Morrey spaces related to certain nonnegative potentials and fractional integrals on the Heisenberg groups

Abstract

Let $\mathcal{L}=-\Delta _{\mathbb{H}^{n}}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb{H}^{n}$, where $\Delta _{\mathbb{H}^{n}}$ is the sub-Laplacian on $\mathbb{H}^{n}$ and the nonnegative potential V belongs to the reverse Hölder class $RH_{s}$ with $s\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^{n}$. For given $\alpha \in (0,Q)$, the fractional integrals associated to the Schrödinger operator $\mathcal{L}$ is defined by $\mathcal{I}_{\alpha }={\mathcal{L}}^{-{\alpha }/2}$. In this article, we first introduce the Morrey space $L^{p,\kappa }_{\rho , \infty }(\mathbb{H}^{n})$ and weak Morrey space $WL^{p,\kappa }_{ \rho ,\infty }(\mathbb{H}^{n})$ related to the nonnegative potential V. Then we establish the boundedness of fractional integrals ${\mathcal{L}}^{-{\alpha }/2}$ on these new spaces. Furthermore, in order to deal with certain extreme cases, we also introduce the spaces $\operatorname{BMO}_{\rho ,\infty }(\mathbb{H}^{n})$ and $\mathcal{C}^{ \beta }_{\rho ,\infty }(\mathbb{H}^{n})$ with exponent $\beta \in (0,1]$.

1 Introduction

1.1 Heisenberg group $\mathbb{H}^{n}$

The Heisenberg group $\mathbb{H}^{n}$ is a nilpotent Lie group with underlying manifold $\mathbb{C}^{n}\times \mathbb{R}$. The group structure (the multiplication law) is given by

$$ (z,t)\cdot \bigl(z',t'\bigr):= \bigl(z+z',t+t'+2\operatorname{Im}\bigl(z\cdot \overline{z'}\bigr) \bigr), $$

where $z=(z_{1},z_{2},\ldots ,z_{n})$, $z'=(z_{1}',z_{2}',\ldots ,z_{n}') \in \mathbb{C}^{n}$, and

$$ z\cdot \overline{z'}:=\sum_{j=1}^{n}z_{j} \overline{z_{j}'}. $$

It can be easily seen that the inverse element of $u=(z,t)$ is $u^{-1}=(-z,-t)$, and the identity is the origin $(0,0)$. The Lie algebra of left-invariant vector fields on $\mathbb{H}^{n}$ is spanned by

$$ \textstyle\begin{cases} X_{j}= \frac{\partial }{\partial x_{j}}+2y_{j}\frac{\partial }{\partial t}, \quad j=1,2,\ldots ,n, \\ Y_{j}= \frac{\partial }{\partial y_{j}}-2x_{j}\frac{\partial }{\partial t}, \quad j=1,2,\ldots ,n, \\ T= \frac{\partial }{\partial t}. \end{cases} $$

All non-trivial commutation relations are given by

$$ [X_{j},Y_{j}]=-4T, \quad j=1,2,\ldots ,n. $$

The sub-Laplacian $\Delta _{\mathbb{H}^{n}}$ is defined by

$$ \Delta _{\mathbb{H}^{n}}:=\sum_{j=1}^{n} \bigl(X_{j}^{2}+Y_{j}^{2} \bigr). $$

The dilations on $\mathbb{H}^{n}$ have the following form:

$$ \delta _{a}(z,t):=\bigl(az,a^{2}t\bigr),\quad a>0. $$

For given $(z,t)\in \mathbb{H}^{n}$, the homogeneous norm of $(z,t)$ is given by

$$ \bigl\vert (z,t) \bigr\vert := \bigl( \vert z \vert ^{4}+t^{2} \bigr)^{1/4}. $$

Observe that $|(z,t)^{-1}|=|(z,t)|$ and

$$ \bigl\vert \delta _{a}(z,t) \bigr\vert = \bigl( \vert az \vert ^{4}+\bigl(a^{2}t\bigr)^{2} \bigr)^{1/4}=a \bigl\vert (z,t) \bigr\vert . $$

In addition, this norm $|\cdot |$ satisfies the triangle inequality and leads to a left-invariant distance $d(u,v)=|u^{-1}\cdot v|$ for $u=(z,t), v=(z',t')\in \mathbb{H}^{n}$. The ball of radius r centered at u is denoted by

$$ B(u,r):= \bigl\{ v\in \mathbb{H}^{n}:d(u,v)< r \bigr\} . $$

The Haar measure on $\mathbb{H}^{n}$ coincides with the Lebesgue measure on $\mathbb{R}^{2n}\times \mathbb{R}$. The measure of any measurable set $E\subset \mathbb{H}^{n}$ is denoted by $|E|$. For $(u,r)\in \mathbb{H}^{n}\times (0,\infty )$, it can be shown that the volume of $B(u,r)$ is

$$ \bigl\vert B(u,r) \bigr\vert =r^{Q}\cdot \bigl\vert B(0,1) \bigr\vert , $$

where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H} ^{n}$ and $|B(0,1)|$ is the volume of the unit ball in $\mathbb{H} ^{n}$. A direct calculation shows that

$$ \bigl\vert B(0,1) \bigr\vert =\frac{2\pi ^{n+\frac{ 1 }{2}}\varGamma (\frac{ n }{2})}{(n+1) \varGamma (n)\varGamma (\frac{n+1}{2})}. $$

Given a ball $B=B(u,r)$ in $\mathbb{H}^{n}$ and $\lambda >0$, we shall use the notation λB to denote $B(u,\lambda r)$. Clearly, we have

$$ \bigl\vert B(u,\lambda r) \bigr\vert =\lambda ^{Q}\cdot \bigl\vert B(u,r) \bigr\vert . $$

(1.1)

For more information about the harmonic analysis on the Heisenberg groups, we refer the reader to [12, Chapter XII] and [13].

Let $V:\mathbb{H}^{n}\rightarrow \mathbb{R}$ be a nonnegative locally integrable function that belongs to the reverse Hölder class $RH_{s}$ for some exponent $1< s< \infty$; i.e., there exists a positive constant $C>0$ such that the reverse Hölder inequality

$$ \biggl(\frac{1}{ \vert B \vert } \int _{B} V(w)^{s} \,dw \biggr)^{1/s} \leq C \biggl(\frac{1}{ \vert B \vert } \int _{B} V(w) \,dw \biggr) $$

holds for every ball B in $\mathbb{H}^{n}$. For given $V\in RH_{s}$ with $s\geq Q/2$, we introduce the critical radius function $\rho (u)=\rho (u;V)$ which is given by

$$ \rho (u):=\sup \biggl\{ r>0:\frac{1}{r^{Q-2}} \int _{B(u,r)}V(w) \,dw \leq 1 \biggr\} ,\quad u\in \mathbb{H}^{n}, $$

(1.2)

where $B(u,r)$ denotes the ball in $\mathbb{H}^{n}$ centered at u and with radius r. It is well known that this auxiliary function satisfies $0<\rho (u)<\infty$ for any $u\in \mathbb{H}^{n}$ under the above assumption on V (see [9]). We need the following well-known result concerning the critical radius function (1.2).

Lemma 1.1

([9])

If $V\in RH_{s}$ with $s\geq Q/2$, then there exist constants $C_{0}\geq 1$ and $N_{0}>0$ such that, for all u and v in $\mathbb{H}^{n}$,

$$ \frac{ 1 }{C_{0}} \biggl(1+\frac{ \vert v^{-1}u \vert }{\rho (u)} \biggr)^{-N _{0}}\leq \frac{\rho (v)}{\rho (u)}\leq C_{0} \biggl(1+\frac{ \vert v^{-1}u \vert }{ \rho (u)} \biggr)^{\frac{N_{0}}{N_{0}+1}}. $$

(1.3)

Lemma 1.1 is due to Lu [9]. In the setting of $\mathbb{R}^{n}$, this result was given by Shen in [10]. As a straightforward consequence of (1.3), we can see that, for each integer $k\geq 1$, the estimate

$$ 1+\frac{2^{k}r}{\rho (v)}\geq \frac{1}{C_{0}} \biggl(1+ \frac{r}{\rho (u)} \biggr)^{-\frac{N_{0}}{N_{0}+1}} \biggl(1+\frac{2^{k}r}{ \rho (u)} \biggr) $$

(1.4)

holds for any $v\in B(u,r)$ with $u\in \mathbb{H}^{n}$ and $r>0$, $C_{0}$ is the same as in (1.3).

1.2 Fractional integrals

First we recall the fractional power of the Laplacian operator on $\mathbb{R}^{n}$. For given $\alpha \in (0,n)$, the classical fractional integral operator $I^{\Delta }_{\alpha }$ (also referred to as the Riesz potential) is defined by

$$ I^{\Delta }_{\alpha }(f):=(-\Delta )^{-\alpha /2}(f), $$

where Δ is the Laplacian operator on $\mathbb{R}^{n}$. If $f\in \mathcal{S}(\mathbb{R}^{n})$, then, by virtue of the Fourier transform, we have

$$ \widehat{I^{\Delta }_{\alpha }f}(\xi )=\bigl(2\pi \vert \xi \vert \bigr)^{-\alpha } \widehat{f}(\xi ), \quad \forall \xi \in \mathbb{R}^{n}. $$

Comparing this to the Fourier transform of $|x|^{-\alpha }$, $0<\alpha <n$, we are led to redefine the fractional integral operator $I^{\Delta }_{\alpha }$ by

$$ I^{\Delta }_{\alpha }f(x):=\frac{1}{\gamma (\alpha )} \int _{\mathbb{R} ^{n}}\frac{f(y)}{ \vert x-y \vert ^{n-\alpha }} \,dy, $$

(1.5)

where

$$ \gamma (\alpha )=\frac{\pi ^{\frac{n}{ 2 }}2^{\alpha }\varGamma (\frac{ \alpha }{ 2 })}{\varGamma (\frac{n-\alpha }{2})} $$

with $\varGamma (\cdot )$ being the usual gamma function. It is well known that the Hardy–Littlewood–Sobolev theorem states that the fractional integral operator $I^{\Delta }_{\alpha }$ is bounded from $L^{p}( \mathbb{R}^{n})$ to $L^{q}(\mathbb{R}^{n})$ for $0<\alpha <n$, $1< p< n/{\alpha }$ and $1/q=1/p-{\alpha }/n$. Also we know that $I^{\Delta }_{\alpha }$ is bounded from $L^{1}(\mathbb{R}^{n})$ to $WL^{q}(\mathbb{R}^{n})$ for $0<\alpha <n$ and $q=n/{(n-\alpha )}$ (see [11]).

Next we are going to discuss the fractional integrals on the Heisenberg group. For given $\alpha \in (0,Q)$ with $Q=2n+2$, the fractional integral operator $I_{\alpha }$ (also referred to as the Riesz potential) is defined by (see [14])

$$ I_{\alpha }(f):=(-\Delta _{\mathbb{H}^{n}})^{-\alpha /2}(f), $$

(1.6)

where $\Delta _{\mathbb{H}^{n}}$ is the sub-Laplacian on $\mathbb{H} ^{n}$ defined above. Let f and g be integrable functions defined on $\mathbb{H}^{n}$. Define the convolution $f*g$ by

$$ (f*g) (u):= \int _{\mathbb{H}^{n}}f(v)g\bigl(v^{-1}u\bigr) \,dv. $$

We denote by $H_{s}(u)$ the convolution kernel of heat semigroup $\{T_{s}=e^{s\Delta _{\mathbb{H}^{n}}}:s>0 \}$. Namely,

$$ e^{s\Delta _{\mathbb{H}^{n}}}f(u)= \int _{\mathbb{H}^{n}}H_{s}\bigl(v^{-1}u \bigr)f(v) \,dv. $$

For any $u=(z,t)\in \mathbb{H}^{n}$, it was proved in [14, Theorem 4.2] that $I_{\alpha }$ can be expressed by the following formula:

$$ \begin{aligned}[b] I_{\alpha }f(u) &= \frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty }e^{s \Delta _{\mathbb{H}^{n}}}f(u) s^{\alpha /2-1}\,ds \\ &=\frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty } (H_{s}*f ) (u) s^{\alpha /2-1}\,ds. \end{aligned} $$

(1.7)

Let $V\in RH_{s}$ for $s\geq Q/2$. For such a potential V, we consider the time independent Schrödinger operator on $\mathbb{H} ^{n}$ (see [8]),

$$ \mathcal{L}:=-\Delta _{\mathbb{H}^{n}}+V, $$

and its associated semigroup

$$ \mathcal{T}^{\mathcal{L}}_{s}f(u):=e^{-s\mathcal{L}}f(u)= \int _{\mathbb{H}^{n}}P_{s}(u,v)f(v) \,dv,\quad f\in L^{2}\bigl(\mathbb{H} ^{n}\bigr), s>0, $$

where $P_{s}(u,v)$ denotes the kernel of the operator $e^{-s \mathcal{L}},s>0$. For any $u=(z,t)\in \mathbb{H}^{n}$, it is well known that the heat kernel $H_{s}(u)$ has the explicit expression

$$ H_{s}(z,t)=(2\pi )^{-1}(4\pi )^{-n} \int _{\mathbb{R}} \biggl(\frac{ \vert \lambda \vert }{\sinh \vert \lambda \vert s} \biggr)^{n}\exp \biggl\{ - \frac{ \vert \lambda \vert \vert z \vert ^{2}}{4}\coth \vert \lambda \vert s-i\lambda t \biggr\} \,d\lambda , $$

and hence it satisfies the following estimate (see [5] for instance):

$$ 0\leq H_{s}(u)\leq C\cdot s^{-Q/2}\exp \biggl(-\frac{ \vert u \vert ^{2}}{As} \biggr), $$

(1.8)

where the constants $C,A>0$ are independent of s and $u\in \mathbb{H}^{n}$. Since $V\geq 0$, by the Trotter product formula and (1.8), one has

$$ 0\leq P_{s}(u,v)\leq H_{s} \bigl(v^{-1}u\bigr)\leq C\cdot s^{-Q/2}\exp \biggl(- \frac{ \vert v ^{-1}u \vert ^{2}}{As} \biggr),\quad s>0. $$

(1.9)

Moreover, this estimate (1.9) can be improved when V belongs to the reverse Hölder class $RH_{s}$ for some $s\geq Q/2$. The auxiliary function $\rho (u)$ arises naturally in this context.

Lemma 1.2

Let $V\in RH_{s}$ with $s\geq Q/2$, and let $\rho (u)$ be the auxiliary function determined by V. For every positive integer $N\geq 1$, there exists a positive constant $C_{N}>0$ such that, for all u and v in $\mathbb{H}^{n}$,

$$ 0\leq P_{s}(u,v)\leq C_{N}\cdot s^{-Q/2}\exp \biggl(- \frac{ \vert v^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{\sqrt{s }}{\rho (u)}+ \frac{\sqrt{s }}{\rho (v)} \biggr)^{-N},\quad s>0. $$

This estimate of $P_{s}(u,v)$ is better than (1.9), which was given by Lin and Liu in [8, Lemma 7].

Inspired by (1.6) and (1.7), for given $\alpha \in (0,Q)$, the $\mathcal{L}$-fractional integral operator or $\mathcal{L}$-Riesz potential on the Heisenberg group is defined by (see [6] and [7])

$$ \begin{aligned} \mathcal{I}_{\alpha }(f) (u) &:={ \mathcal{L}}^{-{\alpha }/2}f(u) \\ &=\frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty }e^{-s\mathcal{L}}f(u) s^{\alpha /2-1}\,ds. \end{aligned} $$

Recall that in the setting of $\mathbb{R}^{n}$, this integral operator was first introduced by Dziubański et al. [3]. In this article we shall be interested in the behavior of the fractional integral operator $\mathcal{I}_{\alpha }$ associated to Schrödinger operator on $\mathbb{H}^{n}$. For $1\leq p<\infty$, the Lebesgue space $L^{p}(\mathbb{H}^{n})$ is defined to be the set of all measurable functions f on $\mathbb{H}^{n}$ such that

$$ \Vert f \Vert _{L^{p}(\mathbb{H}^{n})}:= \biggl( \int _{\mathbb{H}^{n}} \bigl\vert f(u) \bigr\vert ^{p} \,du \biggr)^{1/p}< \infty . $$

The weak Lebesgue space $WL^{p}(\mathbb{H}^{n})$ consists of all measurable functions f on $\mathbb{H}^{n}$ such that

$$ \Vert f \Vert _{WL^{p}(\mathbb{H}^{n})}:= \sup_{\lambda >0}\lambda \cdot \bigl\vert \bigl\{ u\in \mathbb{H}^{n}: \bigl\vert f(u) \bigr\vert >\lambda \bigr\} \bigr\vert ^{1/p}< \infty . $$

Now we are going to establish strong-type and weak-type estimates of the $\mathcal{L}$-fractional integral operator $\mathcal{I}_{\alpha }$ on the Lebesgue spaces. We first claim that the estimate

$$ \bigl\vert \mathcal{I}_{\alpha }f(u) \bigr\vert \leq C \int _{\mathbb{H}^{n}} \bigl\vert f(v) \bigr\vert \frac{1}{ \vert v ^{-1}u \vert ^{Q-\alpha }} \,dv=C \bigl( \vert f \vert * \vert \cdot \vert ^{\alpha -Q} \bigr) (u) $$

(1.10)

holds for all $u\in \mathbb{H}^{n}$. Let us verify (1.10). To do so, denote by $\mathcal{K}_{\alpha }(u,v)$ the kernel of the fractional integral operator $\mathcal{I}_{\alpha }$. Then we have

$$ \begin{aligned} \int _{\mathbb{H}^{n}}\mathcal{K}_{\alpha }(u,v)f(v) \,dv &= \mathcal{I} _{\alpha }f(u)={\mathcal{L}}^{-{\alpha }/2}f(u) \\ &=\frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty }e^{-s\mathcal{L}}f(u) s^{\alpha /2-1}\,ds \\ &= \int _{0}^{\infty } \biggl[\frac{1}{\varGamma (\alpha /2)} \int _{\mathbb{H}^{n}}P_{s}(u,v)f(v) \,dv \biggr]s^{\alpha /2-1}\,ds \\ &= \int _{\mathbb{H}^{n}} \biggl[\frac{1}{\varGamma (\alpha /2)} \int _{0} ^{\infty }P_{s}(u,v) s^{\alpha /2-1}\,ds \biggr]f(v) \,dv. \end{aligned} $$

Hence,

$$ \mathcal{K}_{\alpha }(u,v)=\frac{1}{\varGamma (\alpha /2)} \int _{0}^{ \infty }P_{s}(u,v) s^{\alpha /2-1}\,ds. $$

Moreover, by using (1.9), we can deduce that

$$ \begin{aligned} \bigl\vert \mathcal{K}_{\alpha }(u,v) \bigr\vert &\leq \frac{C}{\varGamma (\alpha /2)} \int _{0}^{\infty }\exp \biggl(- \frac{ \vert v ^{-1}u \vert ^{2}}{As} \biggr)s^{\alpha /2-Q/2-1}\,ds \\ &\leq \frac{C}{\varGamma (\alpha /2)}\cdot \frac{1}{ \vert v^{-1}u \vert ^{Q-\alpha }} \int _{0}^{\infty }e^{-t} t^{(Q/2-\alpha /2)-1}dt \\ &=C\cdot \frac{\varGamma (Q/2-\alpha /2)}{\varGamma (\alpha /2)}\cdot \frac{1}{ \vert v ^{-1}u \vert ^{Q-\alpha }}, \end{aligned} $$

where in the second step we have used a change of variables. Thus (1.10) holds. According to Theorems 4.4 and 4.5 in [14], we get the Hardy–Littlewood–Sobolev theorem on the Heisenberg group.

Theorem 1.3

Let $0<\alpha <Q$ and $1\leq p< Q/{\alpha }$. Define $1< q<\infty$ by the relation $1/q=1/p-{\alpha }/Q$. Then the following statements are valid:

• (1) if $p>1$, then $\mathcal{I}_{\alpha }$ is bounded from $L^{p}( \mathbb{H}^{n})$ to $L^{q}(\mathbb{H}^{n})$;

• (2) if $p=1$, then $\mathcal{I}_{\alpha }$ is bounded from $L^{1}( \mathbb{H}^{n})$ to $WL^{q}(\mathbb{H}^{n})$.

The organization of this paper is as follows. In Sect. 2, we will give the definitions of Morrey space and weak Morrey space and state our main results: Theorems 2.3, 2.4, and 2.5. Section 3 is devoted to proving the boundedness of the fractional integral operator in the context of Morrey spaces. We will study certain extreme cases in Sect. 4. Throughout this paper, C represents a positive constant that is independent of the main parameters, but may be different from line to line, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript. We also use $a\approx b$ to denote the equivalence of a and b; that is, there exist two positive constants $C_{1}$, $C_{2}$ independent of a, b such that $C_{1}a\leq b\leq C_{2}a$.

2 Main results

In this section, we introduce some types of Morrey spaces related to the nonnegative potential V on $\mathbb{H}^{n}$, and then give our main results.

Definition 2.1

Let ρ be the auxiliary function determined by $V\in RH_{s}$ with $s\geq Q/2$. Let $1\leq p<\infty$ and $0\leq \kappa <1$. For given $0<\theta <\infty$, the Morrey space $L^{p,\kappa }_{\rho ,\theta }( \mathbb{H}^{n})$ is defined to be the set of all p-locally integrable functions f on $\mathbb{H}^{n}$ such that

$$ \biggl(\frac{1}{ \vert B \vert ^{\kappa }} \int _{B} \bigl\vert f(u) \bigr\vert ^{p} \,du \biggr)^{1/p} \leq C\cdot \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{\theta } $$

(2.1)

for every ball $B=B(u_{0},r)$ in $\mathbb{H}^{n}$. A norm for $f\in L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})$, denoted by $\|f\|_{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})}$, is given by the infimum of the constants in (2.1), or equivalently,

$$ \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})}:= \sup_{B(u_{0},r)} \biggl(1+ \frac{r}{\rho (u_{0})} \biggr)^{-\theta } \biggl(\frac{1}{ \vert B \vert ^{\kappa }} \int _{B} \bigl\vert f(u) \bigr\vert ^{p} \,du \biggr)^{1/p} < \infty , $$

where the supremum is taken over all balls $B=B(u_{0},r)$ in $\mathbb{H}^{n}$, $u_{0}$ and r denote the center and radius of B, respectively. Define

$$ L^{p,\kappa }_{\rho ,\infty }\bigl(\mathbb{H}^{n}\bigr):= \bigcup_{\theta >0}L ^{p,\kappa }_{\rho ,\theta } \bigl(\mathbb{H}^{n}\bigr). $$

For any given $f\in L^{p,\kappa }_{\rho ,\infty }(\mathbb{H}^{n})$, let

$$ \theta ^{*}:=\inf \bigl\{ \theta >0:f\in L^{p,\kappa }_{\rho ,\theta } \bigl( \mathbb{H}^{n}\bigr) \bigr\} . $$

Now define

$$ \Vert f \Vert _{\star }= \Vert f \Vert _{L^{p,\kappa }_{\rho ,\infty }( \mathbb{H}^{n})}:= \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta ^{*}}( \mathbb{H}^{n})}. $$

It is easy to check that $\|\cdot \|_{\star }$ satisfies the axioms of a norm; i.e., that for $f,g\in L^{p,\kappa }_{\rho ,\infty }( \mathbb{H}^{n})$ and $\lambda \in \mathbb{R}$ we have

• $\|f\|_{\star }\geq 0$;

• $\|f\|_{\star }=0\Leftrightarrow f=0$;

• $\|\lambda f\|_{\star }=|\lambda |\|f\|_{\star }$;

• $\|f+g\|_{\star }\leq \|f\|_{\star }+\|g\|_{\star }$.

Definition 2.2

Let ρ be the auxiliary function determined by $V\in RH_{s}$ with $s\geq Q/2$. Let $1\leq p<\infty$ and $0\leq \kappa <1$. For given $0<\theta <\infty$, the weak Morrey space $WL^{p,\kappa }_{\rho , \theta }(\mathbb{H}^{n})$ is defined to be the set of all measurable functions f on $\mathbb{H}^{n}$ such that

$$ \frac{1}{ \vert B \vert ^{\kappa /p}}\sup_{\lambda >0}\lambda \cdot \bigl\vert \bigl\{ u \in B: \bigl\vert f(u) \bigr\vert >\lambda \bigr\} \bigr\vert ^{1/p} \leq C\cdot \biggl(1+\frac{r}{ \rho (u_{0})} \biggr)^{\theta } $$

for every ball $B=B(u_{0},r)$ in $\mathbb{H}^{n}$, or equivalently,

$$ \Vert f \Vert _{WL^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})}:= \sup_{B(u_{0},r)} \biggl(1+ \frac{r}{\rho (u_{0})} \biggr)^{-\theta }\frac{1}{ \vert B \vert ^{ \kappa /p}} \sup _{\lambda >0}\lambda \cdot \bigl\vert \bigl\{ u\in B: \bigl\vert f(u) \bigr\vert > \lambda \bigr\} \bigr\vert ^{1/p}< \infty . $$

Correspondingly, we define

$$ WL^{p,\kappa }_{\rho ,\infty }\bigl(\mathbb{H}^{n}\bigr):= \bigcup_{\theta >0}WL ^{p,\kappa }_{\rho ,\theta } \bigl(\mathbb{H}^{n}\bigr). $$

For any given $f\in WL^{p,\kappa }_{\rho ,\infty }(\mathbb{H}^{n})$, let

$$ \theta ^{**}:=\inf \bigl\{ \theta >0:f\in WL^{p,\kappa }_{\rho ,\theta } \bigl(\mathbb{H}^{n}\bigr) \bigr\} . $$

Similarly, we define

$$ \Vert f \Vert _{\star \star }= \Vert f \Vert _{WL^{p,\kappa }_{\rho ,\infty }( \mathbb{H}^{n})}:= \Vert f \Vert _{WL^{p,\kappa }_{\rho ,\theta ^{**}}( \mathbb{H}^{n})}. $$

By definition, we can easily show that $\|\cdot \|_{\star \star }$ satisfies the axioms of a (quasi)norm, and $WL^{p,\kappa }_{\rho , \infty }(\mathbb{H}^{n})$ is a (quasi)normed linear space. Obviously, if we take $\theta =0$ or $V\equiv 0$, then this Morrey space (or weak Morrey space) is just the Morrey space $L^{p,\kappa }(\mathbb{H}^{n})$ (or $WL^{p,\kappa }(\mathbb{H}^{n})$), which was defined by Guliyev et al. [4]. Moreover, according to the above definitions, one has

$$ \textstyle\begin{cases} L^{p,\kappa }(\mathbb{H}^{n})\subset L^{p,\kappa }_{\rho ,\theta _{1}}( \mathbb{H}^{n})\subset L^{p,\kappa }_{\rho ,\theta _{2}}(\mathbb{H} ^{n}); \\ WL^{p,\kappa }(\mathbb{H}^{n})\subset WL^{p,\kappa }_{\rho ,\theta _{1}}(\mathbb{H}^{n})\subset WL^{p,\kappa }_{\rho ,\theta _{2}}( \mathbb{H}^{n}), \end{cases} $$

for $0<\theta _{1}<\theta _{2}<\infty$. Hence $L^{p,\kappa }( \mathbb{H}^{n})\subset L^{p,\kappa }_{\rho ,\infty }(\mathbb{H}^{n})$ and $WL^{p,\kappa }(\mathbb{H}^{n})\subset WL^{p,\kappa }_{\rho , \infty }(\mathbb{H}^{n})$ for $(p,\kappa )\in [1,\infty )\times [0,1)$. The space $L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})$ (or $WL^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})$) could be viewed as an extension of Lebesgue (or weak Lebesgue) space on $\mathbb{H}^{n}$ (when $\kappa =\theta =0$). In this article we will extend the Hardy–Littlewood–Sobolev theorem on $\mathbb{H}^{n}$ to the Morrey spaces. We now present our main results.

Theorem 2.3

Let $0<\alpha <Q$, $1< p< Q/{\alpha }$ and $1/q=1/p-{\alpha }/Q$. If $V\in RH_{s}$ with $s\geq Q/2$ and $0<\kappa <p/q$, then the $\mathcal{L}$-fractional integral operator $\mathcal{I}_{\alpha }$ is bounded from $L^{p,\kappa }_{\rho ,\infty }(\mathbb{H}^{n})$ into $L^{q,{(\kappa q)}/p}_{\rho ,\infty }(\mathbb{H}^{n})$.

Theorem 2.4

Let $0<\alpha <Q$, $p=1$ and $q=Q/{(Q-\alpha )}$. If $V\in RH_{s}$ with $s\geq Q/2$ and $0<\kappa <1/q$, then the $\mathcal{L}$-fractional integral operator $\mathcal{I}_{\alpha }$ is bounded from $L^{1, \kappa }_{\rho ,\infty }(\mathbb{H}^{n})$ into $WL^{q,(\kappa q)}_{ \rho ,\infty }(\mathbb{H}^{n})$.

Before stating our next theorem, we need to introduce a new space $\operatorname{BMO}_{\rho ,\infty }(\mathbb{H}^{n})$ defined by

$$ \operatorname{BMO}_{\rho ,\infty }\bigl(\mathbb{H}^{n}\bigr):= \bigcup_{\theta >0} \operatorname{BMO}_{\rho ,\theta } \bigl(\mathbb{H}^{n}\bigr), $$

where for $0<\theta <\infty$ the space $\operatorname{BMO}_{\rho ,\theta }( \mathbb{H}^{n})$ is defined to be the set of all locally integrable functions f satisfying

$$ \frac{1}{ \vert B(u_{0},r) \vert } \int _{B(u_{0},r)} \bigl\vert f(u)-f_{B(u_{0},r)} \bigr\vert \,du \leq C\cdot \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{\theta }, $$

(2.2)

for all $u_{0}\in \mathbb{H}^{n}$ and $r>0$, $f_{B(u_{0},r)}$ denotes the mean value of f on $B(u_{0},r)$, that is,

$$ f_{B(u_{0},r)}:=\frac{1}{ \vert B(u_{0},r) \vert } \int _{B(u_{0},r)}f(v) \,dv. $$

A norm for $f\in \operatorname{BMO}_{\rho ,\theta }(\mathbb{H}^{n})$, denoted by $\|f\|_{\operatorname{BMO}_{\rho ,\theta }}$, is given by the infimum of the constants satisfying (2.2), or equivalently,

$$ \Vert f \Vert _{\operatorname{BMO}_{\rho ,\theta }} :=\sup_{B(u_{0},r)} \biggl(1+ \frac{r}{ \rho (u_{0})} \biggr)^{-\theta } \biggl(\frac{1}{ \vert B(u_{0},r) \vert } \int _{B(u_{0},r)} \bigl\vert f(u)-f_{B(u_{0},r)} \bigr\vert \,du \biggr), $$

where the supremum is taken over all balls $B(u_{0},r)$ with $u_{0}\in \mathbb{H}^{n}$ and $r>0$. Recall that in the setting of $\mathbb{R}^{n}$, the space $\operatorname{BMO}_{\rho ,\theta }(\mathbb{R} ^{n})$ was first introduced by Bongioanni et al. [2] (see also [1]).

Moreover, given any $\beta \in [0,1]$, we introduce the space of Hölder continuous functions on $\mathbb{H}^{n}$, with exponent β.

$$ \mathcal{C}^{\beta }_{\rho ,\infty }\bigl(\mathbb{H}^{n} \bigr):=\bigcup_{\theta >0}\mathcal{C}^{\beta }_{\rho ,\theta } \bigl(\mathbb{H}^{n}\bigr), $$

where for $0<\theta <\infty$ the space $\mathcal{C}^{\beta }_{\rho , \theta }(\mathbb{H}^{n})$ is defined to be the set of all locally integrable functions f satisfying

$$ \frac{1}{ \vert B(u_{0},r) \vert ^{1+\beta /Q}} \int _{B(u_{0},r)} \bigl\vert f(u)-f_{B(u _{0},r)} \bigr\vert \,du \leq C\cdot \biggl(1+\frac{r}{\rho (u_{0})} \biggr) ^{\theta }, $$

(2.3)

for all $u_{0}\in \mathbb{H}^{n}$ and $r\in (0,\infty )$. The smallest bound C for which (2.3) is satisfied is then taken to be the norm of f in this space and is denoted by $\|f\|_{\mathcal{C}^{ \beta }_{\rho ,\theta }}$. When $\theta =0$ or $V\equiv 0$, $\operatorname{BMO}_{\rho ,\theta }(\mathbb{H}^{n})$ and $\mathcal{C}^{ \beta }_{\rho ,\theta }(\mathbb{H}^{n})$ will be simply written as $\operatorname{BMO}(\mathbb{H}^{n})$ and $\mathcal{C}^{\beta }(\mathbb{H} ^{n})$, respectively. Note that when $\beta =0$ this space $\mathcal{C}^{\beta }_{\rho ,\theta }(\mathbb{H}^{n})$ reduces to the space $\operatorname{BMO}_{\rho ,\theta }(\mathbb{H}^{n})$ mentioned above.

For the case $\kappa \geq p/q$ of Theorem 2.3, we will prove the following result.

Theorem 2.5

Let $0<\alpha <Q$, $1< p< Q/{\alpha }$ and $1/q=1/p-{\alpha }/Q$. If $V\in RH_{s}$ with $s\geq Q/2$ and $p/q\leq \kappa <1$, then the $\mathcal{L}$-fractional integral operator $\mathcal{I}_{\alpha }$ is bounded from $L^{p,\kappa }_{\rho ,\infty }(\mathbb{H}^{n})$ into $\mathcal{C}^{\beta }_{\rho ,\infty }(\mathbb{H}^{n})$ with $\beta /Q=\kappa /p-1/q$ and β sufficiently small. To be more precise, $\beta <\delta \leq 1$ and δ is given as in Lemma 4.2.

In particular, for the limiting case $\kappa =p/q$ (or $\beta =0$), we obtain the following result on BMO-type estimate of $\mathcal{I}_{ \alpha }$.

Corollary 2.6

Let $0<\alpha <Q$, $1< p< Q/{\alpha }$ and $1/q=1/p-{\alpha }/Q$. If $V\in RH_{s}$ with $s\geq Q/2$ and $\kappa =p/q$, then the $\mathcal{L}$-fractional integral operator $\mathcal{I}_{\alpha }$ is bounded from $L^{p,\kappa }_{\rho ,\infty }(\mathbb{H}^{n})$ into $\operatorname{BMO}_{\rho ,\infty }(\mathbb{H}^{n})$.

3 Proofs of Theorems 2.3 and 2.4

In this section, we will prove the conclusions of Theorems 2.3 and 2.4. Let us recall that the $\mathcal{L}$-fractional integral operator of order $\alpha \in (0,Q)$ can be written as

$$ \mathcal{I}_{\alpha }f(u)={\mathcal{L}}^{-{\alpha }/2}f(u)= \int _{\mathbb{H}^{n}}\mathcal{K}_{\alpha }(u,v)f(v) \,dv, $$

where

$$ \mathcal{K}_{\alpha }(u,v)=\frac{1}{\varGamma (\alpha /2)} \int _{0}^{ \infty }P_{s}(u,v) s^{\alpha /2-1}\,ds. $$

(3.1)

The following lemma gives the estimate of the kernel $\mathcal{K}_{ \alpha }(u,v)$ related to the Schrödinger operator $\mathcal{L}$, which plays a key role in the proof of our main theorems.

Lemma 3.1

Let $V\in RH_{s}$ with $s\geq Q/2$ and $0<\alpha <Q$. For every positive integer $N\geq 1$, there exists a positive constant $C_{N,\alpha }>0$ such that, for all u and v in $\mathbb{H}^{n}$,

$$ \bigl\vert \mathcal{K}_{\alpha }(u,v) \bigr\vert \leq C_{N,\alpha } \biggl(1+\frac{ \vert v ^{-1}u \vert }{\rho (u)} \biggr)^{-N} \frac{1}{ \vert v^{-1}u \vert ^{Q-\alpha }}. $$

(3.2)

Proof

By Lemma 1.2 and (3.1), we have

$$ \begin{aligned} \bigl\vert \mathcal{K}_{\alpha }(u,v) \bigr\vert &\leq \frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty } \bigl\vert P_{s}(u,v) \bigr\vert s ^{\alpha /2-1}\,ds \\ &\leq \frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty }\frac{C_{N}}{s ^{Q/2}}\cdot \exp \biggl(-\frac{ \vert v^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{\sqrt{s }}{\rho (u)}+ \frac{\sqrt{s }}{\rho (v)} \biggr)^{-N}s^{\alpha /2-1}\,ds \\ &\leq \frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty }\frac{C_{N}}{s ^{Q/2}}\cdot \exp \biggl(-\frac{ \vert v^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{\sqrt{s }}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds. \end{aligned} $$

We consider two cases $s>|v^{-1}u|^{2}$ and $0\leq s\leq |v^{-1}u|^{2}$, respectively. Thus, $|\mathcal{K}_{\alpha }(u,v)|\leq I+\mathit{II}$, where

$$ I=\frac{1}{\varGamma (\alpha /2)} \int _{ \vert v^{-1}u \vert ^{2}}^{\infty }\frac{C _{N}}{s^{Q/2}}\cdot \exp \biggl(-\frac{ \vert v^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{\sqrt{s }}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds $$

and

$$ \mathit{II}=\frac{1}{\varGamma (\alpha /2)} \int _{0}^{ \vert v^{-1}u \vert ^{2}}\frac{C_{N}}{s ^{Q/2}}\cdot \exp \biggl(-\frac{ \vert v^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{\sqrt{s }}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds. $$

When $s>|v^{-1}u|^{2}$, then $\sqrt{s }>|v^{-1}u|$, and hence

$$ \begin{aligned} I &\leq \frac{1}{\varGamma (\alpha /2)} \int _{ \vert v^{-1}u \vert ^{2}}^{\infty }\frac{C _{N}}{s^{Q/2}}\cdot \exp \biggl(-\frac{ \vert v^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{ \vert v ^{-1}u \vert }{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds \\ &\leq C_{N,\alpha } \biggl(1+\frac{ \vert v^{-1}u \vert }{\rho (u)} \biggr)^{-N} \int _{ \vert v^{-1}u \vert ^{2}}^{\infty }s^{\alpha /2-Q/2-1}\,ds \\ &\leq C_{N,\alpha } \biggl(1+\frac{ \vert v^{-1}u \vert }{\rho (u)} \biggr)^{-N} \frac{1}{ \vert v ^{-1}u \vert ^{Q-\alpha }}, \end{aligned} $$

where the last integral converges because $0<\alpha <Q$. On the other hand,

$$ \begin{aligned} \mathit{II} &\leq C_{N,\alpha } \int _{0}^{ \vert v^{-1}u \vert ^{2}}\frac{1}{s^{Q/2}}\cdot \biggl(\frac{ \vert v^{-1}u \vert ^{2}}{s} \biggr)^{-(Q/2+N/2)} \biggl(1+ \frac{\sqrt{s }}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds \\ &=C_{N,\alpha } \int _{0}^{ \vert v^{-1}u \vert ^{2}}\frac{1}{ \vert v^{-1}u \vert ^{Q}}\cdot \biggl(\frac{\sqrt{s }}{ \vert v^{-1}u \vert } \biggr)^{N} \biggl(1+ \frac{\sqrt{s }}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds. \end{aligned} $$

It is easy to see that, when $0\leq s\leq |v^{-1}u|^{2}$,

$$ \frac{\sqrt{s }}{ \vert v^{-1}u \vert }\leq \frac{\sqrt{s }+\rho (u)}{ \vert v ^{-1}u \vert +\rho (u)}. $$

Hence,

$$ \begin{aligned} \mathit{II} &\leq C_{N,\alpha } \int _{0}^{ \vert v^{-1}u \vert ^{2}}\frac{1}{ \vert v^{-1}u \vert ^{Q}} \cdot \biggl(\frac{\sqrt{s }+\rho (u)}{ \vert v^{-1}u \vert +\rho (u)} \biggr)^{N} \biggl(\frac{\sqrt{s }+\rho (u)}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds \\ &=\frac{C_{N,\alpha }}{ \vert v^{-1}u \vert ^{Q}} \biggl(1+ \frac{ \vert v^{-1}u \vert }{\rho (u)} \biggr)^{-N} \int _{0}^{ \vert v^{-1}u \vert ^{2}}s^{ \alpha /2-1}\,ds \\ &=C_{N,\alpha } \biggl(1+\frac{ \vert v^{-1}u \vert }{\rho (u)} \biggr)^{-N} \frac{1}{ \vert v ^{-1}u \vert ^{Q-\alpha }}. \end{aligned} $$

Combining the estimates of I and II yields the desired estimate (3.2) for $\alpha \in (0,Q)$. This concludes the proof of the lemma. □

We are now ready to show our main theorems.

Proof of Theorem 2.3

By definition, we only need to show that, for any given ball $B=B(u_{0},r)$ of $\mathbb{H}^{n}$, there is some $\vartheta >0$ such that

$$ \biggl(\frac{1}{ \vert B \vert ^{\kappa q/p}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f(u) \bigr\vert ^{q} \,du \biggr)^{1/q}\leq C\cdot \biggl(1+ \frac{r}{\rho (u_{0})} \biggr) ^{\vartheta } $$

(3.3)

holds for given $f\in L^{p,\kappa }_{\rho ,\infty }(\mathbb{H}^{n})$ with $(p,\kappa )\in (1,Q/{\alpha })\times (0,p/q)$. Suppose that $f\in L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})$ for some $\theta >0$. We decompose the function f as

$$ \textstyle\begin{cases} f=f_{1}+f_{2}\in L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n}); \\ f_{1}=f\cdot \chi _{2B}; \\ f_{2}=f\cdot \chi _{(2B)^{c}}, \end{cases} $$

where 2B is the ball centered at $u_{0}$ of radius $2r>0$, $\chi _{2B}$ is the characteristic function of 2B and $(2B)^{c}= \mathbb{H}^{n}\backslash (2B)$. Then, by the linearity of $\mathcal{I}_{\alpha }$, we write

$$ \begin{aligned} \biggl(\frac{1}{ \vert B \vert ^{\kappa q/p}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f(u) \bigr\vert ^{q} \,du \biggr)^{1/q} &\leq \biggl( \frac{1}{ \vert B \vert ^{\kappa q/p}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f_{1}(u) \bigr\vert ^{q} \,du \biggr)^{1/q} \\ &\quad {}+ \biggl(\frac{1}{ \vert B \vert ^{\kappa q/p}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f _{2}(u) \bigr\vert ^{q} \,du \biggr)^{1/q} \\ &:=I_{1}+I_{2}. \end{aligned} $$

In what follows, we consider each part separately. By Theorem 1.3 (1), we have

$$ \begin{aligned} I_{1} &= \biggl(\frac{1}{ \vert B \vert ^{\kappa q/p}} \int _{B} \bigl\vert \mathcal{I}_{ \alpha }f_{1}(u) \bigr\vert ^{q} \,du \biggr)^{1/q} \\ &\leq C\cdot \frac{1}{ \vert B \vert ^{\kappa /p}} \biggl( \int _{\mathbb{H}^{n}} \bigl\vert f_{1}(u) \bigr\vert ^{p} \,du \biggr)^{1/p} \\ &=C\cdot \frac{1}{ \vert B \vert ^{\kappa /p}} \biggl( \int _{2B} \bigl\vert f(u) \bigr\vert ^{p} \,du \biggr)^{1/p} \\ &\leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \cdot \frac{ \vert 2B \vert ^{\kappa /p}}{ \vert B \vert ^{\kappa /p}}\cdot \biggl(1+\frac{2r}{ \rho (u_{0})} \biggr)^{\theta }. \end{aligned} $$

Also observe that, for any fixed $\theta >0$,

$$ 1\leq \biggl(1+\frac{2r}{\rho (u_{0})} \biggr)^{\theta } \leq 2^{ \theta } \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{\theta }. $$

(3.4)

This in turn implies that

$$ \begin{aligned} I_{1} &\leq C_{\theta ,n} \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{\theta }. \end{aligned} $$

Next we estimate the other term $I_{2}$. Notice that, for any $u\in B(u_{0},r)$ and $v\in (2B)^{c}$, one has

$$ \bigl\vert v^{-1}u \bigr\vert = \bigl\vert \bigl(v^{-1}u_{0}\bigr)\cdot \bigl(u_{0}^{-1}u \bigr) \bigr\vert \leq \bigl\vert v^{-1}u_{0} \bigr\vert + \bigl\vert u_{0}^{-1}u \bigr\vert $$

and

$$ \bigl\vert v^{-1}u \bigr\vert = \bigl\vert \bigl(v^{-1}u_{0}\bigr)\cdot \bigl(u_{0}^{-1}u \bigr) \bigr\vert \geq \bigl\vert v^{-1}u_{0} \bigr\vert - \bigl\vert u_{0}^{-1}u \bigr\vert . $$

Thus,

$$ \frac{1}{ 2 } \bigl\vert v^{-1}u_{0} \bigr\vert \leq \bigl\vert v^{-1}u \bigr\vert \leq \frac{3}{ 2 } \bigl\vert v^{-1}u_{0} \bigr\vert , $$

i.e., $|v^{-1}u|\approx |v^{-1}u_{0}|$. It then follows from Lemma 3.1 that, for any $u\in B(u_{0},r)$ and any positive integer N,

$$ \begin{aligned}[b] \bigl\vert \mathcal{I}_{\alpha }f_{2}(u) \bigr\vert &\leq \int _{(2B)^{c}} \bigl\vert \mathcal{K}_{\alpha }(u,v) \bigr\vert \cdot \bigl\vert f(v) \bigr\vert \,dv \\ &\leq C_{N,\alpha } \int _{(2B)^{c}} \biggl(1+\frac{ \vert v^{-1}u \vert }{\rho (u)} \biggr)^{-N}\frac{1}{ \vert v^{-1}u \vert ^{Q-\alpha }}\cdot \bigl\vert f(v) \bigr\vert \,dv \\ &\leq C_{N,\alpha ,n} \int _{(2B)^{c}} \biggl(1+\frac{ \vert v^{-1}u_{0} \vert }{ \rho (u)} \biggr)^{-N}\frac{1}{ \vert v^{-1}u_{0} \vert ^{Q-\alpha }}\cdot \bigl\vert f(v) \bigr\vert \,dv \\ &=C_{N,\alpha ,n}\sum_{k=1}^{\infty } \int _{2^{k}r\leq \vert v^{-1}u_{0} \vert < 2^{k+1}r} \biggl(1+\frac{ \vert v^{-1}u_{0} \vert }{ \rho (u)} \biggr)^{-N} \frac{1}{ \vert v^{-1}u_{0} \vert ^{Q-\alpha }}\cdot \bigl\vert f(v) \bigr\vert \,dv \\ &\leq C_{N,\alpha ,n}\sum_{k=1}^{\infty } \frac{1}{ \vert B(u_{0},2^{k+1}r) \vert ^{1-( \alpha /Q)}} \int _{ \vert v^{-1}u_{0} \vert < 2^{k+1}r} \biggl(1+\frac{2^{k}r}{ \rho (u)} \biggr)^{-N} \bigl\vert f(v) \bigr\vert \,dv. \end{aligned} $$

(3.5)

In view of (1.4) and (3.4), we can further obtain

$$\begin{aligned} \bigl\vert \mathcal{I}_{\alpha }f_{2}(u) \bigr\vert &\leq C\sum_{k=1}^{\infty } \frac{1}{ \vert B(u _{0},2^{k+1}r) \vert ^{1-(\alpha /Q)}} \\ &\quad {}\times \int _{ \vert v^{-1}u_{0} \vert < 2^{k+1}r} \biggl(1+\frac{r}{\rho (u_{0})} \biggr) ^{N\cdot \frac{N_{0}}{N_{0}+1}} \biggl(1+\frac{2^{k}r}{\rho (u_{0})} \biggr) ^{-N} \bigl\vert f(v) \bigr\vert \,dv \\ &\leq C\sum_{k=1}^{\infty } \frac{1}{ \vert B(u_{0},2^{k+1}r) \vert ^{1-(\alpha /Q)}} \\ &\quad {}\times \int _{B(u_{0},2^{k+1}r)} \biggl(1+\frac{r}{\rho (u_{0})} \biggr) ^{N\cdot \frac{N_{0}}{N_{0}+1}} \biggl(1+ \frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{-N} \bigl\vert f(v) \bigr\vert \,dv. \end{aligned}$$

(3.6)

We consider each term in the sum of (3.6) separately. By using Hölder’s inequality, we find that, for each integer $k\geq 1$,

$$ \begin{aligned} &\frac{1}{ \vert B(u_{0},2^{k+1}r) \vert ^{1-(\alpha /Q)}} \int _{B(u_{0},2^{k+1}r)} \bigl\vert f(v) \bigr\vert \,dv \\ &\quad \leq \frac{1}{ \vert B(u_{0},2^{k+1}r) \vert ^{1-(\alpha /Q)}} \biggl( \int _{B(u_{0},2^{k+1}r)} \bigl\vert f(v) \bigr\vert ^{p} \,dv \biggr)^{1/p} \biggl( \int _{B(u_{0},2^{k+1}r)}1 \,dv \biggr)^{1/{p'}} \\ &\quad \leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \cdot \frac{ \vert B(u_{0},2^{k+1}r) \vert ^{{\kappa }/p}}{ \vert B(u_{0},2^{k+1}r) \vert ^{1/q}} \biggl(1+\frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{\theta }. \end{aligned} $$

This allows us to obtain

$$ \begin{aligned} I_{2} &\leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H} ^{n})}\cdot \frac{ \vert B(u_{0},r) \vert ^{1/q}}{ \vert B(u_{0},r) \vert ^{{\kappa }/p}} \sum_{k=1}^{\infty } \frac{ \vert B(u_{0},2^{k+1}r) \vert ^{{\kappa }/p}}{ \vert B(u_{0},2^{k+1}r) \vert ^{1/q}} \\ &\quad {}\times\biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N_{0}+1}} \biggl(1+\frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{-N+\theta } \\ &=C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N_{0}+1}} \sum_{k=1}^{\infty } \frac{ \vert B(u_{0},r) \vert ^{1/q-\kappa /p}}{ \vert B(u_{0},2^{k+1}r) \vert ^{1/q- \kappa /p}} \biggl(1+\frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{-N+\theta }. \end{aligned} $$

Thus, by choosing N large enough so that $N>\theta$, and the last series is convergent, then we have

$$ \begin{aligned} I_{2} &\leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H} ^{n})} \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N _{0}+1}}\sum _{k=1}^{\infty } \biggl(\frac{ \vert B(u_{0},r) \vert }{ \vert B(u_{0},2^{k+1}r) \vert } \biggr) ^{{(1/q-\kappa /p)}} \\ &\leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N_{0}+1}}, \end{aligned} $$

where the last inequality follows from the fact that $1/q-\kappa /p>0$. Summing up the above estimates for $I_{1}$ and $I_{2}$ and letting $\vartheta =\max \{\theta ,N\cdot \frac{N_{0}}{N_{0}+1} \}$, we obtain the desired inequality (3.3). This completes the proof of Theorem 2.3. □

Proof of Theorem 2.4

To prove Theorem 2.4, by definition, it suffices to prove that, for each given ball $B=B(u_{0},r)$ of $\mathbb{H}^{n}$, there is some $\vartheta >0$ such that

$$ \frac{1}{ \vert B \vert ^{\kappa }}\sup_{\lambda >0}\lambda \cdot \bigl\vert \bigl\{ u \in B: \bigl\vert \mathcal{I}_{\alpha }f(u) \bigr\vert >\lambda \bigr\} \bigr\vert ^{1/q} \leq C \cdot \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{\vartheta } $$

(3.7)

holds for given $f\in L^{1,\kappa }_{\rho ,\infty }(\mathbb{H}^{n})$ with $0<\kappa <1/q$ and $q=Q/{(Q-\alpha )}$. Now suppose that $f\in L^{1,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})$ for some $\theta >0$. We decompose the function f as

$$ \textstyle\begin{cases} f=f_{1}+f_{2}\in L^{1,\kappa }_{\rho ,\theta }(\mathbb{H}^{n}); \\ f_{1}=f\cdot \chi _{2B}; \\ f_{2}=f\cdot \chi _{(2B)^{c}}. \end{cases} $$

Then, for any given $\lambda >0$, by the linearity of $\mathcal{I} _{\alpha }$, we can write

$$ \begin{aligned} &\frac{1}{ \vert B \vert ^{\kappa }}\lambda \cdot \bigl\vert \bigl\{ u\in B: \bigl\vert \mathcal{I} _{\alpha }f(u) \bigr\vert >\lambda \bigr\} \bigr\vert ^{1/q} \\ &\quad \leq \frac{1}{ \vert B \vert ^{\kappa }}\lambda \cdot \bigl\vert \bigl\{ u\in B: \bigl\vert \mathcal{I}_{\alpha }f_{1}(u) \bigr\vert >\lambda /2 \bigr\} \bigr\vert ^{1/q} \\ &\qquad {}+\frac{1}{ \vert B \vert ^{\kappa }}\lambda \cdot \bigl\vert \bigl\{ u\in B: \bigl\vert \mathcal{I}_{\alpha }f_{2}(u) \bigr\vert >\lambda /2 \bigr\} \bigr\vert ^{1/q} \\ &\quad :=J_{1}+J_{2}. \end{aligned} $$

We first give the estimate for the term $J_{1}$. By Theorem 1.3 (2), we get

$$ \begin{aligned} J_{1} &=\frac{1}{ \vert B \vert ^{\kappa }}\lambda \cdot \bigl\vert \bigl\{ u\in B: \bigl\vert \mathcal{I}_{\alpha } f_{1}(u) \bigr\vert >\lambda /2 \bigr\} \bigr\vert ^{1/q} \\ &\leq C\cdot \frac{1}{ \vert B \vert ^{\kappa }} \biggl( \int _{\mathbb{H}^{n}} \bigl\vert f_{1}(u) \bigr\vert \,du \biggr) \\ &=C\cdot \frac{1}{ \vert B \vert ^{\kappa }} \biggl( \int _{2B} \bigl\vert f(u) \bigr\vert \,du \biggr) \\ &\leq C \Vert f \Vert _{L^{1,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \cdot \frac{ \vert 2B \vert ^{\kappa }}{ \vert B \vert ^{\kappa }} \biggl(1+\frac{2r}{\rho (u _{0})} \biggr)^{\theta }. \end{aligned} $$

Therefore, in view of (3.4),

$$ J_{1}\leq C \Vert f \Vert _{L^{1,\kappa }_{\rho ,\theta }(\mathbb{H} ^{n})}\cdot \biggl(1+ \frac{r}{\rho (u_{0})} \biggr)^{\theta }. $$

As for the second term $J_{2}$, by using the pointwise inequality (3.6) and Chebyshev’s inequality, we can deduce that

$$ \begin{aligned}[b] J_{2} &= \frac{1}{ \vert B \vert ^{\kappa }}\lambda \cdot \bigl\vert \bigl\{ u\in B: \bigl\vert \mathcal{I}_{\alpha }f_{2}(u) \bigr\vert >\lambda /2 \bigr\} \bigr\vert ^{1/q} \\ &\leq \frac{2}{ \vert B \vert ^{\kappa }} \biggl( \int _{B} \bigl\vert \mathcal{I}_{\alpha }f_{2}(u) \bigr\vert ^{q} \,du \biggr)^{1/q} \\ &\leq C\cdot \frac{ \vert B \vert ^{1/q}}{ \vert B \vert ^{\kappa }} \sum_{k=1}^{\infty } \frac{1}{ \vert B(u _{0},2^{k+1}r) \vert ^{1-(\alpha /Q)}} \\ &\quad {}\times \int _{B(u_{0},2^{k+1}r)} \biggl(1+\frac{r}{\rho (u_{0})} \biggr) ^{N\cdot \frac{N_{0}}{N_{0}+1}} \biggl(1+ \frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{-N} \bigl\vert f(v) \bigr\vert \,dv. \end{aligned} $$

(3.8)

We consider each term in the sum of (3.8) separately. For each integer $k\geq 1$, we compute

$$ \begin{aligned} &\frac{1}{ \vert B(u_{0},2^{k+1}r) \vert ^{1-(\alpha /Q)}} \int _{B(u_{0},2^{k+1}r)} \bigl\vert f(v) \bigr\vert \,dv \\ &\quad \leq C \Vert f \Vert _{L^{1,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \cdot \frac{ \vert B(u_{0},2^{k+1}r) \vert ^{\kappa }}{ \vert B(u_{0},2^{k+1}r) \vert ^{1/q}} \biggl(1+\frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{\theta }. \end{aligned} $$

Consequently,

$$ \begin{aligned} J_{2} &\leq C \Vert f \Vert _{L^{1,\kappa }_{\rho ,\theta }(\mathbb{H} ^{n})} \cdot \frac{ \vert B(u_{0},r) \vert ^{1/q}}{ \vert B(u_{0},r) \vert ^{\kappa }}\sum_{k=1}^{\infty } \frac{ \vert B(u_{0},2^{k+1}r) \vert ^{\kappa }}{ \vert B(u_{0},2^{k+1}r) \vert ^{1/q}} \\ &\quad {}\times\biggl(1+\frac{r}{ \rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N_{0}+1}} \biggl(1+\frac{2^{k+1}r}{ \rho (u_{0})} \biggr)^{-N+\theta } \\ &=C \Vert f \Vert _{L^{1,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N_{0}+1}} \sum_{k=1}^{\infty } \frac{ \vert B(u_{0},r) \vert ^{{1/q-\kappa }}}{ \vert B(u_{0},2^{k+1}r) \vert ^{ {1/q-\kappa }}} \biggl(1+\frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{-N+ \theta }. \end{aligned} $$

Therefore, by selecting N large enough so that $N>\theta$, we thus have

$$ \begin{aligned} J_{2} &\leq C \Vert f \Vert _{L^{1,\kappa }_{\rho ,\theta }(\mathbb{H} ^{n})} \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N _{0}+1}} \sum _{k=1}^{\infty } \biggl(\frac{ \vert B(u_{0},r) \vert }{ \vert B(u_{0},2^{k+1}r) \vert } \biggr) ^{{(1/q-\kappa )}} \\ &\leq C \Vert f \Vert _{L^{1,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N_{0}+1}}, \end{aligned} $$

where the last step is due to the fact that $0<\kappa <1/q$. Let $\vartheta =\max \{\theta ,N\cdot \frac{N_{0}}{N_{0}+1} \}$. Here N is an appropriate constant. Summing up the above estimates for $J_{1}$ and $J_{2}$, and then taking the supremum over all $\lambda >0$, we obtain the desired inequality (3.7). This finishes the proof of Theorem 2.4. □

4 Proof of Theorem 2.5

We need the following lemma which establishes the Lipschitz regularity of the kernel $P_{s}(u,v)$. See Lemma 11 and Remark 4 in [8].

Lemma 4.1

([8])

Let $V\in RH_{s}$ with $s\geq Q/2$. For every positive integer $N\geq 1$, there exists a positive constant $C_{N}>0$ such that, for all u and v in $\mathbb{H}^{n}$, and for some fixed $0<\delta \leq 1$,

$$ \bigl\vert P_{s}(u\cdot h,v)-P_{s}(u,v) \bigr\vert \leq C_{N} \biggl(\frac{ \vert h \vert }{\sqrt{s }} \biggr)^{\delta } s^{-Q/2}\exp \biggl(-\frac{ \vert v^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+ \frac{\sqrt{s }}{\rho (u)}+\frac{\sqrt{s }}{ \rho (v)} \biggr)^{-N}, $$

whenever $|h|\leq |v^{-1}u|/2$.

Based on the above lemma, we are able to prove the following result, which plays a key role in the proof of our main theorem.

Lemma 4.2

Let $V\in RH_{s}$ with $s\geq Q/2$ and $0<\alpha <Q$. For every positive integer $N\geq 1$, there exists a positive constant $C_{N,\alpha }>0$ such that, for all u, v and w in $\mathbb{H}^{n}$, and for some fixed $0<\delta \leq 1$,

$$ \bigl\vert \mathcal{K}_{\alpha }(u,w)- \mathcal{K}_{\alpha }(v,w) \bigr\vert \leq C _{N,\alpha } \biggl(1+ \frac{ \vert w^{-1}u \vert }{\rho (u)} \biggr)^{-N}\frac{ \vert v^{-1}u \vert ^{ \delta }}{ \vert w^{-1}u \vert ^{Q-\alpha +\delta }}, $$

(4.1)

whenever $|v^{-1}u|\leq |w^{-1}u|/2$.

Proof

In view of Lemma 4.1 and (3.1), we have

$$ \begin{aligned} & \bigl\vert \mathcal{K}_{\alpha }(u,w)- \mathcal{K}_{\alpha }(v,w) \bigr\vert \\ &\quad =\frac{1}{\varGamma (\alpha /2)} \biggl\vert \int _{0}^{\infty }P_{s}(u,w) s ^{\alpha /2-1}\,ds- \int _{0}^{\infty }P_{s}(v,w) s^{\alpha /2-1}\,ds \biggr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty } \bigl\vert P_{s} \bigl(u \cdot \bigl(u^{-1}v\bigr),w\bigr)-P_{s}(u,w) \bigr\vert s^{\alpha /2-1}\,ds \\ &\quad \leq \frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty }C_{N}\cdot \biggl( \frac{ \vert u^{-1}v \vert }{\sqrt{s }} \biggr)^{\delta } s^{-Q/2} \\ &\qquad {}\times\exp \biggl(- \frac{ \vert w^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{\sqrt{s }}{ \rho (u)}+\frac{\sqrt{s }}{\rho (w)} \biggr)^{-N}s^{\alpha /2-1}\,ds \\ &\quad \leq \frac{1}{\varGamma (\alpha /2)} \int _{0}^{\infty }C_{N}\cdot \biggl( \frac{ \vert u^{-1}v \vert }{\sqrt{s }} \biggr)^{\delta } s^{-Q/2}\exp \biggl(- \frac{ \vert w^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{\sqrt{s }}{ \rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds. \end{aligned} $$

Arguing as in the proof of Lemma 3.1, consider two cases as below: $s>|w^{-1}u|^{2}$ and $0\leq s\leq |w^{-1}u|^{2}$. Then the right-hand side of the above expression can be written as $\mathit{III}+\mathit{IV}$, where

$$ \mathit{III}=\frac{1}{\varGamma (\alpha /2)} \int _{ \vert w^{-1}u \vert ^{2}}^{\infty }\frac{C _{N}}{s^{Q/2}}\cdot \biggl(\frac{ \vert u^{-1}v \vert }{\sqrt{s }} \biggr)^{ \delta }\exp \biggl(- \frac{ \vert w^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{\sqrt{s }}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds $$

and

$$ \mathit{IV}=\frac{1}{\varGamma (\alpha /2)} \int _{0}^{ \vert w^{-1}u \vert ^{2}}\frac{C_{N}}{s ^{Q/2}}\cdot \biggl(\frac{ \vert u^{-1}v \vert }{\sqrt{s }} \biggr)^{\delta } \exp \biggl(- \frac{ \vert w^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{\sqrt{s }}{ \rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds. $$

When $s>|w^{-1}u|^{2}$, then $\sqrt{s }>|w^{-1}u|$, and hence

$$ \begin{aligned} \mathit{III} &\leq \frac{1}{\varGamma (\alpha /2)} \int _{ \vert w^{-1}u \vert ^{2}}^{\infty }\frac{C _{N}}{s^{Q/2}}\cdot \biggl(\frac{ \vert u^{-1}v \vert }{ \vert w^{-1}u \vert } \biggr)^{\delta } \exp \biggl(- \frac{ \vert w^{-1}u \vert ^{2}}{As} \biggr) \biggl(1+\frac{ \vert w^{-1}u \vert }{ \rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds \\ &\leq C_{N,\alpha } \biggl(1+\frac{ \vert w^{-1}u \vert }{\rho (u)} \biggr)^{-N} \biggl(\frac{ \vert u^{-1}v \vert }{ \vert w^{-1}u \vert } \biggr)^{\delta } \int _{ \vert w^{-1}u \vert ^{2}} ^{\infty }s^{\alpha /2-Q/2-1}\,ds \\ &=C_{N,\alpha } \biggl(1+\frac{ \vert w^{-1}u \vert }{\rho (u)} \biggr)^{-N} \frac{ \vert v ^{-1}u \vert ^{\delta }}{ \vert w^{-1}u \vert ^{Q-\alpha +\delta }}, \end{aligned} $$

where the last equality holds since $|u^{-1}v|=|v^{-1}u|$ and $0<\alpha <Q$. On the other hand,

$$ \begin{aligned} \mathit{IV} &\leq C_{N,\alpha } \int _{0}^{ \vert w^{-1}u \vert ^{2}}\frac{1}{s^{Q/2}}\cdot \biggl(\frac{ \vert u^{-1}v \vert }{\sqrt{s }} \biggr)^{\delta } \biggl(\frac{ \vert w ^{-1}u \vert ^{2}}{s} \biggr)^{-(Q/2+N/2+\delta /2)} \biggl(1+\frac{ \sqrt{s }}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds \\ &=C_{N,\alpha } \int _{0}^{ \vert w^{-1}u \vert ^{2}}\frac{ \vert u^{-1}v \vert ^{\delta }}{ \vert w ^{-1}u \vert ^{Q+\delta }} \biggl( \frac{\sqrt{s }}{ \vert w^{-1}u \vert } \biggr)^{N} \biggl(1+\frac{\sqrt{s }}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds. \end{aligned} $$

It is easy to check that when $0\leq s\leq |w^{-1}u|^{2}$,

$$ \frac{\sqrt{s }}{ \vert w^{-1}u \vert }\leq \frac{\sqrt{s }+\rho (u)}{ \vert w ^{-1}u \vert +\rho (u)}. $$

This in turn implies that

$$ \begin{aligned} \mathit{IV} &\leq C_{N,\alpha } \int _{0}^{ \vert w^{-1}u \vert ^{2}}\frac{ \vert u^{-1}v \vert ^{\delta }}{ \vert w^{-1}u \vert ^{Q+\delta }} \biggl( \frac{\sqrt{s }+\rho (u)}{ \vert w^{-1}u \vert + \rho (u)} \biggr)^{N} \biggl(\frac{\sqrt{s }+\rho (u)}{\rho (u)} \biggr)^{-N}s^{\alpha /2-1}\,ds \\ &=C_{N,\alpha }\cdot \frac{ \vert u^{-1}v \vert ^{\delta }}{ \vert w^{-1}u \vert ^{Q+\delta }} \biggl(1+\frac{ \vert w^{-1}u \vert }{\rho (u)} \biggr)^{-N} \int _{0}^{ \vert w^{-1}u \vert ^{2}}s ^{\alpha /2-1}\,ds \\ &=C_{N,\alpha } \biggl(1+\frac{ \vert w^{-1}u \vert }{\rho (u)} \biggr)^{-N} \frac{ \vert v ^{-1}u \vert ^{\delta }}{ \vert w^{-1}u \vert ^{Q-\alpha +\delta }}, \end{aligned} $$

where the last step holds because $|u^{-1}v|=|v^{-1}u|$. Combining the estimates of III and IV produces the desired inequality (4.1) for $\alpha \in (0,Q)$. This concludes the proof of the lemma. □

We are now in a position to give the proof of Theorem 2.5.

Proof of Theorem 2.5

Fix a ball $B=B(u_{0},r)$ with $u_{0}\in \mathbb{H}^{n}$ and $r\in (0,\infty )$, it suffices to prove that the inequality

$$ \frac{1}{ \vert B \vert ^{1+\beta /Q}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f(u)-( \mathcal{I}_{\alpha }f)_{B} \bigr\vert \,du\leq C\cdot \biggl(1+\frac{r}{ \rho (u_{0})} \biggr)^{\vartheta } $$

(4.2)

holds for given $f\in L^{p,\kappa }_{\rho ,\infty }(\mathbb{H}^{n})$ with $1< p< q<\infty$ and $p/q\leq \kappa <1$, where $0<\alpha <Q$ and $(\mathcal{I}_{\alpha }f)_{B}$ denotes the average of $\mathcal{I} _{\alpha }f$ over B. Suppose that $f\in L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})$ for some $\theta >0$. Decompose the function f as $f=f_{1}+f_{2}$, where $f_{1}=f\cdot \chi _{4B}$, $f_{2}=f\cdot \chi _{(4B)^{c}}$, $4B=B(u_{0},4r)$ and $(4B)^{c}=\mathbb{H}^{n}\backslash (4B)$. By the linearity of the $\mathcal{L}$-fractional integral operator $\mathcal{I}_{\alpha }$, the left-hand side of (4.2) can be written as

$$ \begin{aligned} &\frac{1}{ \vert B \vert ^{1+\beta /Q}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f(u)-( \mathcal{I}_{\alpha }f)_{B} \bigr\vert \,du \\ &\quad \leq \frac{1}{ \vert B \vert ^{1+\beta /Q}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f _{1}(u)-(\mathcal{I}_{\alpha }f_{1})_{B} \bigr\vert \,du +\frac{1}{ \vert B \vert ^{1+ \beta /Q}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f_{2}(u)-( \mathcal{I}_{ \alpha }f_{2})_{B} \bigr\vert \,du \\ &\quad :=K_{1}+K_{2}. \end{aligned} $$

Let us consider the first term $K_{1}$. Applying the strong-type $(p,q)$ estimate of $\mathcal{I}_{\alpha }$ (see Theorem 1.3) and Hölder’s inequality, we obtain

$$ \begin{aligned} K_{1} &\leq \frac{2}{ \vert B \vert ^{1+\beta /Q}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f _{1}(u) \bigr\vert \,du \\ &\leq \frac{2}{ \vert B \vert ^{1+\beta /Q}} \biggl( \int _{B} \bigl\vert \mathcal{I}_{\alpha }f _{1}(u) \bigr\vert ^{q} \,du \biggr)^{1/q} \biggl( \int _{B}1 \,du \biggr)^{1/{q'}} \\ &\leq \frac{C}{ \vert B \vert ^{1+\beta /Q}} \biggl( \int _{4B} \bigl\vert f(u) \bigr\vert ^{p} \,du \biggr)^{1/p} \vert B \vert ^{1/ {q'}} \\ &\leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \cdot \frac{ \vert B(u_{0},4r) \vert ^{{\kappa }/p}}{ \vert B(u_{0},r) \vert ^{1/q+\beta /Q}} \biggl(1+\frac{4r}{\rho (u_{0})} \biggr)^{\theta }. \end{aligned} $$

Using the inequalities (1.1) and (3.4), and noting the fact that $\beta /Q=\kappa /p-1/q$, we derive

$$ \begin{aligned} K_{1} &\leq C_{n} \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }( \mathbb{H}^{n})} \biggl(1+\frac{4r}{\rho (u_{0})} \biggr)^{\theta } \\ &\leq C_{n,\theta } \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }( \mathbb{H}^{n})} \biggl(1+ \frac{r}{\rho (u_{0})} \biggr)^{\theta }. \end{aligned} $$

Let us now turn to the estimate of the second term $K_{2}$. For any $u\in B(u_{0},r)$,

$$ \begin{aligned} \bigl\vert \mathcal{I}_{\alpha }f_{2}(u)-( \mathcal{I}_{\alpha }f_{2})_{B} \bigr\vert &= \biggl\vert \frac{1}{ \vert B \vert } \int _{B} \bigl[\mathcal{I}_{\alpha }f_{2}(u)- \mathcal{I}_{\alpha }f_{2}(v) \bigr] \,dv \biggr\vert \\ &= \biggl\vert \frac{1}{ \vert B \vert } \int _{B} \biggl\{ \int _{(4B)^{c}} \bigl[\mathcal{K} _{\alpha }(u,w)- \mathcal{K}_{\alpha }(v,w) \bigr]f(w) \,dw \biggr\} \,dv \biggr\vert \\ &\leq \frac{1}{ \vert B \vert } \int _{B} \biggl\{ \int _{(4B)^{c}} \bigl\vert \mathcal{K} _{\alpha }(u,w)- \mathcal{K}_{\alpha }(v,w) \bigr\vert \cdot \bigl\vert f(w) \bigr\vert \,dw \biggr\} \,dv. \end{aligned} $$

By using the same arguments as for Theorem 2.3, we find that

$$ \bigl\vert v^{-1}u \bigr\vert \leq \bigl\vert w^{-1}u \bigr\vert /2 \quad \mbox{and}\quad \bigl\vert w^{-1}u \bigr\vert \approx \bigl\vert w ^{-1}u_{0} \bigr\vert , $$

whenever $u,v\in B$ and $w\in (4B)^{c}$. This fact along with Lemma 4.2 yields

$$\begin{aligned} & \bigl\vert \mathcal{I}_{\alpha }f_{2}(u)-( \mathcal{I}_{\alpha }f_{2})_{B} \bigr\vert \\ &\quad \leq \frac{C_{N,\alpha }}{ \vert B \vert } \int _{B} \biggl\{ \int _{(4B)^{c}} \biggl(1+\frac{ \vert w ^{-1}u \vert }{\rho (u)} \biggr)^{-N} \frac{ \vert v^{-1}u \vert ^{\delta }}{ \vert w^{-1}u \vert ^{Q- \alpha +\delta }}\cdot \bigl\vert f(w) \bigr\vert \,dw \biggr\} \,dv \\ &\quad \leq C_{N,\alpha ,n} \int _{(4B)^{c}} \biggl(1+\frac{ \vert w^{-1}u_{0} \vert }{ \rho (u)} \biggr)^{-N}\frac{r^{\delta }}{ \vert w^{-1}u_{0} \vert ^{Q-\alpha + \delta }}\cdot \bigl\vert f(w) \bigr\vert \,dw \\ &\quad =C_{N,\alpha ,n}\sum_{k=2}^{\infty } \int _{2^{k}r\leq \vert w^{-1}u_{0} \vert < 2^{k+1}r} \biggl(1+\frac{ \vert w^{-1}u_{0} \vert }{ \rho (u)} \biggr)^{-N}\frac{r^{\delta }}{ \vert w^{-1}u_{0} \vert ^{Q-\alpha + \delta }}\cdot \bigl\vert f(w) \bigr\vert \,dw \\ &\quad \leq C_{N,\alpha ,n}\sum_{k=2}^{\infty } \frac{1}{2^{k\delta }} \cdot \frac{1}{ \vert B(u_{0},2^{k+1}r) \vert ^{1-({\alpha }/Q)}} \int _{B(u_{0},2^{k+1}r)} \biggl(1+\frac{2^{k}r}{\rho (u)} \biggr)^{-N} \bigl\vert f(w) \bigr\vert \,dw. \end{aligned}$$

(4.3)

Furthermore, by using Hölder’s inequality and (1.4), we deduce that, for any $u\in B(u_{0},r)$,

$$\begin{aligned} & \bigl\vert \mathcal{I}_{\alpha }f_{2}(u)-( \mathcal{I}_{\alpha }f_{2})_{B} \bigr\vert \\ &\quad \leq C\sum_{k=2}^{\infty } \frac{1}{2^{k\delta }}\cdot \frac{1}{ \vert B(u _{0},2^{k+1}r) \vert ^{1-({\alpha }/Q)}} \biggl(1+\frac{r}{\rho (u_{0})} \biggr) ^{N\cdot \frac{N_{0}}{N_{0}+1}} \biggl(1+ \frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{-N} \\ &\qquad {}\times \biggl( \int _{B(u_{0},2^{k+1}r)} \bigl\vert f(w) \bigr\vert ^{p} \,dw \biggr)^{1/p} \biggl( \int _{B(u_{0},2^{k+1}r)}1 \,dw \biggr)^{1/{p'}} \\ &\quad \leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \sum_{k=2}^{\infty } \frac{1}{2^{k\delta }}\cdot \biggl(1+\frac{r}{ \rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N_{0}+1}} \biggl(1+\frac{2^{k+1}r}{ \rho (u_{0})} \biggr)^{-N} \\ &\qquad {}\times \frac{ \vert B(u_{0},2^{k+1}r) \vert ^{{\kappa }/p}}{ \vert B(u_{0},2^{k+1}r) \vert ^{1/q}} \biggl(1+\frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{\theta } \\ &\quad =C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \sum_{k=2}^{\infty } \frac{ \vert B(u_{0},2^{k+1}r) \vert ^{\beta /Q}}{2^{k\delta }}\cdot \biggl(1+\frac{r}{\rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N _{0}+1}} \biggl(1+\frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{-N+\theta }, \end{aligned}$$

(4.4)

where the last equality is due to the assumption $\beta /Q=\kappa /p-1/q$. From the pointwise estimate (4.4) and (1.1), it readily follows that

$$ \begin{aligned} K_{2} &=\frac{1}{ \vert B \vert ^{1+\beta /Q}} \int _{B} \bigl\vert \mathcal{I}_{\alpha }f _{2}(u)-(\mathcal{I}_{\alpha }f_{2})_{B} \bigr\vert \,du \\ &\leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \sum_{k=2}^{\infty } \frac{1}{2^{k\delta }}\cdot \biggl(\frac{ \vert B(u_{0},2^{k+1}r) \vert }{ \vert B(u _{0},r) \vert } \biggr)^{\beta /Q} \biggl(1+\frac{r}{\rho (u_{0})} \biggr) ^{N\cdot \frac{N_{0}}{N_{0}+1}} \biggl(1+ \frac{2^{k+1}r}{\rho (u_{0})} \biggr)^{-N+\theta } \\ &\leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H}^{n})} \sum_{k=2}^{\infty } \frac{1}{2^{k(\delta -\beta )}}\cdot \biggl(1+\frac{r}{ \rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N_{0}+1}}, \end{aligned} $$

where $N>0$ is a sufficiently large number so that $N>\theta$. Also observe that $\beta <\delta \leq 1$, and hence the last series is convergent. Therefore,

$$ K_{2}\leq C \Vert f \Vert _{L^{p,\kappa }_{\rho ,\theta }(\mathbb{H} ^{n})} \biggl(1+ \frac{r}{\rho (u_{0})} \biggr)^{N\cdot \frac{N_{0}}{N _{0}+1}}. $$

Fix this N and set $\vartheta =\max \{\theta ,N\cdot \frac{N _{0}}{N_{0}+1} \}$. Finally, combining the above estimates for $K_{1}$ and $K_{2}$, the inequality (4.2) is proved and then the proof of Theorem 2.5 is finished. □

In the end of this article, we discuss the corresponding estimates of the fractional integral operator $I_{\alpha }=(-\Delta _{\mathbb{H} ^{n}})^{-\alpha /2}$ (under $0<\alpha <Q$). We denote by $K^{*}_{ \alpha }(u,v)$ the kernel of $I_{\alpha }=(-\Delta _{\mathbb{H}^{n}})^{- \alpha /2}$. In (1.10), we have already shown that

$$ \bigl\vert K^{*}_{\alpha }(u,v) \bigr\vert \leq C_{\alpha ,n}\cdot \frac{1}{ \vert v^{-1}u \vert ^{Q- \alpha }}. $$

(4.5)

Using the same methods and steps as we deal with (4.1) in Lemma 4.2, we can also show that, for some fixed $0<\delta \leq 1$ and $0<\alpha <Q$, there exists a positive constant $C_{\alpha ,n}>0$ such that, for all u, v and w in $\mathbb{H}^{n}$,

$$ \bigl\vert K^{*}_{\alpha }(u,w)-K^{*}_{\alpha }(v,w) \bigr\vert \leq C_{\alpha ,n} \cdot \frac{ \vert v^{-1}u \vert ^{\delta }}{ \vert w^{-1}u \vert ^{Q-\alpha +\delta }}, $$

(4.6)

whenever $|v^{-1}u|\leq |w^{-1}u|/2$. Following along the lines of the proof of Theorems 2.3–2.5 and using the inequalities (4.5) and (4.6), we can obtain the following estimates of $I_{\alpha }$ with $\alpha \in (0,Q)$.

Theorem 4.3

Let $0<\alpha <Q$, $1< p< Q/{\alpha }$ and $1/q=1/p-{\alpha }/Q$. If $0<\kappa <p/q$, then the fractional integral operator $I_{\alpha }$ is bounded from $L^{p,\kappa }(\mathbb{H}^{n})$ into $L^{q,{(\kappa q)}/p}( \mathbb{H}^{n})$.

Theorem 4.4

Let $0<\alpha <Q$, $p=1$ and $q=Q/{(Q-\alpha )}$. If $0<\kappa <1/q$, then the fractional integral operator $I_{\alpha }$ is bounded from $L^{1,\kappa }(\mathbb{H}^{n})$ into $WL^{q,(\kappa q)}(\mathbb{H} ^{n})$.

Here, we remark that Theorems 4.3 and 4.4 have been proved by Guliyev et al. [4].

Theorem 4.5

Let $0<\alpha <Q$, $1< p< Q/{\alpha }$ and $1/q=1/p-{\alpha }/Q$. If $p/q\leq \kappa <1$, then the fractional integral operator $I_{\alpha }$ is bounded from $L^{p,\kappa }(\mathbb{H}^{n})$ into $\mathcal{C} ^{\beta }(\mathbb{H}^{n})$ with $\beta /Q=\kappa /p-1/q$ and $\beta <\delta \leq 1$, where δ is given as in (4.6).

As an immediate consequence we have the following corollary.

Corollary 4.6

Let $0<\alpha <Q$, $1< p< Q/{\alpha }$ and $1/q=1/p-{\alpha }/Q$. If $\kappa =p/q$, then the fractional integral operator $I_{\alpha }$ is bounded from $L^{p,\kappa }(\mathbb{H}^{n})$ into $\operatorname{BMO}( \mathbb{H}^{n})$.

Upon taking $\alpha =1$ in Theorem 4.5, we get Morrey’s lemma on the Heisenberg group.

Corollary 4.7

Let $\alpha =1$, $1< p< Q$ and $1/q=1/p-1/Q$. If $p/q<\kappa <1$, then the fractional integral operator $I_{1}$ is bounded from $L^{p,\kappa }( \mathbb{H}^{n})$ into $\mathcal{C}^{\beta }(\mathbb{H}^{n})$ with $\beta /Q=\kappa /p-1/q$ and $\beta <\delta \leq 1$, where δ is given as in (4.6). From this, it follows that, for any given $f\in C^{1}_{c}(\mathbb{H}^{n})$, i.e., f is $C^{1}$-smooth with compact support in $\mathbb{H}^{n}$,

$$ \Vert f \Vert _{\mathcal{C}^{\beta }(\mathbb{H}^{n})}\leq C \Vert \nabla _{\mathbb{H}^{n}}f \Vert _{L^{p,\kappa }(\mathbb{H}^{n})}, $$

where $0<\kappa <1$, $p>(1-\kappa )Q$, $\beta =1-{(1-\kappa )Q}/p$ and the gradient $\nabla _{\mathbb{H}^{n}}$ is defined by

$$ \nabla _{\mathbb{H}^{n}}= (X_{1},\ldots ,X_{n},Y_{1}, \ldots ,Y_{n} ). $$