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