

   $$ \begin{align*}\Vert T^{L^{\alpha}}_N f\Vert_{BMO_L({\mathbb R}^n)}\le C\Vert f\Vert_{BMO_L({\mathbb R}^n)}.\end{align*} $$

   The constants C in (i) and (ii) are independent of $N.$

Proof. Since $L^\infty ({\mathbb R}^n) \subset BMO_{\mathcal L}({\mathbb R}^n)$ and $\|f\|_{BMO_{\mathcal L}({\mathbb R}^n)} \le \|f\|_{L^\infty ({\mathbb R}^n)}$ , it is enough to prove that (ii) of Theorem 3.6 holds. We first show that $T_N^{L^{\alpha }} f<\infty $ if $f\in BMO_L({\mathbb R}^n)$ . Let ${B=B(x_0, \rho (x_0))}$ and $B^*=B(x_0, 2C_0\rho (x_0))$ with some $x_0\in {\mathbb R}^{n}$ and $C_0>1$ . We decompose f as

$$ \begin{align*}f=f\chi_{B^*}+f\chi_{(B^*)^c}=:f_1+f_2.\end{align*} $$

The function $f_1$ is integrable, and hence $T^{L^{\alpha }}_N f_1< \infty\ \text{for almost every}\ x\in {\mathbb R}^{n}.$ When ${x\in B}$ and $y\in (B^{*})^{c}$ , we have $|y-x_0|/2\le |x-y|\le 2|y-x_0|$ and $\rho (x)\sim \rho (x_0).$ It can be deduced from Proposition 3.5 and Hölder’s inequality that

$$ \begin{align*} \vert T^{L^{\alpha}}_N f_2(x)\vert&=\bigg\vert\!\int_{{\mathbb R}^n} K^\alpha_N(x, y)f_2(y)\,dy\bigg\vert\nonumber \\ &\lesssim {\int_{(B^{*})^{c}} {1\over |x-y|^n}{\rho^2(x)\over |x-y|^2 } \vert f_2(y)\vert \,dy}\nonumber \\ &\lesssim {\int_{(B^{*})^{c}} {\rho^2(x_0)\over |x_0-y|^{n+2} }\vert f(y)\vert \,dy}\nonumber\\ &\lesssim\bigg(\int_{(B^{*})^{c}} {\rho^2(x_0)\over |x_0-y|^{n+2} } \,dy\bigg)^{1/2}\bigg(\int_{(B^{*})^{c}} {\rho^2(x_0)\over |x_0-y|^{n+2} }\vert f(y)\vert^2 \,dy\bigg)^{1/2}\nonumber\\ &\lesssim\bigg(\sum_{k=1}^\infty\int_{2^{k+1}B\setminus 2^kB} {\rho^2(x_0)\,dy\over |x_0-y|^{n+2} } \bigg)^{1/2}\bigg(\sum_{k=1}^\infty\int_{2^{k+1}B\setminus 2^kB} {\rho^2(x_0)\vert f(y)\vert^2\over |x_0-y|^{n+2} } \,dy\bigg)^{1/2}\nonumber\\ &\lesssim\bigg(\sum_{k=1}^\infty{\rho^2(x_0)\over (2^kC_0\rho(x_0))^2}{1\over \vert2^{k+1}B\vert}\int_{2^{k+1}B} \vert f(y)\vert^2 \,dy\bigg)^{1/2}\nonumber\\ &\lesssim \bigg(\sum_{k=1}^\infty2^{-2k}\Vert f\Vert^2_{BMO_L({\mathbb R}^n)}\bigg)^{1/2}\nonumber\\ &\lesssim\Vert f\Vert_{BMO_L({\mathbb R}^n)}<\infty.\nonumber \end{align*} $$

Hence, we have proved that $T^{L^{\alpha }}_N f(x)<\infty\ \text{for almost every}\ x\in {\mathbb R}^n.$ Then the proof of

$$ \begin{align*}\Vert T^{L^{\alpha}}_N f\Vert_{BMO_L({\mathbb R}^n)}\lesssim\Vert f\Vert_{BMO_L({\mathbb R}^n)}\end{align*} $$

is similar to that for the maximal operator $T^*_{L^{\alpha }}$ , which can be seen from Theorem 4.6. So we omit the details here.

Theorem 3.7. Let $\{a_j\}_{j\in \mathbb Z}$ be a positive increasing sequence, let $0<\kappa <1$ and let $T^{L^{\alpha }}_N$ be the operator defined in (1-4) with $N=(N_1,N_2)$ . The following statements hold.

1. (i) There exists a positive constant C, depending on n, $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ , $\theta $ and $\alpha $ , such that, for all functions $f\in L^{\rho ,\infty }_{p,\kappa }(\mathbb {R}^{n})$ ,

   $$ \begin{align*}\Vert T^{L^{\alpha}}_N f\Vert_{L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})}\leq C\Vert f\Vert_{L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})}.\end{align*} $$

2. (ii) There exists a positive constant C, depending on n, $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ , $\theta $ and $\alpha $ , such that, for all functions $f\in L^{\rho ,\infty }_{1,\kappa }(\mathbb {R}^{n})$ ,

   $$ \begin{align*}\Vert T^{L^{\alpha}}_N f\Vert_{WL^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n})}\le C\Vert f\Vert_{L^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n})}.\end{align*} $$

   The constants C appearing above are all independent of $N.$

Proof. We have previously remarked that the constants in the $L^{p}$ -boundedness only depend on the initial constant in $L^{p_{0}}(\mathbb {R}^{n})$ (in our case, $p_{0}=2$ ), the size constant and the smoothness constant of the kernel. Hence, the uniform $L^{p}$ -boundedness of the operators $T^{L^{\alpha }}_N $ is a direct consequence of the vector-valued Calderón–Zygmund theory.

(i) For any given $f\in L^{\rho ,\infty }_{p,\kappa }(\mathbb {R}^{n})$ with $1\leq p<\infty $ and $0<\kappa <1$ , suppose that $f\in L^{\rho ,\theta ^{\ast }}_{p,\kappa }(\mathbb {R}^{n})$ for some $\theta ^{\ast }>0$ , where

$$ \begin{align*}\theta^{\ast}=\inf\{\theta>0:f\in L^{\rho,\theta}_{p,\kappa}(\mathbb{R}^{n})\}\quad \text{and}\quad\|f\|_{L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})} =\|f\|_{L^{\rho,\theta^{\ast}}_{p,\kappa}(\mathbb{R}^{n})}.\end{align*} $$

Then we only need to prove that, for each ball $B(x_{0},r)\subset \mathbb {R}^{n}$ , there is some $\nu>0$ such that

(3-1) $$ \begin{align} \bigg(\frac{1}{|B(x_{0},r)|^{\kappa}}\int_{B(x_{0},r)}|T^{L^{\alpha}}_{N}(f)(x)|^{p}\,dx\bigg)^{1/p}\lesssim \bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{\nu} \end{align} $$

holds for all functions $f\in L^{\rho ,\theta ^{\ast }}_{p,\kappa }(\mathbb {R}^{n})$ with $(p,\kappa )\in (1,\infty )\times (0,1)$ .

We first decompose the function f as

$$ \begin{align*} \begin{cases} f=f_{1}+f_{2}\in L^{\rho,\theta^{\ast}}_{p,\kappa}(\mathbb{R}^{n}),\\ f_{1}=f\cdot\chi_{2B},\\ f_{2}=f\cdot\chi_{(2B)^{c}}, \end{cases} \end{align*} $$

where $2B$ is the ball centered at $x_{0}$ of radius $2r$ , $(2B)^{c}=\mathbb {R}^{n}\setminus (2B)$ and $\chi _{E}$ denotes the characteristic function of the set E. Then we can write

$$ \begin{align*}\bigg(\frac{1}{|B(x_{0},r)|^{\kappa}}\int_{B(x_{0},r)}|T^{L^{\alpha}}_{N}(f)(x)|^{p}\,dx\bigg)^{1/p}\leq I_{1}+I_{2},\end{align*} $$

where

$$ \begin{align*} \left\{ \begin{aligned} I_{1}:=\bigg(\frac{1}{|B(x_{0},r)|^{\kappa}}\int_{B(x_{0},r)}|T^{L^{\alpha}}_{N}(f_{1})(x)|^{p}\,dx\bigg)^{1/p},\\ I_{2}:=\bigg(\frac{1}{|B(x_{0},r)|^{\kappa}}\int_{B(x_{0},r)}|T^{L^{\alpha}}_{N}(f_{2})(x)|^{p}\,dx\bigg)^{1/p}. \end{aligned} \right.\end{align*} $$

In what follows, we estimate $I_{1}$ and $I_{2}$ separately. For the term $I_{1}$ ,

$$ \begin{align*} I_{1} &\lesssim \frac{1}{|B|^{\kappa/p}}\bigg(\int_{\mathbb{R}^{n}}|f_{1}(x)|^{p}\,dx\bigg)^{1/p} \\ &= \frac{1}{|B|^{\kappa/p}}\bigg(\int_{2B}|f(x)|^{p}\,dx\bigg)^{1/p} \\ &\lesssim \|f\|_{L^{\rho,\theta^{\ast}}_{p,\kappa}(\mathbb{R}^{n})}\frac{|2B|^{\kappa/p}}{|B|^{\kappa/p}}\bigg(1+\frac{2r}{\rho(x_{0})}\bigg)^{\theta^{\ast}}. \end{align*} $$

Moreover, for any fixed $\theta ^{\ast }>0$ , the inequalities

(3-2) $$ \begin{align} 1\leq \bigg(1+\frac{2r}{\rho(x_{0})}\bigg)^{\theta^{\ast}}\leq 2^{\theta^{\ast}}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{\theta^{\ast}} \end{align} $$

further indicate that

$$ \begin{align*}I_{1}\lesssim \|f\|_{L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{\theta^{\ast}}.\end{align*} $$

Next, we estimate $I_{2}$ . From Proposition 3.5, we deduce that

(3-3) $$ \begin{align} T^{L^{\alpha}}_{N}(f_{2})(x)\lesssim \int_{(2B)^{c}}\frac{1}{|x-y|^{n}}\bigg(1+\frac{|x-y|}{\rho(x)}\bigg)^{-N}|f(y)|\,dy. \end{align} $$

Since $x\in B(x_{0},r)$ and $y\in (2B)^{c}$ , we have $|x-y|\sim |x_{0}-y|$ . This fact, along with (3-3), implies that, for any $x\in B(x_{0},r)$ ,

$$ \begin{align*} |T^{L^{\alpha}}_{N}(f_{2})(x)| &\lesssim \sum^{\infty}_{k=1}\int_{2^{k}r\leq |x_{0}-y|<2^{k+1}r}\bigg(1+\frac{|x_{0}-y|}{\rho(x)}\bigg)^{-N}\frac{1}{|x_{0}-y|^{n}}|f(y)|\,dy \\ \notag &\lesssim \sum^{\infty}_{k=1}\frac{1}{|B(x_{0},2^{k+1}r)|}\int_{|x_{0}-y|<2^{k+1}r}\bigg(1+\frac{2^{k}r}{\rho(x)}\bigg)^{-N}|f(y)|\,dy. \end{align*} $$

In view of (2-3) and (3-2), we deduce that

(3-4) $$ \begin{align} & |T^{L^{\alpha}}_{N}(f_{2})(x)|\\ \notag &\quad\lesssim \sum^{\infty}_{k=1}\frac{1}{|B(x_{0},2^{k+1}r)|}\int_{B(x_{0},2^{k+1}r)}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{N\cdot({N_{0}}/{N_{0}+1})} \bigg(1+\frac{2^{k+1}r}{\rho(x_{0})}\bigg)^{-N}|f(y)|\,dy.\notag \end{align} $$

We consider each term in the sum of (3-4) separately. Using Hölder’s inequality, we deduce that, for each fixed $k\in \mathbb {N}$ ,

$$ \begin{align*} \frac{1}{|B(x_{0},2^{k+1}r)|}\int_{B(x_{0},2^{k+1}r)}|f(y)|\,dy \lesssim \|f\|_{L^{\rho,\theta^{\ast}}_{p,\kappa}(\mathbb{R}^{n})}\frac{|B(x_{0},2^{k+1}r)|^{\kappa/p}}{|B(x_{0},2^{k+1}r)|^{1/p}} \bigg(1+\frac{2^{k+1}r}{\rho(x_{0})}\bigg)^{\theta^{\ast}}. \end{align*} $$

This allows us to obtain

$$ \begin{align*} I_{2} \lesssim \|f\|_{L^{\rho,\theta^{\ast}}_{p,\kappa}(\mathbb{R}^{n})} \bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{N\cdot({N_{0}}/{N_{0}+1})}\sum_{k=1}^{\infty}\frac{|B(x_{0},r)|^{(1-\kappa)/p}}{|B(x_{0},2^{k+1}r)|^{(1-k)/p}} \bigg(1+\frac{2^{k+1}r}{\rho(x_{0})}\bigg)^{-N+\theta^{\ast}}. \end{align*} $$

We can choose N large enough such that $N\geq \theta ^{\ast }$ and, consequently, the last series is convergent. Then we have

$$ \begin{align*} I_{2} &\lesssim \|f\|_{L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{N\cdot({N_{0}}/{N_{0}+1})}\sum_{k=1}^{\infty} \bigg(\frac{|B(x_{0},r)|}{|B(x_{0},2^{k+1}r)|}\bigg)^{(1-\kappa)/p} \\ &\lesssim \|f\|_{L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{N\cdot({N_{0}}/{N_{0}+1})}, \end{align*} $$

where the last inequality follows from the fact that $1-\kappa>0$ . Summing the above estimates for $I_{1}$ and $I_{2}$ , and letting $\nu =\max \{\theta ^{\ast },N\cdot ({N_{0}}/{N_{0}+1})\}$ with $N\geq \theta ^{\ast }$ , we obtain the desired inequality (3-1).

(ii) According to Definition 2.4, it suffices to prove that, for each given ball ${B=B(x_{0},r)\subset \mathbb {R}^{n}}$ , there exists some $\nu>0$ such that

(3-5) $$ \begin{align} \frac{1}{|B(x_{0},r)|^{\kappa}}\sup_{\lambda>0}\lambda\cdot|\{x\in B(x_{0},r):\ |T^{L^{\alpha}}_{N}(f)(x)|>\lambda\}|\lesssim \bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{\nu} \end{align} $$

holds for any function $f\in L^{\rho ,\theta ^{\ast }}_{1,\kappa }(\mathbb {R}^{n})$ with some $\theta ^{\ast }>0$ and $0<\kappa <1$ . We decompose the function f as

$$ \begin{align*} \begin{cases} f=f_{1}+f_{2}\in L^{\rho,\theta^{\ast}}_{1,\kappa}(\mathbb{R}^{n}),\\ f_{1}=f\cdot\chi_{2B},\\ f_{2}=f\cdot\chi_{(2B)^{c}}. \end{cases} \end{align*} $$

Then, for any given $\lambda>0$ , we can write

$$ \begin{align*}\frac{1}{|B(x_{0},r)|^{\kappa}}\lambda\cdot|\{x\in B(x_{0},r):|T^{L^{\alpha}}_{N}(f)(x)|>\lambda\}|\leq J_{1}+J_{2},\end{align*} $$

where

$$ \begin{align*} \left\{ \begin{aligned} J_{1}:= \frac{1}{|B(x_{0},r)|^{\kappa}}\lambda\cdot|\{x\in B(x_{0},r):|T^{L^{\alpha}}_{N}(f_{1})(x)|>\lambda/2\}|,\\ J_{2}:= \frac{1}{|B(x_{0},r)|^{\kappa}}\lambda\cdot|\{x\in B(x_{0},r):|T^{L^{\alpha}}_{N}(f_{2})(x)|>\lambda/2\}|. \end{aligned} \right.\end{align*} $$

With the aid of (3-2), we first estimate the term $J_{1}$ as

$$ \begin{align*} J_{1} &\lesssim \frac{1}{|B|^{\kappa}}\bigg(\int_{2B}|f(x)|\,dx\bigg) \\ &\lesssim \|f\|_{L^{\rho,\theta^{\ast}}_{1,\kappa}(\mathbb{R}^{n})}\frac{|2B|^{\kappa}}{|B|^{\kappa}}\bigg(1+\frac{2r}{\rho(x_{0})}\bigg)^{\theta^{\ast}}\\ &\lesssim \|f\|_{L^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n})}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{\theta^{\ast}}. \end{align*} $$

For the second term $J_{2}$ , using (3-4) and Chebyshev’s inequality, we have

(3-6) $$ \begin{align}\notag \notag J_{2} &\leq \frac{2}{|B(x_{0},r)|^{\kappa}}\bigg(\int_{B(x_{0},r)}|T^{L^{\alpha}}_{N}(f_{2})(x)|\,dx\bigg) \\ \notag &\lesssim \frac{|B(x_{0},r)|}{|B(x_{0},r)|^{\kappa}}\sum^{\infty}_{k=1}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{N\cdot({N_{0}}/{N_{0}+1})}\bigg(1+\frac{2^{k+1}r}{\rho(x_{0})}\bigg)^{-N}\\ &\quad \times\frac{1}{|B(x_{0},2^{k+1}r)|}\int_{B(x_{0},2^{k+1}r)} |f(y)|\,dy. \end{align} $$

Note that, in (3-6), for each fixed $k\in \mathbb {N}$ ,

$$ \begin{align*} \frac{1}{|B(x_{0},2^{k+1}r)|}\int_{B(x_{0},2^{k+1}r)}|f(y)|\,dy \lesssim C\|f\|_{L^{\rho,\theta^{\ast}}_{1,\kappa}(\mathbb{R}^{n})}\frac{|B(x_{0},2^{k+1}r)|^{\kappa}}{|B(x_{0},2^{k+1}r)|} \bigg(1+\frac{2^{k+1}r}{\rho(x_{0})}\bigg)^{\theta^{\ast}}. \end{align*} $$

Consequently,

$$ \begin{align*} J_{2} \lesssim \|f\|_{L^{\rho,\theta^{\ast}}_{1,\kappa}(\mathbb{R}^{n})} \bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{N\cdot({N_{0}}/{N_{0}+1})}\sum_{k=1}^{\infty}\frac{|B(x_{0},r)|}{|B(x_{0},r)|^{k}} \bigg(1+\frac{2^{k+1}r}{\rho(x_{0})}\bigg)^{-N+\theta^{\ast}}\frac{|B(x_{0},2^{k+1}r)|^{\kappa}}{|B(x_{0},2^{k+1}r)|}. \end{align*} $$

Therefore, by choosing N large enough such that $N\geq \theta ^{\ast }$ , we obtain

$$ \begin{align*} J_{2} &\lesssim \|f\|_{L^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n})}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{N\cdot({N_{0}}/{N_{0}+1})}\sum_{k=1}^{\infty} \bigg(\frac{|B(x_{0},r)|}{|B(x_{0},2^{k+1}r)|}\bigg)^{1-\kappa} \\[4pt] &\lesssim \|f\|_{L^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n})}\bigg(1+\frac{r}{\rho(x_{0})}\bigg)^{N\cdot({N_{0}}/{N_{0}+1})}, \end{align*} $$

since $0<\kappa <1$ . Now we choose $\nu =\max \{\theta ^{\ast },N\cdot ({N_{0}}/{N_{0}+1})\}$ with $N\geq \theta ^{\ast }$ . Summing the above estimates for $J_{1}$ and $J_{2}$ and taking the supremum over all $\lambda>0$ , we obtain the desired inequality (3-5).

4 Boundedness of the maximal operator $T^*_{L^{\alpha }}$

In this section, we give the proof of Theorem 4.4 related to the boundedness of the maximal operator $T^*_{L^{\alpha }}$ . We first prove the following lemma by adopting the ideas of [20, Lemma 2.3], which is parallel to [3, Proposition 3.2] or [19, Proposition 3.1]. Furthermore, the following lemma implies that, without loss of generality, we may assume that

(4-1) $$ \begin{align}1<\lambda \leq {a_{j+1} \over a_j}\leq \lambda^2 \quad \mbox{for all }j\in \mathbb Z. \end{align} $$

Lemma 4.1. Given a $\lambda $ -lacunary sequence $\{a_j\}_{j\in \mathbb Z}$ and a multiplying sequence $\{v_j\}_{j\in \mathbb Z}\in \ell ^\infty (\mathbb Z)$ , we can define a $\lambda $ -lacunary sequence $\{\eta _j\}_{j\in \mathbb Z}$ and $\{\omega _j\}_{j\in \mathbb Z}\in \ell ^\infty (\mathbb Z)$ verifying the following properties.

1. (i) $1<\lambda \leq \eta _{j+1}/\eta _j\leq \lambda ^2,\quad \Vert \{\omega _j\}\Vert _{\ell ^\infty (\mathbb Z)}=\Vert \{v_j\}\Vert _{\ell ^\infty (\mathbb Z)}$ .

2. (ii) For all $N=(N_1, N_2)$ , there exists $N'=(N_1', N_2')$ with $T^{L^{\alpha }}_N=\widetilde {T}^{L^{\alpha }}_{N'},$ where $\widetilde {T}_{N'}^{L^{\alpha }}$ is the operator defined in (1-4) for the new sequences $\{\eta _j\}_{j\in {\mathbb Z}}$ and $\{\omega _j\}_{j\in {\mathbb Z}}.$

Owing to Lemma 4.1, in the rest of this article, we always assume that the lacunary sequence $\{a_j\}_{j\in \mathbb Z}$ satisfies (4-1) without saying it explicitly.

Lemma 4.2. Let $\{a_j\}_{j\in \mathbb Z}$ be a $\lambda $ -lacunary sequence and let $\{v_j\}_{j\in \mathbb Z} \in \ell ^\infty (\mathbb Z)$ .

1. (i) For $x,y\in \mathbb R^{n}$ ,

   $$ \begin{align*}\displaystyle \bigg\vert\!\sum_{j=m}^{M}v_j (K^{L}_{\alpha, a_{j+1}}(x,y)- K^{L}_{\alpha, a_{j}}(x,y)) \bigg\vert \le { {C_{v, \lambda,\alpha, n}} \over a_m^{n/(2\alpha)}}.\end{align*} $$

2. (ii) If $k\ge m$ and $z,y \in \mathbb {R}^n$ with $|z-y|\ge a_k^{1/(2\alpha )}$ , then

   $$ \begin{align*} \bigg|\sum_{j=-M}^{m-1}v_j (K^{L}_{\alpha, a_{j+1}}(z,y)-K^{L}_{\alpha, a_{j}}(z,y) ) \bigg|\, \le C_{\lambda, v,\alpha, n}\frac1{a_k^{n/(2\alpha)}}\lambda^{-(k-m+1)}. \end{align*} $$

Proof. For (i), by the mean value theorem and Lemma 3.2, there exists $\xi _j\in [a_j, a_{j+1}]$ such that

$$ \begin{align*}&\bigg\vert\!\sum_{j=m}^{M}v_j (K^{L}_{\alpha, a_{j+1}}(x,y)- K^{L}_{\alpha, a_{j}}(x,y) ) \bigg\vert\\ &\quad\le C\Vert v\Vert_{l^\infty(\mathbb Z)} \sum_{j=m}^{M} (a_{j+1}-a_j)\vert\partial_t K^{L}_{\alpha, t}(x,y)|_{t=\xi_j}\vert\\ &\quad\le C_{v} \sum_{j=m}^{M} (a_{j+1}-a_j) {\frac 1{(\xi_j^{{1}/{(2\alpha)}}+|x-y|)^{n+2\alpha}}} \\ &\quad\le C_{v} \sum_{j=m}^{M} (\lambda^2-1)a_j ^{-{n/(2\alpha)}}\\ &\quad \le C_{v, \lambda} \frac1{a_m^{n/(2\alpha)}} \sum_{j=m}^{M} \frac1{\lambda^{n(j-m)/(2\alpha)}}\\ &\quad \le C_{v, \lambda,\alpha, n} {1\over a_m^{n/(2\alpha)}}, \end{align*} $$

where we have used the fact that $\displaystyle \lambda \le {a_{j+1}/ a_j}\le \lambda ^2.$

Now we prove (ii). Using the mean value theorem again, we can see that there exists $a_j\le \xi _j\le a_{j+1}$ such that

$$ \begin{align*} & \bigg\vert\!\sum_{j=-M}^{m-1}v_j (K^{L}_{\alpha, a_{j+1}}(z-y)-K^{L}_{\alpha, a_{j}}(z-y) ) \bigg\vert\\ &\quad\le C\Vert v\Vert_{l^\infty(\mathbb Z)} \sum_{j=m}^{M} (a_{j+1}-a_j)\vert\partial_t K^{L}_{\alpha, t}(x,y)|_{t=\xi_j}\vert\\&\quad\le C_{v,\alpha, n}\sum_{j=-M}^{m-1}(\lambda^2-1)a_j \frac 1{(\xi_j^{ 1/{(2\alpha)}}+|x-y|)^{n+2\alpha}} \\&\quad\le C_{\lambda, v,\alpha, n}\sum_{j=-M}^{m-1}{a_j \over (\xi_j^{ 1/{(2\alpha)}}+|x-y|)^{n+2\alpha}} \\ &\quad\le C_{\lambda, v,\alpha, n} \sum_{j=-M}^{m-1} {\frac{ a_j }{a_k}}\bigg({1\over a_k^{n/(2\alpha)}}\bigg) \le C_{\lambda, v,\alpha, n} {1\over a_k^{n/(2\alpha)}}\lambda^{-(k-m+1)}, \end{align*} $$

since $k \ge m.$

To prove Theorem 4.4 for $T^*_{L^{\alpha }}$ , we need the following Cotlar-type inequality to control the operator $T^*_{L^{\alpha },M}$ . This is given precisely in the following theorem.

Theorem 4.3. For each $q\in (1, +\infty ),$ there exists a positive constant C, depending on $n, \Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ and $\lambda $ , such that, for every $x\in {\mathbb R}^n$ and for every $M\in \mathbb Z^+$ ,

$$ \begin{align*} T_{L^{\alpha},M}^*f(x)\le C\{{\mathcal M}(T^{L^{\alpha}}_{(-M, M)} f)(x)+{\mathcal M}_q f(x)\}, \end{align*} $$

where

$$ \begin{align*}T_{L^{\alpha},M}^*f(x)=\sup_{-M\le N_1<N_2\le M}\vert T^{L^{\alpha}}_N f(x)\vert\end{align*} $$

and

$$ \begin{align*} {\mathcal M}_qf(x)=\sup_{r>0} \bigg(\frac{1}{|B(x, r)|}\int_{B(x, r)}\vert f(y)\vert^qdy\bigg)^{1/q}. \end{align*} $$

Proof. Observe that, for any $x_0\in {\mathbb R}^n$ and $N=(N_1, N_2),$

$$ \begin{align*}T^{L^{\alpha}}_N f(x_0)=T^{L^{\alpha}}_{(N_1, M)} f(x_0)-T^{L^{\alpha}}_{(N_2+1, M)} f(x_0)\end{align*} $$

with $-M\le N_1<N_2\le M.$ Then it suffices to estimate $|{T^{L^{\alpha }}_{(m, M)} f(x_0)}|$ for $\vert m\vert \le M$ with constants independent of m and $M.$ Denote $B_k=B(x_0, a_k^{1/(2\alpha )})$ for each $k\in \mathbb N$ . We split f as $f=f\chi _{B_m}+f\chi _{B_m^c}:= f_1+f_2$ . Then $|T^{L^{\alpha }}_{(m,M)} f(x_0)|\le I+II,$ where

$$ \begin{align*} \left\{\begin{aligned} I&:=\vert T^{L^{\alpha}}_{(m,M)} f_1(x_0)\vert,\\ II&:=\vert T^{L^{\alpha}}_{(m,M)} f_2(x_0)\vert. \end{aligned}\right.\end{align*} $$

By Lemma 4.2(i),

$$ \begin{align*} I&=\bigg\vert\!\int_{{\mathbb R}^n} \sum_{j=m}^{M}v_j (K^{L}_{\alpha, a_{j+1}}(x_0,y)- K^{L}_{\alpha, a_{j}}(x_0,y)) f_1(y)\,dy\bigg\vert\\ &\le {C_{n, v, \lambda, \alpha}\over |{a^{n/(2\alpha)}_m}|} \int_{{\mathbb R}^n} \vert f_1(y)\vert \,dy \le C_{v, \lambda, n, \alpha}{\mathcal M} f(x_0). \end{align*} $$

Also,

$$ \begin{align*} II&=\frac{2^{n/2}}{a_{m-1}^{n/(2\alpha)}}\int_{B(x_0, {1/ 2}a_{m-1}^{1/(2\alpha)})} \vert T^{L^{\alpha}}_{(m,M)} f_2(x_0)\vert \,dz\leq \sum^{4}_{i=1}A_i, \end{align*} $$

where

$$ \begin{align*}\left\{\begin{aligned} A_{1}&:=\frac{2^{n/2}}{a_{m-1}^{n/(2\alpha)}}\int_{B(x_0, {1/ 2}a_{m-1}^{1/(2\alpha) })}\vert T^{L^{\alpha}}_{(-M,M)} f(z)\vert \,dz;\\ A_{2}&:=\frac{2^{n/2}}{a_{m-1}^{n/(2\alpha)}}\int_{B(x_0, {1/2}a_{m-1}^{1/(2\alpha)})}\vert T^{L^{\alpha}}_{(-M,M)} f_1(z)\vert \,dz;\\ A_{3}&:=\frac{2^{n/2}}{a_{m-1}^{n/(2\alpha)}}\int_{B(x_0, {1/2}a_{m-1}^{1/(2\alpha)})}\vert T^{L^{\alpha}}_{(m,M)} f_2(z)-T^{L^{\alpha}}_{(m,M)} f_2(x_0)\vert \,dz;\\ A_{4}&:=\frac{2^{n/2}}{a_{m-1}^{n/(2\alpha)}}\int_{B(x_0, {1/2}a_{m-1}^{1/(2\alpha)})}\vert T^{L^{\alpha}}_{(-M,m-1)} f_2(z)\vert \,dz. \end{aligned}\right.\end{align*} $$

It should be noted that $A_4=0$ if $m+1=-M$ . It is obvious that

$$ \begin{align*} A_1\le {\mathcal M} (T^{L^{\alpha}}_{(-M,M)} f)(x_0). \end{align*} $$

It follows from the uniform boundedness of $T^{L^{\alpha }}_{N}$ that

$$ \begin{align*} A_2&\le \bigg(\frac{2^{n/2}}{a_{m-1}^{n/(2\alpha)}}\int_{B_{m-1}}\vert T^{L^{\alpha}}_{(-M,M)} f_1(z)\vert^q\,dz\bigg)^{1/q}\\ &\lesssim \bigg(\frac{1}{a_{m-1}^{n/(2\alpha)}}\int_{\mathbb{R}^n}\vert f_1(z)\vert^q\,dz\bigg)^{1/q}\\ &\thicksim\bigg(\frac{1}{a_{m-1}^{n/(2\alpha)}}\int_{B_{m}}\vert f(z)\vert^q\,dz\bigg)^{1/q}\\ &\lesssim \bigg(\frac{\lambda^{n/(2\alpha)}}{a_{m}^{n/(2\alpha)}}\int_{B_{m}}\vert f(z)\vert^q\,dz\bigg)^{1/q}\\ &\lesssim {\mathcal M}_q f(x_0). \end{align*} $$

For the third term $A_3$ , we can see that $ A_3\lesssim {\mathcal M} f(x_0)$ , since, for $z\in B(x_0, {1\over 2}a_{m-1}^{1/(2\alpha )})$ ,

$$ \begin{align*} &\vert T^{L^{\alpha}}_{(m,M)} f_2(z)-T^{L^{\alpha}}_{(m,M)} f_2(x_0)\vert\\ &\quad\le \int_{B_m^c} \vert K^\alpha_{(m,M)}(z,y)-K^\alpha_{(m,M)}(x_0,y)\vert\vert f(y)\vert \,dy\\ &\quad\le \sum_{j=1}^{+\infty}\int_{B_{2^jm} \setminus B_{2^{j-1}m}}\frac{\vert z-x_0\vert^{\delta'}}{\vert x_0-y\vert^{n+\delta'}}\vert f(y)\vert \,dy\\ &\quad\lesssim\sum_{j=1}^{+\infty}\frac 1{2^{j\delta'}}\frac{a_{m-1}^{\delta'/(2\alpha)}}{a_m^{\delta'/(2\alpha)}}{1\over |2^ja_{m}|^{n/(2\alpha)}}\int_{B_{2^jm}} \vert f(y)\vert \,dy\lesssim {\mathcal M} f(x_0), \end{align*} $$

where $B_{2^jm}=B(x_0, 2^ja_m^{1/ (2\alpha )})$ for any $j\ge 1$ .

For $A_4,$ we have

(4-2) $$ \begin{align} A_4\le \frac{2^{n/2}}{a_{m-1}^{n/(2\alpha)}}\int_{B(x_0, {1/ 2}a_{m-1}^{1/(2\alpha)})}\int_{B_m^c}\vert K^\alpha_{(-M,m-1)}(z,y) f(y)\vert \,dy\,dz. \end{align} $$

For the inner integral in (4-2), since $z\in B(x_0, {1\over 2}a_{m-1}^{1/(2\alpha )})$ , $y\in B_m^c$ and the sequence $\{a_j\}_{j\in \mathbb Z}$ is a $\lambda $ -lacunary sequence, we have $\vert z-y\vert \sim \vert y-x_0\vert .$ From the previous arguments and by Lemma 4.2(ii), we get

$$ \begin{align*} &\int_{B_m^c}\vert K^\alpha_{(-M,m-1)}(z, y) f(y)\vert \,dy\\ &\quad\le C_{\lambda, v,\alpha, n} \sum_{k=m}^{+\infty}\lambda^{-(k-m+1)}\bigg({1\over a_k^{n/(2\alpha)}}\int_{B_{k+1}\setminus B_{k}} \vert f(y)\vert \,dy\bigg)\\ &\quad\le C_{\lambda, v,\alpha, n} {\mathcal M} f(x_0)\sum_{k=m}^{+\infty}\lambda^{-(k-m+1)}\\ &\quad\le C_{\lambda, v,\alpha, n}{\mathcal M} f(x_0), \end{align*} $$

which implies $A_4\lesssim {\mathcal M} f(x_0)$ . Combining the above estimates for $A_i, i=1,2,3,4$ , we get

$$ \begin{align*}II\lesssim {\mathcal M} ( T^{L^{\alpha}}_{(-M,M)} f)(x_0)+ {\mathcal M}_q f(x_0) \end{align*} $$

and

$$ \begin{align*} \vert T^{L^{\alpha}}_{(m,M)} f(x_0)\vert\lesssim( {\mathcal M} (T^{L^{\alpha}}_{(-M,M)} f)(x_0)+{\mathcal M}_q f(x_0)). \end{align*} $$

Since the constants C appearing above only depend on $\Vert v\Vert _{l^\infty (\mathbb Z)}$ , $\lambda $ , $\alpha $ and n,

$$ \begin{align*} T^{*}_{{L^{\alpha}},M} f(x_0)\lesssim\{{\mathcal M}( T^{L^{\alpha}}_{(-M, M)} f)(x_0)+{\mathcal M}_q f(x_0)\}.\end{align*} $$

This completes the proof of Theorem 4.3.

We now prove the main result of this paper.

Theorem 4.4. Assume that the sequence $\{a_j\}_{j\in \mathbb Z}$ is a $\lambda $ -lacunary sequence with $\lambda>1$ . Let $\mathcal {S}$ be either the operator $T^*_{L^{\alpha }}$ or the operator $T^*_{(-\Delta )^{\alpha }}$ defined in (1-5) or (2-3).

1. (i) For any $1<p<\infty $ , there exists a positive constant C, depending on $n, p, \lambda $ , $\Vert v\Vert _{l^\infty (\mathbb Z)}$ and $\alpha $ , such that, for all functions $f\in L^p({\mathbb R}^{n})$ ,

   $$ \begin{align*}\Vert\mathcal{S} f\Vert_{L^p(\mathbb R^{n})}\leq C\Vert f\Vert_{L^p(\mathbb R^{n})}.\end{align*} $$

2. (ii) There exists a positive constant C, depending on $n, \lambda $ , $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ and $\alpha $ , such that, for all functions $f\in L^1({\mathbb R}^{n})$ ,

   $$ \begin{align*}|\{x\in {\mathbb R}^{n}:\vert\mathcal{S} f(x)\vert>\sigma\}| \le\frac{ C}{\sigma}\Vert f\Vert_{L^1(\mathbb R^{n})}, \quad \sigma>0,\end{align*} $$

   Here the constants C appearing in (i) and (ii) above are all independent of N.

Proof. First, it follows from [4, Theorem 2.5] that the maximal operators ${\mathcal M}$ and $ {\mathcal M}_q$ are bounded on $L^p({\mathbb R}^n) $ . On the other hand, since the operators $T^{L^{\alpha }}_N$ are uniformly bounded on $L^p({\mathbb R}^n)$ ,

$$ \begin{align*} \Vert T_{L^{\alpha},M}^*f\Vert_{L^p({\mathbb R}^n)}&\leq C \Vert{\mathcal M} (T^{L^{\alpha}}_{(-M, M)} f)\Vert_{L^p({\mathbb R}^n)}+\Vert{\mathcal M}_q f\Vert_{L^p({\mathbb R}^n)}\\&\leq C \Vert T^{L^{\alpha}}_{(-M, M)} f\Vert_{L^p({\mathbb R}^n)}+\Vert f\Vert_{L^p({\mathbb R}^n)}\\ &\leq C\Vert f\Vert_{L^p({\mathbb R}^n)}. \end{align*} $$

Note that the constants C appearing above are independent of M. Consequently, letting M increase to infinity yields to the $L^p$ -boundedness of $T^*_{L^{\alpha }}$ . This completes the proof of part (i) of this theorem.

In order to prove (ii), we consider the $\ell ^\infty (\mathbb Z^2)$ -valued operator

$$ \begin{align*}\mathcal{T}f(x) = \{ T^{L^{\alpha}}_N f(x) \}_{N\in \mathbb Z^2}.\end{align*} $$

Since

(4-3) $$ \begin{align}\|\mathcal{T}f(x) \|_{\ell^\infty(\mathbb Z^2)}= T^*f(x),\end{align} $$

using (i), we conclude that the operator $\mathcal {T}$ is bounded from $L^p({\mathbb R}^n) $ into $L^p_{\ell ^\infty (\mathbb Z^2)}(\mathbb {R}^n)$ for every $1<p<\infty $ . The kernel of $\mathcal {T}$ is given by $\mathcal {K}(x) = \{ K^\alpha _N(x)\} _{N\in \mathbb Z^2}$ . Therefore, by the vector-valued Calderón–Zygmund theory, the operator $\mathcal {T}$ is bounded from $L^1(\mathbb {R}^n)$ into weak- $L^1_{\ell ^\infty (\mathbb Z^2)}(\mathbb {R}^n)$ . Hence, using (4-3) again, we get the proof of (ii).

We also need the following result established in [11] to prove the boundedness of the maximal operator $T^*_{L^{\alpha }}$ in the space $BMO_{L}(\mathbb R^{n})$ . Let f be a locally integrable function on $\mathbb {R}^{n}$ and let $B=B(x,R)$ be a ball. Let

$$ \begin{align*} f(B,V):= \begin{cases} \dfrac{1}{|B|}\displaystyle\int_{B}f(y)\,dy & \mbox{if }R<\rho(x),\\[5pt] 0 & \mbox{if }R\geq\rho(x), \end{cases} \end{align*} $$

and let $0<\gamma \leq 1$ . The space $BMO_{L}^{\gamma }(\mathbb {R}^{n})$ is defined as the set of all locally integrable functions f satisfying

$$ \begin{align*}\|f\|_{BMO_{L}^{\gamma}(\mathbb{R}^{n})}:=\sup_{B\subset\mathbb{R}^{n}}\bigg\{ \frac{1}{|B|^{1+{\gamma}/{n}}}\int_{B}|f(x)-f(B,V)|\,dx\bigg\}<\infty.\end{align*} $$

Lemma 4.5 [11, Theorem 1.2].

Let T be a $\gamma $ -Schrödinger–Calderón–Zygmund operator for $0\le \gamma <\min \{1,\delta \}$ with the smoothness exponent $\delta>0$ . Then T is a bounded operator from $BMO_L(\mathbb R^{n})$ into $BMO_L^\gamma (\mathbb R^{n})$ if and only if there exists a constant C such that

(4-4) $$ \begin{align} \displaystyle\log\bigg(\frac{\rho(x_0)}{t}\bigg)\frac{1}{|B|^{1+{\gamma}/{n}}}\int_B|T1(x) -(T1)_B|\,dx\leq C \end{align} $$

for every ball $B=B(x_0,t)$ , $x_0\in {\mathbb R}^n$ and $0<t\leq \rho (x_0)/2$ . $T1$ means that the operator T acts on the constant function 1.

Theorem 4.6. Let $\{a_j\}_{j\in \mathbb Z}$ be a $\lambda $ -lacunary sequence and let $T^*_{L^{\alpha }}$ be the operator defined in (1-5). We have the following statements.

1. (i) There exists a positive constant C, depending on n, $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ and $\alpha $ , such that, for all functions $f\in L^\infty ({\mathbb R}^{n})$ ,

   $$ \begin{align*}\Vert T^*_{L^{\alpha}} f\Vert_{BMO_L(\mathbb R^{n})}\leq C\Vert f\Vert_{L^\infty(\mathbb R^{n})}.\end{align*} $$

2. (ii) There exists a positive constant C, depending on n, $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ and $\alpha $ , such that, for all functions $f\in BMO_L({\mathbb R}^n)$ ,

   $$ \begin{align*}\Vert T^*_{L^{\alpha}} f\Vert_{BMO_L({\mathbb R}^n)}\le C\Vert f\Vert_{BMO_L({\mathbb R}^n)}.\end{align*} $$

   The constants C appearing in (i) and (ii) are all independent of $N.$

Proof. By an argument similar to that used in Theorem 3.6, we know that $ T^*_{L^{\alpha }} f<\infty \text{for almost every}\ x \in \mathbb {R}^{n}$ . Since (i) can be deduced from (ii), we only need to prove (ii). By Theorem 4.4(i) and Proposition 3.5, $T^{L^{\alpha }}_N$ and ${T}^{*} =\| \, \{ T^{L^{\alpha }}_N\}_{N=(N_1, N_2)}\, \|_{\ell ^\infty (\mathbb Z^2)}$ are $\gamma $ -Schrödinger–Calderón–Zygmund operators with $\gamma =0$ . Then it is enough to verify that both of the operators satisfy (4-4). More precisely,

$$ \begin{align*} | {T}^{*}1(x)-({T}^{*}1)_B | &\le \bigg| {T}^{*}1(x) - \frac1{|B|} \int_B {T}^{*}1(y) \,dy \bigg| \\ & \le \frac1{|B|} \int_B | {T}^{*}1(x) - {T}^{*}1(y) | \,dy \\ &= \frac1{|B|} \int_B | \| \, \{ T^{L^{\alpha}}_N 1(x)\}_N\, \|_{\ell^\infty(\mathbb Z^2)} - \| \, \{ T^{L^{\alpha}}_N 1(y) \}_N\, \|_{\ell^\infty(\mathbb Z^2)} | \, dy \\ & \le \frac1{|B|} \int_B \| \, T^{L^{\alpha}}_N 1(x) - T^{L^{\alpha}}_N 1(y)\, \|_{\ell^\infty(\mathbb Z^2)}\, dy\\ &= \frac1{|B|} \int_B \sup_N | T^{L^{\alpha}}_N 1(x) - T^{L^{\alpha}}_N 1(y)\, | \, dy. \end{align*} $$

Hence, to prove (4-4), it is sufficient to prove that, for $0<\delta <\min \{1, 2\alpha , 2-n/q\} $ and for any $x, y \in B=B(x_0,t),$ if $t\le \rho (x_0)/2$ , then

(4-5) $$ \begin{align} \vert T^{L^{\alpha}}_N 1(x)-T^{L^{\alpha}}_N1(y)\vert\lesssim(t/ {\rho(x_0)})^\delta. \end{align} $$

Note that $\rho (x)\sim \rho (y)\sim \rho (x_0)$ when $x, y \in B$ . Since $e^{- {t(-\Delta )^\alpha }}f$ is a convolution operator, it follows from (1-1) that $(\mathcal {F}K^{-\Delta }_{\alpha , s}(\cdot ))(\xi )=e^{-t|\xi |^{2\alpha }}$ and

(4-6) $$ \begin{align} \int_{{\mathbb R}^{n}} K^{-\Delta}_{\alpha, s}(x-z)\,dz=\int_{{\mathbb R}^{n}} K^{-\Delta}_{\alpha, s}(y-z)\,dz=e^{-t|\xi|^{2\alpha}}|_{\xi=0}=1 \end{align} $$

for any $s>0.$ Then we can write

$$ \begin{align*} &\vert T^{L^{\alpha}}_N 1(x)-T^{L^{\alpha}}_N 1(y)\vert\\ &\quad= \bigg|\! \int_{{\mathbb R}^{n}} \sum_{j=N_1}^{N_2}v_j[( (K^{L}_{\alpha, a_{j+1}}(x,z)- K^{-\Delta}_{\alpha, a_{j+1}}(x-z)) -(K^{L}_{\alpha, a_{j}}(x,z) - K^{-\Delta}_{\alpha, a_{j}}(x-z) ) ) \\ &\qquad - ((K^{L}_{\alpha, a_{j+1}}(y,z)- K^{-\Delta}_{\alpha, a_{j+1}}(y-z))- (K^{L}_{\alpha, a_{j}}(y,z) - K^{-\Delta}_{\alpha, a_{j}}(y-z))) ] \,dz\bigg| \\ &\quad\lesssim\Vert v\Vert_{\ell^\infty(\mathbb Z)} \int_{{\mathbb R}^{n}}\int_0^\infty|\partial_s((K^{L}_{\alpha, s}(x,z)-K^{-\Delta}_{\alpha, s}(x-z)) -(K^{L}_{\alpha, s}(y,z)- K^{-\Delta}_{\alpha, s}(y-z) )\, | \,ds \,dz\\ &\quad\lesssim I+II+III, \end{align*} $$

where

$$ \begin{align*}\left\{\begin{aligned} I&:=\int_{{\mathbb R}^{n}}\int_0^{(2t)^{2\alpha}} |\partial_s((K^{L}_{\alpha, s}(x,z)-K^{-\Delta}_{\alpha, s}(x-z))\\ &\quad -(K^{L}_{\alpha, s}(y,z)- K^{-\Delta}_{\alpha, s}(y-z) )\, | \,ds \,dz;\\ II&:=\int_{{\mathbb R}^{n}}\int_{(2t)^{2\alpha}}^{ \rho^{2\alpha}(x_0)}|\partial_s((K^{L}_{\alpha, s}(x,z)-K^{-\Delta}_{\alpha, s}(x-z))\\ &\quad -(K^{L}_{\alpha, s}(y,z)- K^{-\Delta}_{\alpha, s}(y-z) )\, | \,ds \,dz;\\ III&:=\int_{{\mathbb R}^{n}}\int_{\rho^{2\alpha}(x_0)}^\infty |\partial_s((K^{L}_{\alpha, s}(x,z)-K^{-\Delta}_{\alpha, s}(x-z))\\ &\quad -(K^{L}_{\alpha, s}(y,z)- K^{-\Delta}_{\alpha, s}(y-z) )\, | \,ds \,dz. \end{aligned}\right.\end{align*} $$

For $I,$ by Lemma 3.4, we get $I\leq C(I_{1}+I_{2})$ , where

$$ \begin{align*}\left\{\begin{aligned} I_{1}&:=\int_{{\mathbb R}^{n}} \int_0^{(2t)^{2\alpha}} |\partial_s(K^{L}_{\alpha, s}(x,z)-K^{-\Delta}_{\alpha, s}(x-z) ) |\,ds \,dz;\\ I_{2}&:=\int_{{\mathbb R}^{n}} \int_0^{(2t)^{2\alpha}}| \partial_s(K^{L}_{\alpha, s}(y,z)-K^{-\Delta}_{\alpha, s}(y-z)) |\,ds \,dz. \end{aligned}\right.\end{align*} $$

For $I_{1}$ , when $\sqrt {s^{1/\alpha }}\leq |x-z|$ , by choosing $\delta _{0}<2\alpha $ , we get

$$ \begin{align*} I_{1} &\lesssim \int^{(2t)^{2\alpha}}_{0}\int_{\mathbb{R}^{n}}\bigg(\frac{|x-z|}{\rho(x)}\bigg)^{\delta_{0}}\frac{1}{(|x-z|+s^{1/(2\alpha)})^{n+2\alpha}}\,dz\,ds \\ &\lesssim \int^{(2t)^{2\alpha}}_{0}\frac{1}{\rho(x_{0})^{\delta_{0}}}\int_{\mathbb{R}^{n}}\frac{|x-z|^{\delta_{0}}}{(|x-z|+s^{1/(2\alpha)})^{n+2\alpha}}\,dz\,ds \\ &\lesssim \int^{(2t)^{2\alpha}}_{0}\bigg(\frac{s^{1/(2\alpha)}}{\rho(x_{0})}\bigg)^{\delta_{0}}\frac{1}{s}\,ds \\ &\lesssim \bigg(\frac{t}{\rho(x_{0})}\bigg)^{\delta_{0}}. \end{align*} $$

When $\sqrt {s^{1/\alpha }}>|x-z|$ , we obtain

$$ \begin{align*} I_{1} &\lesssim \int^{(2t)^{2\alpha}}_{0}\int_{\mathbb{R}^{n}}\bigg(\frac{\sqrt{s^{1/\alpha}}}{\rho(x)}\bigg)^{\delta_{0}}\frac{1}{(|x-z|+s^{1/(2\alpha)})^{n+2\alpha}}\,dz\,ds \\ &\lesssim \int^{(2t)^{2\alpha}}_{0}\bigg(\frac{s^{1/(2\alpha)}}{\rho(x_{0})}\bigg)^{\delta_{0}}\frac{1}{s}\,ds \\ &\lesssim \bigg(\frac{t}{\rho(x_{0})}\bigg)^{\delta_{0}}. \end{align*} $$

The proof for $I_{2}$ is similar to that for $I_{1}$ , so we omit the details.

We have

$$ \begin{align*} II &= \int_{{\mathbb R}^{n}} \int_{(2t)^{2\alpha}}^{ \rho^{2\alpha}(x_0)} |\partial_s(K^{L}_{\alpha, s}(x,z)-K^{L}_{\alpha, s}(y,z) )\\ &\quad-\partial_s(K^{-\Delta}_{\alpha, s}(x-z)-K^{-\Delta}_{\alpha, s}(y-z)) | \,ds \,dz\\ &\lesssim II_1+II_2+II_3, \end{align*} $$

where

$$ \begin{align*}\left\{\begin{aligned} II_{1}&:=\int_{\vert x-z\vert>c\rho(x)}\int_{(2t)^{2\alpha}}^{ \rho^{2\alpha}(x_0)}\vert\partial_s(K^{L}_{\alpha, s}(x,z)-K^{L}_{\alpha, s}(y,z)) \vert\\ &\quad +\vert\partial_s(K^{-\Delta}_{\alpha, s}(x-z)-K^{-\Delta}_{\alpha, s}(y-z))\vert\,ds \,dz; \\ II_{2}&:= \int_{4|x-y|<\vert x-z\vert\le c\rho(x)}\int_{(2t)^{2\alpha}}^{ \rho^{2\alpha}(x_0)}|\partial_s K^{L}_{\alpha, s}(x,z)-K^{-\Delta}_{\alpha, s}(x-z))\\ &\quad -\partial_s(K^{L}_{\alpha, s}(y,z)-K^{-\Delta}_{\alpha, s}(y-z)) |\,ds \,dz;\\ II_{3}&:=\int_{\vert x-z\vert\le 4|x-y|}\int_{(2t)^{2\alpha}}^{ \rho^{2\alpha}(x_0)}|\partial_s(K^{L}_{\alpha, s}(x,z)-K^{-\Delta}_{\alpha, s}(x-z))\\ &\quad -\partial_s(K^{L}_{\alpha, s}(y,z)-K^{-\Delta}_{\alpha, s}(y-z))| \,ds \,dz. \end{aligned}\right.\end{align*} $$

For $II_1$ , noting that $\vert x-y\vert \le 2t\le s^{1/(2\alpha )},$ by Lemma 3.3 and a similar result for the Laplacian as in Lemma 3.3, we have

$$ \begin{align*} II_{11} &\lesssim\int_{\vert x-z\vert>c\rho(x)}{\vert x-y\vert^\delta\over \vert x-z\vert^{2\alpha+n}} \int_{(2t)^{2\alpha}}^ { \rho^{2\alpha}(x_0)}~\frac{1}{s^{\delta/(2\alpha)}}\,ds\,dz\\ &\lesssim\bigg(\frac{t}{\rho(x_{0})}\bigg)^{\delta}\int_{\vert x-z\vert>c\rho(x)}\frac{1}{|x-z|^{n+2\alpha}}\rho^{2\alpha}(x_0)\,dz\\ &\lesssim\bigg(\frac{t}{\rho(x_{0})}\bigg)^{\delta}. \end{align*} $$

For the term $II_2$ , in this case, $|x-y|< c\rho (x)$ and $\displaystyle |x-y|<{|x-z|/4}$ . By Lemma 3.4, we conclude that, for some $0<\varepsilon <\min \{\alpha ,\delta \}$ small enough,

$$ \begin{align*} II_2&\lesssim\int_{4|x-y|<\vert x-z\vert\le c\rho(x)}\int_{(2t)^{2\alpha}}^{\rho^{2\alpha}(x_0)} \bigg({\vert x-y\vert\over \rho(z)}\bigg)^\delta \frac{1}{(s^{1/(2\alpha)}+|x-z|)^{n+2\alpha}}\,ds \,dz\\ &\lesssim\bigg({\vert x-y\vert\over \rho(x_0)}\bigg)^\delta \int_{4|x-y|<\vert x-z\vert\le c\rho(x)}\int_{(2t)^{2\alpha}}^{ \rho^{2\alpha}(x_0)} \frac{1}{s^{n/(2\alpha)+1}} \frac{|x-z|^{\epsilon}\,ds\,dz}{(1+|x-z|/s^{1/(2\alpha)})^{n+2\epsilon}|x-z|^{\epsilon}}\\ &\lesssim\bigg({\vert x-y\vert\over \rho(x_0)}\bigg)^\delta\bigg({\rho(x_0) \over\vert x-y\vert}\bigg)^{\varepsilon}\\ &\lesssim\bigg({\vert x-y\vert\over \rho(x_0)}\bigg)^{\delta'}. \end{align*} $$

For the term $II_3,$ since $|x-y|\leq 2t\leq s^{1/(2\alpha )}$ , we have $|x-z|\leq s^{1/(2\alpha )}$ . By Lemma 3.4,

$$ \begin{align*} II_3 &\le\int_{\vert x-z\vert\le 4|x-y|}\int_{(2t)^{2\alpha}}^{ \rho^{2\alpha}(x_0)}\\&\quad \times \vert{\partial_s(K^{L}_{\alpha, s}(x,z)-K^{-\Delta}_{\alpha, s}(x-z))- \partial_s(K^{L}_{\alpha, s}(y,z)-K^{-\Delta}_{\alpha, s}(y-z)) }\vert\,ds \,dz\\&\lesssim\int_{\vert x-z\vert\le 4|x-y|}\int_{(2t)^{2\alpha}}^{\rho^{2\alpha}(x_0)}\bigg\{\bigg(\frac{\sqrt{s^{1/\alpha}}}{\rho(x)}\bigg)^\delta \frac{1}{(s^{1/(2\alpha)}+|x-z|)^{n+2\alpha}}\\&\quad +\bigg(\frac{\sqrt{s^{1/\alpha}}}{\rho(y)}\bigg)^\delta \frac{1}{(s^{1/(2\alpha)}+|y-z|)^{n+2\alpha}}\bigg\}\,ds \,dz\\&\lesssim\bigg\{\!\int_{\vert x-z\vert\le 4|x-y|}\int_{(2t)^{2\alpha}}^{ \rho^{2\alpha}(x_0)}\bigg(\frac{\sqrt{s^{1/\alpha}}}{\rho(x)}\bigg)^\delta \frac{1}{(s^{1/(2\alpha)}+|x-z|)^{n+2\alpha}}\,ds\,dz\\&\quad +\int_{\vert y-z\vert\le 5|x-y|}\int_{(2t)^{2\alpha}}^{\rho^{2\alpha}(x_0)}\bigg(\frac{\sqrt{s^{1/\alpha}}}{\rho(y)}\bigg)^\delta \frac{1}{(s^{1/(2\alpha)}+|y-z|)^{n+2\alpha}}\,ds \,dz\bigg\}\\&\lesssim \int_{(2t)^{2\alpha}}^{\rho^{2\alpha}(x_0)} \int_{\vert\xi\vert\le 5{|x-y|/ c}}\bigg(\frac{\sqrt{s^{1/\alpha}}}{\rho(x)}\bigg)^\delta \frac{1}{(s^{1/(2\alpha)}+|\xi|)^{n+2\alpha}}\, d\xi \,ds\\&\lesssim \int_{(2t)^{2\alpha}}^{+\infty} \bigg(\frac{\sqrt{s^{1/\alpha}}}{\rho(x_0)}\bigg)^\delta {1\over s^{n/(2\alpha)+1} } |x-y|^n \,ds \\&\lesssim\bigg({t\over \rho(x_0)}\bigg)^{\delta}, \end{align*} $$

which gives $II\lesssim ({t/ \rho (x_0)})^{\delta }$ with $0<\delta <2-n/q.$

We next consider the last term $III$ . Now

$$ \begin{align*} III&\lesssim \int_{{\mathbb R}^{n}} \int_{ \rho^{2\alpha}(x_0)}^{+\infty}\vert\partial_s(K^{L}_{\alpha, s}(x,z)-K^{L}_{\alpha, s}(y,z))\vert\,ds \,dz\\ &\quad +\int_{{\mathbb R}^{n}} \int_{ \rho^{2\alpha}(x_0)}^{+\infty} \vert\partial_s(K^{-\Delta}_{\alpha, s}(x-z)-K^{-\Delta}_{\alpha, s}(y-z)) \vert \,ds \,dz. \end{align*} $$

In this case, since $s\ge \rho ^{2\alpha }(x_0)>(2t)^{2\alpha }$ , we have $s^{1/(2\alpha )}>2t>|x-y|$ . Regarding the first term in the summand above, we can apply Lemma 3.3 to deduce that

$$ \begin{align*} &{ \int_{{\mathbb R}^{n}}\int_{ \rho^{2\alpha}(x_0)}^{+\infty}{\vert\partial_s(K^{L}_{\alpha, s}(x,z)-K^{L}_{\alpha, s}(y,z))\vert}\,ds \,dz}\\& \quad \lesssim \int_{\rho^{2\alpha}(x_0)}^{+\infty}{1\over s} \bigg({|x-y|\over s^{1/(2\alpha)}}\bigg)^\delta\bigg(\int_{{\mathbb R}^{n}}\frac{s}{(s^{1/(2\alpha)}+|x-z|)^{n+2\alpha}} \,dz\bigg)\,ds \\& \quad \lesssim \int_{\rho^{2\alpha}(x_0)}^{+\infty}{1\over s} \bigg({|x-y|\over s^{1/(2\alpha)}}\bigg)^\delta\,ds \\& \quad \lesssim\bigg({|x-y|\over \rho(x_0)}\bigg)^\delta \le C\bigg({t\over \rho(x_0)}\bigg)^\delta, \quad 0<\delta<2-n/q. \end{align*} $$

For the second part of the summand, by Lemma 3.3, which is also valid for the Laplacian, when $4|x-y| \le |x-z|$ , we obtain the same estimate as before but with $0<\delta <\min \{1, 2-n/q\}$ . If $|x-z| \le 4|x-y|$ , then $|y-z|\le |x-y|+|x-z|\le 5|x-y|$ . Hence,

$$ \begin{align*} &\int_{{\mathbb R}^{n}} \int_{ \rho^{2\alpha}(x_0)}^{+\infty} |\partial_s K^{-\Delta}_{\alpha, s}(x-z)-\partial_s K^{-\Delta}_{\alpha, s}(y-z)| \,ds \,dz\\ &\quad\lesssim\int_{{\mathbb R}^{n}} \int_{ \rho^{2\alpha}(x_0)}^{+\infty} \frac{1}{(s^{1/(2\alpha)}+|x-z|)^{n+2\alpha}} \,ds \,dz\\ &\quad\lesssim \int_{ \rho^{2\alpha}(x_0)}^{+\infty} \frac{|x-y|^n}{s^{n/(2\alpha)+1}} \,ds \le C \frac{|x-y|^n}{\rho^n(x_0)}\\ &\quad\lesssim \bigg({t\over \rho(x_0)}\bigg)^\delta. \end{align*} $$

Combining the above estimates for $I, II$ and $III,$ we have proved (4-5). Hence, we get the estimation (4-4) and the $BMO$ -boundedness of $T^{L^{\alpha }}_N$ and $T^{*}_{L^{\alpha }}.$

For the case $f\in BMO_L({\mathbb R}^n),$ by (4-5),

$$ \begin{align*} \bigg(\frac{\rho(x_0)}{t}\bigg)^{\gamma}\frac{1}{|B|}\int_B| T_N^{L^{\alpha}} 1(x)-( T_N^{L^{\alpha}} 1)_B|\,dx\lesssim \bigg(\frac{\rho(x_0)}{t}\bigg)^{\delta-\gamma}\le C \end{align*} $$

when $\delta -\gamma \ge 0,$ which can be guaranteed by $\gamma <\min \{1, 2\alpha , 2-n/q\}$ and choosing $\delta \ge \gamma .$ This completes the proof of Theorem 4.6.

Theorem 4.7. Let $\{a_j\}_{j\in \mathbb Z}$ a $\lambda $ -lacunary sequence, let $0<\kappa <1$ and let $T^*_{L^{\alpha }}$ be the operator defined in (1-5). We have the following statements.

1. (i) There exists a positive constant C, depending on n, $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ , $\theta $ and $\alpha $ , such that for all functions $f\in L^{\rho ,\infty }_{p,\kappa }(\mathbb {R}^{n})$ ,

   $$ \begin{align*}\Vert T^*_{L^{\alpha}} f\Vert_{L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})}\leq C\Vert f\Vert_{L^{\rho,\infty}_{p,\kappa}(\mathbb{R}^{n})}.\end{align*} $$

2. (ii) There exists a positive constant C, depending on n, $\Vert v\Vert _{\ell ^\infty (\mathbb Z)}$ , $\theta $ and $\alpha $ , such that, for all functions $f\in L^{\rho ,\infty }_{1,\kappa }(\mathbb {R}^{n})$ ,

   $$ \begin{align*}\Vert T^*_{L^{\alpha}} f\Vert_{WL^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n})}\le C\Vert f\Vert_{L^{\rho,\infty}_{1,\kappa}(\mathbb{R}^{n})}.\end{align*} $$

   The constants C appearing in (i) and (ii) are independent of $N.$

Proof. Using Theorem 4.4 and Proposition 3.5, we deduce the desired results. Since the proofs are similar to those of Theorem 3.6, we omit the details.

The following theorem analyzes the behavior of the functions in $L^{\infty }$ more deeply.

Lemma 4.8. Let $0<\gamma <1/2$ .

1. (i) There exists $t_{0}>0$ such that

   $$ \begin{align*}\int^{1}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz=1/4+\gamma/2.\end{align*} $$

2. (ii) Given $t_{0}>0$ as before, there exists $a>1$ large enough such that

   $$ \begin{align*}\left\{\begin{aligned} &\frac{1+\gamma}{4}<\int^{1}_{1/a}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz<\int^{1}_{0} K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz=1/4+\gamma/2;\\[4pt]&\int^{1/a}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz=\bigg(\int^{1}_{0}-\int^{1}_{1/a}\bigg) K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz<\gamma/4;\\[4pt]&\int^{\infty}_{1/a}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz>1/2-\gamma/4. \end{aligned}\right.\end{align*} $$

3. (iii) There exists $\eta _{0}>0$ such that

   $$ \begin{align*}\int^{\infty}_{1-\eta_{0}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz<1/4-\gamma/4.\end{align*} $$

Proof. By (4-6), we know that

$$ \begin{align*}K_{\alpha,t^{2\alpha}}^{-\Delta}=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-t^{2\alpha}|\xi|^{2\alpha}}e^{ix\xi}d\xi =\frac{1}{t}K_{\alpha,1}^{-\Delta}(x/t)\end{align*} $$

and

$$ \begin{align*}\int_{\mathbb{R}}K_{\alpha,1}^{-\Delta}(x)\,dx=1.\end{align*} $$

Moreover,

$$ \begin{align*} K_{\alpha,1}^{-\Delta}(x) &= \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-|\xi|^{2\alpha}}e^{ix\xi}d\xi \\ &= \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-|\eta|^{2\alpha}}e^{i(-x)\eta}d\eta= K_{\alpha,1}^{-\Delta}(-x), \end{align*} $$

where $\eta =-\xi $ . Therefore,

$$ \begin{align*}\int^{\infty}_{0}K_{\alpha,1}^{-\Delta}(x)\,dx=\frac{1}{2}\int_{\mathbb{R}}K_{\alpha,1}^{-\Delta}(x)\,dx=\frac{1}{2}.\end{align*} $$

Then, there exists $A>0$ such that $\int ^{A}_{0}K_{\alpha ,1}^{-\Delta }(w)dw=1/4+\gamma /2.$ Let $w=Az$ . Then,

$$ \begin{align*} K_{\alpha,1}^{-\Delta}(Az) &= \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{iAz\xi}e^{-|\xi|^{2\alpha}}d\xi \\ &= \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{iz\eta}e^{-A^{-2\alpha}|\eta|^{2\alpha}}A^{-1}d\eta=A^{-1}K_{\alpha,A^{-2\alpha}}^{-\Delta}(z), \end{align*} $$

where $\eta =A\xi $ . So,

$$ \begin{align*}\int^{1}_{0}K_{\alpha,A^{-2\alpha}}^{-\Delta}(z)\,dz=1/4+\gamma/2.\end{align*} $$

Assuming that $A=t_{0}^{-1}$ ,

$$ \begin{align*}\int^{1}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz=1/4+\gamma/2.\end{align*} $$

This completes the proof of (i). The proofs of (ii) and (iii) follow easily.

Proposition 4.9. There exist a bounded sequence $\{v_{j}\}_{j\in \mathbb {Z}}$ , a $\lambda $ -lacunary sequence $\{a_{j}\}_{j\in \mathbb {Z}}$ and $f\in L^{\infty }(\mathbb {R})$ such that $T^*_{(-\Delta )^{\alpha }}f(x)=\infty $ for all $x\in \mathbb {R}$ .

Proof. Let $t_{0}$ and a be as in Lemma 4.8 and let f be the function defined as

$$ \begin{align*}f(x)=\sum_{k\in\mathbb{Z}}(-1)^{k+1}\chi_{(a^{k},a^{k+1}]}(x).\end{align*} $$

It is easy to see that

(4-7) $$ \begin{align} f(a^{j}x)=(-1)^{j}f(x). \end{align} $$

Considering the sequence $\{a_{j}\}$ with $a_{j}=(t_{0}a^{j})^{2\alpha }$ , we have

$$ \begin{align*} e^{-a_{j}(-\Delta)^{\alpha}}f(x) = \int_{\mathbb{R}}K_{\alpha,(t_{0}a^{j})^{2\alpha}}^{-\Delta}(y)f(x-y)\,dy = \int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(x-a^{j}z)\,dz. \end{align*} $$

By (4-7),

$$ \begin{align*} e^{-a_{j}(-\Delta)^{\alpha}}f(0) =\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(-a^{j}z)\,dz =(-1)^{\hspace{1.2pt}j}\int^{\infty}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(z)\,dz. \end{align*} $$

On the other hand, using Lemma 4.8, we get

$$ \begin{align*} & \int^{\infty}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(z)\,dz \\ &\quad= \int^{1/a}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(z)\,dz+\int^{1}_{1/a}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(z)\,dz + \int^{\infty}_{1}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(z)\,dz \\ &\quad\geq \int^{1}_{1/a}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz-\bigg(\int^{1/a}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz+ \int^{\infty}_{1}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz\bigg) \\ &\quad> \frac{1+\gamma}{4}-\bigg(\frac{\gamma}{4}+\frac{1}{4}-\frac{\gamma}{2}\bigg)=\frac{\gamma}{2}. \end{align*} $$

By (4-7), we get

$$ \begin{align*} &e^{-a_{j+1}(-\Delta)^{\alpha}}f(x)-e^{-a_{j}(-\Delta)^{\alpha}}f(x) \\ &\quad= (-1)^{j+1}\bigg\{\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(x/a^{j+1}-z)\,dz+ \int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(x/a^{j}-z)\,dz\bigg\}. \end{align*} $$

By the dominated convergence theorem, we obtain

$$ \begin{align*} \lim_{h\rightarrow 0}\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(h-z)\,dz &=\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(-z)\,dz \\ &= \int^{\infty}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(z)\,dz>\frac{\gamma}{2}. \end{align*} $$

So, there exists $0<\eta _{0}<1$ such that, for $|h|<\eta _{0}$ ,

$$ \begin{align*}\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(h-z)\,dz\geq \gamma/4.\end{align*} $$

Then, for each $x\neq 0$ , we can choose $j\in \mathbb {Z}$ such that $|x/a^{j}|<\eta _{0}$ ; at this time, there are infinitely many j satisfying this condition, and we get

$$ \begin{align*}\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(x/a^{j+1}-z)\,dz+\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z) f(x/a^{j}-z)\,dz\geq\gamma/2.\end{align*} $$

Choosing $v_{j}=(-1)^{j+1}$ with $j\in \mathbb {Z}$ , we have, for any $x\in \mathbb {R}$ ,

$$ \begin{align*} T^{\ast}_{L^{\alpha},N}f(x) &\geq \sum_{|t/a^{j}|<\eta_{0}}(-1)^{j+1}(e^{-a_{j+1}(-\Delta)^{\alpha}}f(x)-e^{-a_{j}(-\Delta)^{\alpha}}f(x)) \\ &\geq \sum_{|t/a^{j}|<\eta_{0}}(-1)^{j+1}\bigg(\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(x/a^{j+1}-z)\,dz \\ &\quad+\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(x/a^{j}-z)\,dz\bigg)=\infty.\\[-42pt] \end{align*} $$

Theorem 4.10. Let $\{v_{j}\}_{j\in \mathbb {Z}}\in l^{p}(\mathbb {Z})$ for some $1\leq p\leq \infty $ . We have the following statements.

1. (i) Let $\{a_{j}\}_{j\in \mathbb {Z}}$ be any increasing sequence and let $S^{\ast }$ be either $T^{\ast }_{(-\Delta )^{\alpha }}$ or $T^*_{L^{\alpha }}$ , which is defined in (2-3) or (1-5). Then, for every $f\in L^{\infty }(\mathbb {R}^{n})$ with support in the unit ball $B=B(0,1)$ , for any ball $B_{r}\subset B$ with $2r<1$ ,

   $$ \begin{align*}\frac{1}{|B_{r}|}\int_{B_{r}}|S^{\ast}f(x)|\,dx\lesssim\bigg(\log\frac{2}{r}\bigg)^{1/p'}\|v\|_{l^{p}(\mathbb{Z})}\|f\|_{L^{\infty}(\mathbb{R}^{n})}.\end{align*} $$

2. (ii) When $1< p <\infty $ , for any $\epsilon>0$ , there exist a $\lambda $ -lacunary sequence $\{a_{j}\}_{j\in \mathbb {Z}}$ , a sequence $\{v_{j}\}_{j\in \mathbb {Z}}\in l^{p}(\mathbb {Z})$ and a function $f\in L^{\infty }(\mathbb {R}^{n})$ with support in the unit ball $B=B(0,1)$ such that the following statement holds: for any ball $B_{r}\subset B$ with $2r<1$ ,

   $$ \begin{align*}\frac{1}{|B_{r}|}\int_{B_{r}}|T^{\ast}_{(-\Delta)^{\alpha}}f(x)|\,dx\gtrsim\bigg(\log \frac{2}{r}\bigg)^{1/(p-\epsilon)'}\|v\|_{l^{p}(\mathbb{Z})}\|f\|_{L^{\infty}(\mathbb{R}^{n})}.\end{align*} $$

3. (iii) When $p=\infty $ , there exist a $\lambda $ -lacunary sequence $\{a_{j}\}_{j\in \mathbb {Z}}$ , a sequence $\{v_{j}\}_{j\in \mathbb {Z}}\in l^{\infty }(\mathbb {Z})$ and $f\in L^{\infty }(\mathbb {R}^{n})$ with support in the unit ball $B=B(0,1)$ such that the following statement holds: for any ball $B_{r}\subset B$ with $2r<1$ ,

   $$ \begin{align*}\frac{1}{|B_{r}|}\int_{B_{r}}|T^{\ast}_{(-\Delta)^{\alpha}}f(x)|\,dx\gtrsim\bigg(\log\frac{2}{r}\bigg)\|v\|_{l^{\infty}(\mathbb{Z})}\|f\|_{L^{\infty}(\mathbb{R}^{n})}.\end{align*} $$

In the statements above, $p'={p}/{(p-1)}$ , and if $p=1$ , then $p'=\infty $ .

Proof. For (i), we prove this theorem only for the case $1<p<\infty $ and in the Schrödinger setting. For the cases $p=1$ and $p=\infty $ , the proof follows by introducing the obvious changes. Since $2r<1$ , we know that $B\setminus B_{2r}\neq \varnothing $ . Let $f=f_{1}+f_{2}$ , where $f_{1}=f\chi _{B_{2r}}$ and $f_{2}=f\chi _{B\setminus B_{2r}}$ . Then,

$$ \begin{align*}|T^*_{L^{\alpha}} f(x)|\leq | T^*_{L^{\alpha}} f_{1}(x)|+|T^*_{L^{\alpha}} f_{2}(x)|.\end{align*} $$

By Theorem 4.4,

$$ \begin{align*} \frac{1}{|B_{r}|}\int_{B_{r}}| T^*_{L^{\alpha}} f_{1}(x)|\,dx &\leq\bigg(\frac{1}{|B_{r}|}\int_{B_{r}} | T^*_{L^{\alpha}} f_{1}(x)|^{2}\,dx\bigg)^{1/2} \\ &\lesssim \bigg(\frac{1}{|B_{r}|}\int_{\mathbb{R}^{n}}|f_{1}(x)|^{2}\,dx\bigg)^{1/2}\lesssim\|f\|_{L^{\infty}(\mathbb{R}^{n})}. \end{align*} $$

We should note that, for any $j\in \mathbb {Z}$ , by (1-2),

(4-8) $$ \begin{align} &\int_{\mathbb{R}^{n}}|K^{L}_{\alpha, a_{j+1}}(x,y)-K^{L}_{\alpha, a_{j}}(x,y)|\,dy \notag \\\notag &\quad\leq \int_{\mathbb{R}^{n}}|K^{L}_{\alpha, a_{j+1}}(x,y)|\,dy+\int_{\mathbb{R}^{n}}|K^{L}_{\alpha, a_{j}}(x,y)|\,dy \\\notag &\quad\lesssim\int_{\mathbb{R}^{n}}\frac{a_{j+1}}{(a_{j+1}^{1/(2\alpha)}+|x-y|)^{n+2\alpha}}\,dy+ \int_{\mathbb{R}^{n}}\frac{a_{j}}{(a_{j}^{1/(2\alpha)}+|x-y|)^{n+2\alpha}}\,dy \\&\quad= C. \end{align} $$

For $1<p<\infty $ and any $N=(N_{1},N_{2})$ , by Hölder’s inequality, (4-8) and Fubini’s theorem, we can use Proposition 3.5 to obtain

$$ \begin{align*} &\bigg|\hspace{-1.4pt}\sum^{N_{2}}_{j=N_{1}}v_{j}(e^{-a_{j+1}L^{\alpha}}f_{2}(x)-e^{-a_{j}L^{\alpha}}f_{2}(x))\bigg| \\&\quad\lesssim \sum^{N_{2}}_{j=N_{1}}\bigg|v_{j}\int_{\mathbb{R}^{n}}(K^{L}_{\alpha, a_{j+1}}(x,y)-K^{L}_{\alpha, a_{j}}(x,y))f_{2}(y)\,dy\bigg| \\&\quad\lesssim \|v\|_{l^{p}(\mathbb{Z})}\bigg(\sum^{N_{2}}_{j=N_{1}}\bigg(\int_{\mathbb{R}^{n}} |K^{L}_{\alpha, a_{j+1}}(x,y)-K^{L}_{\alpha, a_{j}}(x,y)||f_{2}(y)|\,dy\bigg)^{p'}\bigg)^{1/p'} \\&\quad\lesssim \|v\|_{l^{p}(\mathbb{Z})}\bigg(\sum^{N_{2}}_{j=N_{1}}\bigg(\int_{\mathbb{R}^{n}} |K^{L}_{\alpha, a_{j+1}}(x,y)-K^{L}_{\alpha, a_{j}}(x,y)||f_{2}(y)|^{p'}\,dy\bigg) \\&\qquad\times \bigg(\int_{\mathbb{R}^{n}} |K^{L}_{\alpha, a_{j+1}}(x,y)-K^{L}_{\alpha, a_{j}}(x,y)|\,dy\bigg)^{p'/p}\bigg)^{1/p'} \\&\quad\lesssim \|v\|_{l^{p}(\mathbb{Z})}\bigg(\sum^{N_{2}}_{j=N_{1}}\int_{\mathbb{R}^{n}} |K^{L}_{\alpha, a_{j+1}}(x,y)-K^{L}_{\alpha, a_{j}}(x,y)||f_{2}(y)|\bigg)\,dy^{1/p'}\\&\quad\lesssim \|v\|_{l^{p}(\mathbb{Z})}\bigg(\int_{\mathbb{R}^{n}}\sum^{+\infty}_{j=-\infty}|K^{L}_{\alpha, a_{j+1}}(x,y)- K^{L}_{\alpha, a_{j}}(x,y)||f_{2}(y)|\,dy\bigg)^{1/p'} \\&\quad\lesssim\|v\|_{l^{p}(\mathbb{Z})}\bigg(\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n}}|f_{2}(y)|^{p'}\,dy\bigg)^{1/p'}. \end{align*} $$

For $y\in B\setminus B_{2r}$ and $x\in B_{r}$ , we have $r\leq |x-y|\leq 2$ . Then,

$$ \begin{align*} \frac{1}{|B_{r}|}\int_{B_{r}}| T^*_{L^{\alpha}} f_{2}(x)|\,dx &\lesssim\frac{1}{|B_{r}|}\int_{B_{r}}\bigg(\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n}}|f_{2}(y)|^{p'}\,dy\bigg)^{1/p'}\,dx \\ &\lesssim \frac{\|f\|_{L^{\infty}(\mathbb{R}^{n})}}{|B_{r}|}\int_{B_{r}}\bigg(\int_{r\leq |x-y|\leq 2}\frac{1}{|x-y|^{n}}\,dy\bigg)^{1/p'}\,dx \\ &\sim \bigg(\log\frac{2}{r}\bigg)^{1/p'} \|f\|_{L^{\infty}(\mathbb{R}^{n})}. \end{align*} $$

Hence,

$$ \begin{align*} \frac{1}{|B_{r}|}\int_{B_{r}}| T^*_{L^{\alpha}} f(x)|\,dx\ {\lesssim}\ \, \bigg(1+\bigg(\log\frac{2}{r}\bigg)^{1/p'}\bigg)\|f\|_{L^{\infty}(\mathbb{R}^{n})} \lesssim\bigg(\log\frac{2}{r}\bigg)^{1/p'}\|f\|_{L^{\infty}(\mathbb{R}^{n})}. \end{align*} $$

For (ii), we only need to consider the case $n=1$ as the multidimensional cases are similar. In what follows, we assume that $1<p<\infty $ and we choose $\epsilon $ such that $0<\epsilon <p-1$ . Let

$$ \begin{align*}f(x)=\sum^{0}_{k=-\infty}(-1)^{k}\chi_{(-a^{k},-a^{k-1}]}(x),\end{align*} $$

where $a>1$ is the constant defined in Lemma 4.8. We check that

$$ \begin{align*}f(a^{j}x)=(-1)^{j}f(x)+(-1)^{j}\sum^{-j}_{\mathrm{l}=1}(-1)^{\mathrm{l}}\chi_{(-a^{\mathrm{l}},-a^{\mathrm{l}-1}]}(x)\end{align*} $$

when $j\leq 0$ . We always assume that $j\leq 0$ in what follows.

Choose a lacunary sequence $\{a_{j}\}_{j\in \mathbb {Z}}$ with $a_{j}=(t_{0}a^{\,j})^{2\alpha }$ , $j\in \mathbb {Z}$ . Then, similarly to the proof of Proposition 4.9, we have

$$ \begin{align*} &e^{-a_{j+1}(-\Delta)^{\alpha}}f(x)-e^{-a_{j}(-\Delta)^{\alpha}}f(x) \\&\quad= C(-1)^{j+1}\bigg(\int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(x/a^{j+1}-z)\,dz+ \int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(x/a^{j}-z)\,dz \\&\qquad+ \int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\sum^{-j-1}_{\mathrm{l}=1}(-1)^{\mathrm{l}} \chi_{(-a^{\mathrm{l}},-a^{\mathrm{l}-1}]}\bigg(\frac{x}{a^{j+1}}-z\bigg)\,dz \\&\qquad+ \int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\sum^{-j}_{\mathrm{l}=1}(-1)^{\mathrm{l}} \chi_{(-a^{\mathrm{l}},-a^{\mathrm{l}-1}]}\bigg(\frac{x}{a^{j}}-z\bigg)\,dz\bigg). \end{align*} $$

Using Lemma 4.8, we obtain

$$ \begin{align*} & \int^{\infty}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(-z)\,dz \\ &\quad= \bigg\{\int^{1/a}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(-z)\,dz+\int^{1}_{1/a}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(-z)\,dz\bigg\}\\ &\quad\geq \int^{1}_{1/a}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz-\int^{1/a}_{0}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz\\ & \quad>{(1+\gamma)}/{4}-{\gamma}/{4}={1}/{4}. \end{align*} $$

Therefore, there exists $0<\eta _{0}<1$ small enough such that, for $|h|<\eta _{0}$ ,

(4-9) $$ \begin{align} \int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)f(h-z)\,dz>1/4. \end{align} $$

Moreover, using Lemma 4.8 again, we can choose the last $\eta _{0}$ small enough such that

(4-10) $$ \begin{align} \int^{\infty}_{1-\eta_{0}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz<{(1-\gamma)}/{4}. \end{align} $$

For the given $\eta _{0}$ in (4-10), let $2r<1$ be such that $r<\eta _{0}^{2}$ and $r\sim a^{\mathrm {J}_{0}}\eta _{0}$ for a certain negative integer $\mathrm {J}_{0}$ . If $\mathrm {J}_{0}\leq j\leq 0$ , we have $r/a^{j}<\eta _{0}$ . And, for any $-r\leq x\leq r$ , we get

$$ \begin{align*}-1\cdot\chi_{[1-\eta_{0},+\infty)}(z)\leq\sum^{-j-1}_{\mathrm{l}=1}(-1)^{\mathrm{l}}\chi_{(-a^{\mathrm{l}},-a^{\mathrm{l}-1}]} \bigg(\frac{x}{a^{j+1}}-z\bigg)\leq \chi_{[a-\eta_{0},+\infty)}(z)\end{align*} $$

and

$$ \begin{align*}-1\cdot\chi_{[1-\eta_{0},+\infty)}(u)\leq\sum^{-j}_{\mathrm{l}=1}(-1)^{\mathrm{l}}\chi_{(-a^{\mathrm{l}},-a^{\mathrm{l}-1}]} \bigg(\frac{x}{a^{j}}-z\bigg)\leq \chi_{[a-\eta_{0},+\infty)}(z).\end{align*} $$

Then, by (4-9),

$$ \begin{align*} & \int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\sum^{-j-1}_{\mathrm{l}=1}(-1)^{\mathrm{l}} \chi_{(-a^{\mathrm{l}},-a^{\mathrm{l}-1}]} \bigg(\frac{x}{a^{j+1}}-z\bigg)\,dz \\[5pt] &\qquad+ \int_{\mathbb{R}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\sum^{-j}_{\mathrm{l}=1}(-1)^{\mathrm{l}} \chi_{(-a^{\mathrm{l}},-a^{\mathrm{l}-1}]} \bigg(\frac{x}{a^{j}}-z\bigg)\,dz \\[5pt] &\quad\geq -2\int^{\infty}_{1-\eta_{0}}K_{\alpha,t_{0}^{2\alpha}}^{-\Delta}(z)\,dz\geq \frac{\gamma-1}{2}. \end{align*} $$

So, for any $t\in [-r,r]$ and $\mathrm {J}_{0}\leq j\leq 0$ , combining (4-9) with (4-10) yields

$$ \begin{align*} |e^{-a_{j+1}(-\Delta)^{\alpha}}f(x)-e^{-a_{j}(-\Delta)^{\alpha}}f(x)| \geq C\frac{\gamma}{2}>0. \end{align*} $$

Choose a sequence $\{v_{j}\}_{j\in \mathbb {Z}}\in l^{p}(\mathbb {Z})$ , where $v_{j}=(-1)^{j+1}(-j)^{-{1}/{(p-\epsilon )}}$ . Then, for $N=(\mathrm {J}_{0},0)$ ,

$$ \begin{align*} \frac{1}{2r}\int_{[-r,r]}|T^{\ast}_{(-\Delta)^{\alpha}}f(x)|\,dx &\geq \frac{1}{2r}\int_{[-r,r]}|T^{(-\Delta)^{\alpha}}_{N}f(x)|\,dx \\[5pt] &\geq\frac{C\gamma}{4r}\int_{[-r,r]}\sum^{-1}_{j=\mathrm{J}_{0}}((-j)^{-{1}{(p-\epsilon)}})\,dx \\[5pt] &\geq C\cdot(-\mathrm{J}_{0})^{{1}/{(p-\epsilon)^{'}}}\sim \bigg(\log\frac{2}{r}\bigg)^{{1}/{(p-\epsilon)^{'}}}. \end{align*} $$

For (iii), let $v_{j}=(-1)^{j+1}, a_{j}=(t_{0}a^{j})^{2}$ and $0<\eta _{0}<1$ as fixed in the proof of (ii). Consider the same function f as in (ii). Then $\|v\|_{l^{\infty }(\mathbb {Z})}=1$ and $\|f\|_{L^{\infty }(\mathbb {R})}=1$ . By the same arguments as in (ii), with $N=(\mathrm {J}_{0},0)$ ,

$$ \begin{align*} \frac{1}{2r}\int_{[-r,r]}|T^{\ast}_{(-\Delta)^{\alpha}}f(x)|\,dx &\geq \frac{1}{2r}\int_{[-r.r]}|T^{(-\Delta)^{\alpha}}_{N}f(x)|\,dx\\&\geq \frac{C}{2r}\int_{[-r,r]}\sum^{0}_{j=\mathrm{J}_{0}}C_{1}\,dx \\[-2pt]&\geq C\cdot C_{1}\cdot(-\mathrm{J}_{0})\sim \log\frac{2}{r}.\\[-38.5pt] \end{align*} $$