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

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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 \(\beta \in (0,1]\).

Introduction

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).

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. (1)

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

  2. (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 \(C_{1}a\leq b\leq C_{2}a\).

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 \(\operatorname{BMO}_{\rho ,\infty }(\mathbb{H}^{n})\).

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. □

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 2.32.5 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

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

References

  1. 1.

    Bongioanni, B., Harboure, E., Salinas, O.: Commutators of Riesz transforms related to Schrödinger operators. J. Fourier Anal. Appl. 17, 115–134 (2011)

  2. 2.

    Bongioanni, B., Harboure, E., Salinas, O.: Weighted inequalities for commutators of Schrödinger–Riesz transforms. J. Math. Anal. Appl. 392, 6–22 (2012)

  3. 3.

    Dziubański, J., Garrigós, G., Martínez, T., Torrea, J.L., Zienkiewicz, J.: BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality. Math. Z. 249, 329–356 (2005)

  4. 4.

    Guliyev, V.S., Eroglu, A., Mammadov, Y.Y.: Riesz potential in generalized Morrey spaces on the Heisenberg group. J. Math. Sci. (N.Y.) 189, 365–382 (2013)

  5. 5.

    Jerison, D., Sanchez-Calle, A.: Estimates for the heat kernel for a sum of squares of vector fields. Indiana Univ. Math. J. 35, 835–854 (1986)

  6. 6.

    Jiang, Y.S.: Some properties of the Riesz potential associated to the Schrödinger operator on the Heisenberg groups. Acta Math. Sin. (Chin. Ser.) 53, 785–794 (2010)

  7. 7.

    Jiang, Y.S.: Endpoint estimates for fractional integral associated to Schrödinger operators on the Heisenberg groups. Acta Math. Sci. Ser. B 31, 993–1000 (2011)

  8. 8.

    Lin, C.C., Liu, H.P.: \(\mathit{BMO}_{L}(\mathbb{H}^{n})\) spaces and Carleson measures for Schrödinger operators. Adv. Math. 228, 1631–1688 (2011)

  9. 9.

    Lu, G.Z.: A Fefferman–Phong type inequality for degenerate vector fields and applications. Panam. Math. J. 6, 37–57 (1996)

  10. 10.

    Shen, Z.W.: \(L^{p}\) estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45, 513–546 (1995)

  11. 11.

    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

  12. 12.

    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

  13. 13.

    Thangavelu, S.: Harmonic Analysis on the Heisenberg Group. Progress in Mathematics, vol. 159. Birkhäuser, Basel (1998)

  14. 14.

    Xiao, J.S., He, J.X.: Riesz potential on the Heisenberg group. J. Inequal. Appl. 2011, Article ID 498638 (2011)

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The author would like to express his deep gratitude to the referee for his valuable comments and suggestions.

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Correspondence to Hua Wang.

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MSC

  • 42B20
  • 35J10
  • 22E25
  • 22E30

Keywords

  • Schrödinger operator
  • Fractional integrals
  • Heisenberg group
  • Morrey spaces
  • Reverse Hölder class