An. Şt. Univ. Ovidius Constanţa
Vol. 20(1), 2012, 189–212
Riesz potential on the Heisenberg group and
modified Morrey spaces
Vagif S. Guliyev and Yagub Y. Mammadov
Abstract
In this paper we study the fractional maximal operator Mα , 0 ≤ α <
Q and the Riesz potential operator Iα , 0 < α < Q on the Heisenberg
e p,λ (Hn ), where Q = 2n + 2 is
group in the modified Morrey spaces L
the homogeneous dimension on Hn . We prove that the operators Mα
e 1,λ (Hn ) to the
and Iα are bounded from the modified Morrey space L
e q,λ (Hn ) if and only if, α/Q ≤ 1 − 1/q ≤
weak modified Morrey space W L
e p,λ (Hn ) to L
e q,λ (Hn ) if and only if, α/Q ≤ 1/p −
α/(Q − λ) and from L
1/q ≤ α/(Q − λ).
In the limiting case Q−λ
≤p≤ Q
we prove that the operator Mα is
α
α
e
bounded from Lp,λ (Hn ) to L∞ (Hn ) and the modified fractional integral
e p,λ (Hn ) to BM O(Hn ).
operator Ieα is bounded from L
As applications of the properties of the fundamental solution of subLaplacian L on Hn , we prove two Sobolev-Stein embedding theorems on
modified Morrey and Besov-modified Morrey spaces in the Heisenberg
group setting. As an another application, we prove the boundedness of
e spθ,λ (Hn ) to B L
e sqθ,λ (Hn ).
Iα from the Besov-modified Morrey spaces B L
1
Introduction
Heisenberg group appear in quantum physics and many parts of mathematics,
including Fourier analysis, several complex variables, geometry and topology.
Key Words: Heisenberg group; Riesz potential; fractional maximal function; fractional
integral; modified Morrey space; BMO space
2010 Mathematics Subject Classification: 42B35, 43A15, 43A80, 47H50
Received: December, 2010.
Revised: January, 2012.
Accepted: February, 2012.
189
190
Vagif S. Guliyev and Yagub Y. Mammadov
Analysis on the groups is also motivated by their role as the simplest and
the most important model in the general theory of vector fields satisfying
Hormander’s condition. In the present paper we will prove the boundedness
of the Riesz potential on the Heisenberg group in modified Morrey spaces.
We state some basic results about Heisenberg group. More detailed information can be found in [9, 11, 30] and the references therein. Let Hn be the
2n + 1-dimensional Heisenberg group. That is, Hn = Cn × R, with multiplication
(z, t) · (w, s) = (z + w, t + s + 2Im(z · w̄)),
∑n
where z · w̄ = j=1 zj w̄j . The inverse element of u = (z, t) is u−1 = (−z, −t)
and we write the identity of Hn as 0 = (0, 0). The Heisenberg group is a
connected, simply connected nilpotent Lie group. We define one-parameter
dilations on Hn , for r > 0, by δr (z, t) = (rz, r2 t). These dilations are group
automorphisms and the Jacobian determinant is rQ , where Q = 2n + 2 is
the homogeneous dimension of Hn . A homogeneous norm on Hn is given
by |(z, t)| = (|z|2 + |t|)1/2 . With this norm, we define the Heisenberg ball
centered at u = (z, t) with radius r by B(u, r) = {v ∈ Hn : |u−1 v| < r},
{
B(u, r) = Hn \ B(u, r), and we denote by Br = B(0, r) = {v ∈ Hn : |v| < r}
the open ball centered at 0, the identity element of Hn , with radius r. The
volume of the ball B(u, r) is CQ rQ , where CQ is the volume of the unit ball
B1 .
Using coordinates u = (z, t) = (x+iy, t) for points in Hn , the left-invariant
∂
∂
, ∂y∂ j and ∂t
at the origin are
vector fields Xj , Yj and T on Hn equal to ∂x
j
given by
∂
∂
∂
∂
∂
Xj =
+ 2yj , Yj =
− 2xj , T = ,
∂xj
∂t
∂yj
∂t
∂t
respectively. These 2n + 1 vector fields form a basis for the Lie algebra of Hn
with commutation relations
[Yj , Xj ] = 4T
for j = 1, ..., n, and all other commutators equal to 0.
Given a function f which is integrable on any ball B(u, r) ⊂ Hn , the
fractional maximal function Mα f , 0 ≤ α < Q of f is defined by
∫
α
Mα f (u) = sup |B(u, r)|−1+ Q
|f (v)|dV (v).
r>0
B(u,r)
The fractional maximal function Mα f coincides for α = 0 with the HardyLittlewood maximal function M f ≡ M0 f (see [9, 30]) and is intimately related
Riesz potential on the Heisenberg group and modified Morrey spaces
to the fractional integral
∫
Iα f (u) =
Hn
|v −1 u|α−Q f (v)dV (v),
191
0 < α < Q.
The operators Mα and Iα play important role in real and harmonic analysis
(see, for example [8, 9, 30]).
The classical Riesz potential is an important technical tool in harmonic
analysis, theory of functions and partial differential equations. The classical
Riesz potential Iα is defined on Rn by
Iα f = (−∆)−α/2 f,
0 < α < n,
∫
where ∆ is the Laplacian operator. It is known, that Iα f (x) = γ(α)−1 Rn |x−
y|α−n f (y)dy ≡ Iα f (x).
The potential and related topics on the Heisenberg group we consider the
sub-Laplacian defined by
L=−
n
∑
(
)
Xj2 + Yj2 .
j=1
The Riesz potential on the Heisenberg group is defined in terms of the
sub-Laplacian L.
Definition 1.1. For 0 < α < Q, the Riesz potential Iα is defined, initially on
the Schwartz space S(Hn ), by
Iα f (z, t) = L− 2 f (z, t).
α
In [33] the relation between Riesz potential and heat kernel on the Heisenberg group is studied. The following theorem give expression of Iα , which
provides a bridge to discuss the boundedness of the Riesz potential (see [33],
Theorem 1).
Theorem A. Let qs (z, t) be the heat kernel on Hn . For 0 ≤ α < Q, we
have for f ∈ S(Hn )
∫∞
Iα f (z, t) =
Γ
( α )−1 α −1
s 2 qs (·)ds ∗ f (z, t).
2
0
The Riesz potential Iα satisfies the following estimate (see [33], Theorem
2)
|Iα f (z, t)| . Iα f (z, t).
(1)
192
Vagif S. Guliyev and Yagub Y. Mammadov
Inequality (1) gives a suitable estimate for the Riesz potential on the Heisenberg group.
In the theory of partial differential equations, together with weighted Lp,w (Rn )
spaces, Morrey spaces Lp,λ (Rn ) play an important role. Morrey spaces were
introduced by C. B. Morrey in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations (see [22]). Later,
Morrey spaces found important applications to Navier-Stokes ([20], [32]) and
Schrödinger ([23], [24], [25], [27], [28]) equations, elliptic problems with discontinuous coefficients ([3], [6]), and potential theory ([1], [2]). An exposition
of the Morrey spaces can be found in the book [19].
Definition 1.2. Let 1 ≤ p < ∞, 0 ≤ λ ≤ Q, [t]1 = min{1, t}. We denote by
e p,λ (Hn ) the modified Morrey space, the
Lp,λ (Hn ) the Morrey space, and by L
set of locally integrable functions f (u), u ∈ Hn , with the finite norms
(
∥f ∥Lp,λ =
t
sup
−λ
u∈Hn , t>0
|f (y)| dV (v)
p
,
B(u,t)
(
∥f ∥Lep,λ =
)1/p
∫
sup
u∈Hn , t>0
[t]−λ
1
)1/p
∫
|f (y)|p dV (v)
B(u,t)
respectively.
Note that
e p,0 (Hn ) = Lp,0 (Hn ) = Lp (Hn ),
L
e p,λ (Hn ) ⊂≻ Lp,λ (Hn )∩Lp (Hn ) and
L
max{∥f ∥Lp,λ , ∥f ∥Lp } ≤ ∥f ∥Lep,λ (2)
e p,λ (Hn ) = Θ, where Θ is the set of
and if λ < 0 or λ > Q, then Lp,λ (Hn ) = L
all functions equivalent to 0 on Hn .
Definition 1.3. [15] Let 1 ≤ p < ∞, 0 ≤ λ ≤ Q. We denote by W Lp,λ (Hn )
e p,λ (Hn ) the modified weak Morrey space the
the weak Morrey space and by W L
set of locally integrable functions f (u), u ∈ Hn with finite norms
∥f ∥W Lp,λ = sup r
r>0
∥f ∥W Lep,λ = sup r
r>0
respectively.
sup
( −λ
)1/p
t
|{v ∈ B(u, t) : |f (v)| > r}|
,
u∈Hn , t>0
sup
u∈Hn , t>0
(
[t]−λ
|{v ∈ B(u, t) : |f (v)| > r}|
1
)1/p
Riesz potential on the Heisenberg group and modified Morrey spaces
193
Note that
e p,0 (Hn ),
W Lp (Hn ) = W Lp,0 (Hn ) = W L
Lp,λ (Hn ) ⊂ W Lp,λ (Hn ) and ∥f ∥W Lp,λ ≤ ∥f ∥Lp,λ ,
e p,λ (Hn ) ⊂ W L
e p,λ (Hn ) and ∥f ∥ e ≤ ∥f ∥ e .
L
W Lp,λ
Lp,λ
The classical result by Hardy-Littlewood-Sobolev states that if 1 <(p < q <
)
∞, then Iα is bounded from Lp (Hn ) to Lq (Hn ) if and only if α = Q p1 − 1q
and for p(= 1 <)q < ∞, Iα is bounded from L1 (Hn ) to W Lq (Hn ) if and only
if α = Q 1 − 1q .
Spanne (see [29]) and Adams [1] studied boundedness of Iα on Rn in Morrey
spaces Lp,λ (Rn ). Later on Chiarenza and Frasca [5] was reproved boundedness
of Iα in these spaces Lp,λ (Rn ). By more general results of Guliyev [12] (see,
also [13, 14, 15]) one can obtain the following generalization of the results in
[1, 5, 29] to the Heisenberg group case (see, also [21]).
Theorem B. Let 0 < α < Q and 0 ≤ λ < Q − α, 1 ≤ p < Q−λ
α .
1
1
α
1) If 1 < p < Q−λ
,
then
condition
−
=
is
necessary
and
sufficient
α
p
q
n−λ
for the boundedness of the operator Iα from Lp,λ (Hn ) to Lq,λ (Hn ).
α
2) If p = 1, then condition 1 − 1q = Q−λ
is necessary and sufficient for the
boundedness of the operator Iα from L1,λ (Hn ) to W Lq,λ (Hn ).
Q
If α = Q
p − q , then λ = 0 and the statement of Theorem B reduces to the
aforementioned result by Hardy-Littlewood-Sobolev.
Recall that, for 0 < α < Q,
α
Mα f (u) ≤ vnQ
−1
Iα (|f |)(u),
(3)
hence Theorem B also implies the boundedness of the fractional maximal
operator Mα , where vn = |B(e, 1| is the volume of the unit ball in Hn .
We define BM O(Hn ), the set of locally integrable functions f with finite
norms
∫
∥f ∥∗ =
sup |Br |−1
|f (v −1 u) − fBr (u)|dV (v) < ∞,
r>0, u∈Hn
∫
−1
Br
where fBr (u) = |Br |
f (v −1 u)dV (v).
Br
In this paper we study the fractional maximal operator Mα and the Riesz
potential operator Iα on the Heisenberg group in the modified Morrey space.
In the case p = 1 we prove that the operators Mα and Iα are bounded from
e 1,λ (Hn ) to W L
e q,λ (Hn ) if and only if, α/Q ≤ 1 − 1/q ≤ α/(Q − λ). In the
L
case 1 < p < (Q − λ)/α we prove that the operators Mα and Iα are bounded
e p,λ (Hn ) to L
e q,λ (Hn ) if and only if, α/Q ≤ 1/p − 1/q ≤ α/(Q − λ).
from L
194
Vagif S. Guliyev and Yagub Y. Mammadov
In the limiting case Q−λ
≤ p ≤ Q
α
α we prove that the operator Mα is
e
bounded from Lp,λ (Hn ) to L∞ (Hn ) and the modified fractional integral opere p,λ (Hn ) to BM O(Hn ).
ator Ieα is bounded from L
As an application of the properties of the fundamental solution of subLaplacian L on Hn , in Theorem 3.3 we prove the following modified Morrey
version of Sobolev inequality on Hn :
∥u∥Leq,λ ≤ C∥∇L u∥Lep,λ , for every u ∈ C0∞ (Hn ),
1
1
where 0 ≤ λ < Q, 1 < p < Q − λ and Q
≤ p1 − 1q ≤ Q−λ
.
In Theorem 7 we obtain boundedness of the operator Iα from the Besove s (Hn ) to B L
e s (Hn ), 1 < p < q < ∞, α/Q ≤
modified Morrey spaces B L
pθ,λ
qθ,λ
1/p − 1/q ≤ α/(Q − λ), 1 ≤ θ ≤ ∞ and 0 < s < 1.
As an another application, in Theorem 3.5 we obtain the following SobolevStein embedding inequality in Besov-modified Morrey space on Hn .
∥u∥B Les
qθ,λ
≤ C∥∇L u∥B Les
pθ,λ
, for every u ∈ C0∞ (Hn )
where 0 ≤ λ < Q − 1, 1 < p < Q − λ, 1 ≤ θ ≤ ∞, 0 < s < 1 and
1/Q ≤ 1/p − 1/q ≤ 1/(Q − λ).
By A . B we mean that A ≤ CB with some positive constant C independent of appropriate quantities. If A . B and B . A, we write A ≈ B and say
that A and B are equivalent.
2
Main Results
The following Hardy-Littlewood-Sobolev inequality in modified Morrey spaces
on the Heisenberg group is valid.
Theorem 2.1. Let 0 < α < Q, 0 ≤ λ < Q − α and 1 ≤ p < Q−λ
α .
α
1
1
α
,
then
condition
≤
−
≤
is
necessary and
1) If 1 < p < Q−λ
α
Q
p
q
Q−λ
e p,λ (Hn ) to
sufficient for the boundedness of the operators Iα and Iα from L
e q,λ (Hn ).
L
2) If p = 1 <
Q−λ
α ,
then condition
α
Q
≤ 1−
1
q
≤
α
Q−λ
sufficient for the boundedness of the operators Iα and Iα
e q,λ (Hn ).
WL
is necessary and
e 1,λ (Hn ) to
from L
Corollary 2.1. Let 0 < α < Q, 0 ≤ λ < Q − α and 1 ≤ p ≤ Q−λ
α .
Q−λ
α
1
1
α
1) If 1 < p < α , then condition Q ≤ p − q ≤ Q−λ is necessary and
e p,λ (Hn ) to L
e q,λ (Hn ).
sufficient for the boundedness of the operator Mα from L
Riesz potential on the Heisenberg group and modified Morrey spaces
2) If p = 1 <
Q−λ
α ,
then condition
α
Q
≤ 1−
1
q
≤
195
α
Q−λ
is necessary and sufe
e q,λ (Hn ).
ficient for the boundedness of the operator Mα from L1,λ (Hn ) to W L
e
3) If Q−λ
≤p≤ Q
α
α , then the operator Mα is bounded from Lp,λ (Hn ) to
L∞ (Hn ).
The examples show that the operator Iα are not defined for all functions
e p,λ (Hn ), 0 ≤ λ < Q − α, if p ≥ Q−λ .
f ∈L
α
We consider the modified fractional integral
∫ (
)
e
Iα f (u) =
|uv −1 |α−Q − |v|α−Q χ {B(e,1) (v) f (v)dV (v),
Hn
{
where χ {B(e,1) is the characteristic function of the set B(e, 1) = Hn \ B(e, 1).
≤ p ≤ Q
Note that in the limiting case Q−λ
α
α statement 1) in Theorem
e
B does not hold. Moreover, there exists f ∈ Lp,λ (Hn ) such that Iα f (u) =
∞ for all u ∈ Hn . However, as will be proved, statement 1) holds for the
modified fractional integral Ieα if the space L∞ (Hn ) is replaced by a wider
space BM O(Hn ).
The following theorem we obtain conditions ensuring that the operator Ieα
e p,λ (Hn ) to BM O(Hn ).
is bounded from the space L
Theorem 2.2. Let 0 < α < Q, 0 ≤ λ < Q − α, and Q−λ
≤p≤ Q
α
α , then
e
e
the operator Iα is bounded from Lp,λ (Hn ) to BM O(Hn ). Moreover, if the
e p,λ (Hn ), Q−λ ≤ p ≤ Q , then
integral Iα f exists almost everywhere for f ∈ L
α
α
Iα f ∈ BM O(Hn ) and the following inequality is valid
∥Iα f ∥BM O ≤ C∥f ∥Lep,λ ,
where C > 0 is independent of f .
3
Some applications
It is known that (see [4], p. 247) if |·| is a homogeneous norm on Hn , then there
exists a positive constant C0 such that Γ(x) = C0 |x|2−Q is the fundamental
solution of L.
From Theorem 2.1, one easily obtains an inequality extending the classical
Sobolev embedding theorem to the Heisenberg groups.
196
Vagif S. Guliyev and Yagub Y. Mammadov
Theorem 3.3. (Sobolev-Stein embedding on modified Morrey spaces)
1
1
Let 0 ≤ λ < Q, 1 < p < Q − λ and Q
≤ p1 − 1q ≤ Q−λ
. Then there exists a
positive constant C such that
∥u∥Leq,λ ≤ C∥∇L u∥Lep,λ , for every u ∈ C0∞ (Hn ).
Proof. Let u ∈ C0∞ (Hn ). By using the integral representation formula for the
fundamental solution (see [4], p. 237), we have
∫
u(x) =
Γ(x−1 y)Lu(y)dy
(4)
∑n
Hn
Keeping in mind that L = i=1 (Xi2 + Yi2 ) and Xi∗ = −Xi , Yi∗ = −Yi , by
integrating by parts at the right-hand side (4), we obtain
∫
u(x) =
(∇L Γ)(x−1 y)∇L u(y)dy.
(5)
Hn
On the other hand, out of the origin, we have
(
)
∇L Γ(x) = C0 ∇L |x|2−Q = (2 − Q)C0 |x|1−Q ∇L |x|,
so that, since ∇L | · | is smooth in Hn \ {0} and δλ -homogeneous of degree zero,
∇L Γ(x) ≤ C|x|1−Q ,
for a suitable constant C > 0 depending only on L. Using this inequality in
(5), we get
∫
|u(x)| ≤ C
|∇L u(y)||x|1−Q dy = CI1 (|∇L u|)(x).
(6)
Hn
Then, by Theorem 2.1,
∥u∥Leq,λ ≤ C∥I1 (|∇L u|)∥Leq,λ ≤ C∥∇L u∥Lep,λ .
In the following theorem we prove the boundedness of Iα in the Besovmodified Morrey spaces on Hn
{
e spθ,λ (Hn ) = f : ∥f ∥ es
BL
BL
pθ,λ
= ∥f ∥Lep,λ +
(∫
∥f (x·) − f (·)∥θLe
p,λ
Hn
|x|Q+sθ
dx
) θ1
<∞
}
(7)
where 1 ≤ p, θ ≤ ∞ and 0 < s < 1.
s
Besov spaces Bpθ
(Hn ) in the setting Lie groups were studied by many
authors (see, for example [8, 10, 16, 26, 31]).
Riesz potential on the Heisenberg group and modified Morrey spaces
197
Theorem 3.4. Let 0 < α < Q, 0 ≤ λ < Q − α and 1 ≤ p < Q−λ
α . If
α
1
1
α
≤
−
≤
,
1
≤
θ
≤
∞
and
0
<
s
<
1,
then
the
operator
Iα is
Q
p
q
Q−λ
s
s
e
e
bounded from the spaces B Lpθ,λ (Hn ) to B Lqθ,λ (Hn ). More precisely, there is
a constant C > 0 such that
∥Iα f ∥B Les
qθ,λ
≤ C∥f ∥B Les
pθ,λ
e s (Hn ).
holds for all f ∈ B L
pθ,λ
Proof. By the definition of the Besov-modified Morrey spaces on Hn it suffices
to show that
∥τy Iα f − Iα f ∥Leq,λ ≤ C∥τy f − f ∥Lep,λ ,
where τy f (x) = f (yx).
It is easy to see that τy f commutes with Iα , i.e., τy Iα f = Iα (τy f ). Hence
we obtain
|τy Iα f − Iα f | = |Iα (τy f ) − Iα f | ≤ Iα (|τy f − f |).
e p,λ -norm on both sides of the last inequality, we obtain the desired
Taking L
e p,λ (Hn ) to L
e q,λ (Hn ).
result by using the boundedness of Iα from L
Thus the proof of the Theorem 3.4 is completed.
From Theorem 3.4 we obtain the following Sobolev-Stein embedding inequality on Besov-modified Morrey space.
Theorem 3.5. (Sobolev-Stein embedding on Besov-modified Morrey
space) Let 0 ≤ λ < Q − 1, 1 < p < Q − λ, 1 ≤ θ ≤ ∞ and 0 < s < 1. Then
there exists a positive constant C such that
∥u∥B Les
qθ,λ
≤ C∥∇L u∥B Les
pθ,λ
, for every u ∈ C0∞ (Hn )
where 1/Q ≤ 1/p − 1/q ≤ 1/(Q − λ).
The Dirichlet problem for the Kohn-Laplacian on Hn belongs to Jerison
[17, 18]. In particular, our results lead to the following apriori estimate for
the sub-Laplacian equation Lf = g.
e s (Hn ) and Lf = g
Theorem 3.6. Let 0 < s < 1, 1 ≤ θ ≤ ∞, g ∈ B L
pθ,λ
2
1
1
2
1) If 0 ≤ λ < Q − 2, 1 < p < Q−λ
and
≤
−
2
Q
p
q ≤ Q−λ , then there exists
a constant C > 0 such that
∥f ∥B Les
qθ,λ
≤ C∥g∥B Les
pθ,λ
.
198
Vagif S. Guliyev and Yagub Y. Mammadov
, 2) If 0 ≤ λ < Q − 1, 1 < p < Q − λ and
a constant C > 0 such that
1
Q
≤
− 1q ≤
1
p
1
Q−λ ,
∥Xi f ∥B Les
≤ C∥g∥B Les
, i = 1, 2, · · · , n,
∥Yi f ∥B Les
≤ C∥g∥B Les
, i = 1, 2, · · · , n.
qθ,λ
qθ,λ
pθ,λ
pθ,λ
then there exists
The proof of Theorems 3.5 and 3.6 are similar to Theorem 3.3.
4
Preliminaries
Define ft (u) =: f (δt u) and [t]1,+ = max{1, t}. Then
(
∥ft ∥Lp,λ = t
−Q
p
r−λ
sup
B(δt u,tr)
[r]−λ
1
sup
u∈Hn , r>0
=t
−Q
p
(
sup
u∈Hn , r>0
=t
|f (v)|p dV (v)
u∈Hn , r>0
(
∥ft ∥Lep,λ =
−Q
p
(
sup
r>0
Q
)1/p
∫
[tr]1
[r]1
=t
λ−Q
p
∥f ∥Lp,λ ,
)1/p
∫
p
|ft (v)| dV (v)
B(u,r)
[r]−λ
1
)1/p
∫
|f (v)| dV (v)
p
B(δt u,tr)
(
)λ/p
sup
u∈Hn , r>0
[tr]−λ
1
)1/p
∫
p
|f (v)| dV (v)
B(δt u,tr)
λ
p
∥f ∥Lep,λ .
= t− p [t]1,+
e p,λ -boundedness of the maximal operator M .
In this section we study the L
Lemma 4.1. Let 1 ≤ p < ∞, 0 ≤ λ ≤ Q. Then
e p,λ (Hn ) = Lp,λ (Hn ) ∩ Lp (Hn )
L
and
{
}
∥f ∥Lep,λ = max ∥f ∥Lp,λ , ∥f ∥Lp .
e
Proof. Let
{ f ∈ Lp,λ (Hn ). }Then from (2) we have that f ∈ Lp,λ (Hn ) ∩ Lp (Hn )
and max ∥f ∥Lp,λ , ∥f ∥Lp ≤ ∥f ∥Lep,λ .
Riesz potential on the Heisenberg group and modified Morrey spaces
199
Let now f ∈ Lp,λ (Hn ) ∩ Lp (Hn ). Then
(
∥f ∥Lep,λ =


= max
sup
u∈Hn ,t>0
[t]−λ
1
)1/p
∫
|f (v)| dV (v)
p
B(u,t)
(
−λ
)1/p
∫
sup
t
u∈Hn ,0<t≤1
|f (v)| dV (v)
p
,
B(u,t)
)1/p 

sup
|f (v)|p dV (v)

u∈Hn ,t>1
B(u,t)
{
}
≤ max ∥f ∥Lp,λ , ∥f ∥Lp .
(∫
e p,λ (Hn ) and the embedding Lp,λ (Hn ) ∩ Lp (Hn ) ⊂≻ L
e p,λ (Hn )
Therefore, f ∈ L
is valid.
{
}
e p,λ (Hn ) = Lp,λ (Hn )∩Lp (Hn ) and ∥f ∥ e = max ∥f ∥
Thus L
Lp,λ , ∥f ∥Lp .
Lp,λ
Analogously proved the following statement.
Lemma 4.2. Let 1 ≤ p < ∞, 0 ≤ λ ≤ Q. Then
e p,λ (Hn ) = W Lp,λ (Hn ) ∩ W Lp (Hn )
WL
and
{
}
∥f ∥W Lep,λ = max ∥f ∥W Lp,λ , ∥f ∥W Lp .
Theorem 4.7. [21] 1. If f ∈ L1,λ (Hn ), 0 ≤ λ < Q, then M f ∈ W L1,λ (Hn )
and
∥M f ∥W L1,λ ≤ Cλ ∥f ∥L1,λ ,
where Cλ depends only on n and λ.
2. If f ∈ Lp,λ (Hn ), 1 < p < ∞, 0 ≤ λ < Q, then M f ∈ Lp,λ (Hn ) and
∥M f ∥Lp,λ ≤ Cp,λ ∥f ∥Lp,λ ,
where Cp,λ depends only on n, p and λ.
The following statement is valid:
200
Vagif S. Guliyev and Yagub Y. Mammadov
e 1,λ (Hn ), 0 ≤ λ < Q, then M f ∈ W L
e 1,λ (Hn ) and
Theorem 4.8. 1. If f ∈ L
∥M f ∥W Le1,λ ≤ C 1,λ ∥f ∥Le1,λ ,
where C1,λ depends only on λ.
e p,λ (Hn ), 1 < p < ∞,0 ≤ λ < Q, then M f ∈ L
e p,λ (Hn ) and
2. If f ∈ L
∥M f ∥Lep,λ ≤ C p,λ ∥f ∥Lep,λ ,
where Cp,λ depends only on p and λ.
Proof. It is obvious that (see Lemmas 4.1 and 4.2)
{
}
∥M f ∥Lep,λ = max ∥M f ∥Lp,λ , ∥M f ∥Lp
for 1 < p < ∞ and
{
}
∥M f ∥W Le1,λ = max ∥M f ∥W L1,λ , ∥M f ∥W L1
for p = 1.
Let 1 < p < ∞. By the boundedness of M on Lp (Hn ) and from Theorem
4.7 we have
∥M f ∥Lep,λ ≤ max {Cp , Cp,λ } ∥f ∥Lep,λ .
Let p = 1. By the boundedness of M from L1 (Hn ) to W L1 (Hn ) and from
Theorem 4.7 we have
∥M f ∥W Le1,λ ≤ max {C1 , C1,λ } ∥f ∥Le1,λ .
Lemma 4.3. Let 0 < α < Q. Then for 2|u| ≤ |v|, u, v ∈ Hn , the following
inequality is valid
−1 α−Q
|v u|
− |v|α−Q ≤ 2Q−α+1 |v|α−Q−1 |u|.
(8)
Proof. From the mean value theorem we get
−1 α−Q
|v u|
− |v|α−Q ≤ |v −1 u| − |v| · ξ α−Q−1 ,
{
}
{
}
where min |v −1 u|, |v| ≤ ξ ≤ max |v −1 u|, |v| .
We note that 12 |v| ≤ |v −1 u| ≤ 32 |v|, and |v −1 u| − |v| ≤ |u|.
Thus the proof of the lemma is completed.
Riesz potential on the Heisenberg group and modified Morrey spaces
5
201
Proof of the Theorems
Proof of Theorem 2.1.
e p,λ (Hn ) and
1) Sufficiency. Let 0 < α < Q, 0 < λ < Q − α, f ∈ L
Q−λ
1 < p < α . Then
(∫
)
∫
Iα f (u) =
+
f (v)|uv −1 |α−Q dV (v) ≡ A(u, t) + C(u, t).
{
B(u,t)
B(u,t)
For A(u, t) we have
∫
|f (v)||uv −1 |α−Q dV (v)
|A(u, t)| ≤
B(u,t)
≤
∞
∑
(
2−j t
)α−Q
∫
B(u,2−j+1 t)\B(u,2−j t)
j=1
|f (v)|dV (v).
Hence
|A(u, t)| . tα M f (u).
(9)
In the second integral by the Hölder’s inequality we have
(∫
|C(u, t)| ≤
)1/p
{
|uv
−1 −β
|
|f (v)| dV (v)
p
B(u,t)
(∫
×
{
|uv
−1
)1/p′
β
′
+α−Q
p
(
)
| p
dV (v)
= J1 · J2 .
B(u,t)
Let λ < β < Q − αp. For J1 we get
202
J1 =
Vagif S. Guliyev and Yagub Y. Mammadov
∞ ∫
(∑
j=0
)1/p
|f (v)|p |uv −1 |−β dV (v)
B(u,2j+1 t)\B(u,2j t)
β
≤ t− p ∥f ∥Lep,λ
∞
(∑
2−βj [2j+1 t]λ1
)1/p
j=0
=t
−β
p
∥f ∥Lep,λ

1
(
[log2 2t

∑ ] (λ−β)j

λ λ

2
+
2
t

j=0




{ (
≈t
−β
p
∥f ∥Lep,λ
{
≈ ∥f ∥Lep,λ
λ
tλ + tβ
1,
)1/p
(∑
∞
∞
∑
2−βj
)1/p
1
j=[log2 2t
]+1
2−βj
)1/p
, 0 < t < 12 ,
t≥
,
j=0
1
2
, 0 < t < 12 ,
t ≥ 12
λ−β
t p , 0 < t < 12 ,
β
t− p ,
t ≥ 12
β
= [2t]1p t− p ∥f ∥Lep,λ .
(10)
For J2 we obtain
(∫
∫
J2 =
dξ
S
∞
r
′
Q−1+( β
p +α−Q)p
) 1′
p
dr
β
Q
≈ t p +α− p .
(11)
t
From (10) and (11) we have
λ
Q
|C(u, t)| . [t]1p tα− p ∥f ∥Lep,λ .
Thus for all t > 0 we get
(
)
λ
Q
|Iα f (u)| . tα M f (u) + [t]1p tα− p ∥f ∥Lep,λ
{
}
Q
Q−λ
. min tα M f (u) + tα− p ∥f ∥Lep,λ , tα M f (u) + tα− p ∥f ∥Lep,λ .
Minimizing with respect to t, at
[
]p/(Q−λ)
−1
t = (M f (u)) ∥f ∥Lep,λ
and
]p/Q
[
−1
t = (M f (u)) ∥f ∥Lep,λ
(12)
203
Riesz potential on the Heisenberg group and modified Morrey spaces
we have
(

) pα (
)1− pα
Q 
 M f (u) 1− Q−λ
M f (u)
|Iα f (u)| ≤ C11 min
,
∥f ∥Lep,λ .
 ∥f ∥Lep,λ

∥f ∥Lep,λ
Then
|Iα f (u)| . (M f (u))
1−p/q
p/q
∥f ∥Le
Hence, by Theorem 4.8, we have
∫
∫
q
q−p
|Iα f (v)| dV (v) . ∥f ∥Le
p,λ
B(u,t)
.
p,λ
p
(M f (v)) dV (v)
B(u,t)
q
. [t]λ1 ∥f ∥Le
,
p,λ
e p,λ (Hn ) to L
e q,λ (Hn ).
which implies that Iα is bounded from L
Q−λ
e p,λ (Hn ) and the operators Iα and Iα
Necessity. Let 1 < p < α , f ∈ L
e
e
are bounded from Lp,λ (Hn ) to Lq,λ (Hn ).
Define ft (u) =: f (δt u), [t]1,+ = max{1, t}. Then
(
∥ft ∥Lep,λ =
[r]−λ
1
sup
r>0, u∈Hn
−Q
p
=t
=t
sup
r>0
Q
|ft (v)| dV (v)
[r]−λ
1
sup
(
p
B(u,r)
(
u∈Hn , r>0
−Q
p
)1/p
∫
)1/p
∫
|f (v)| dV (v)
p
B(u,tr)
[tr]1
[r]1
)λ/p
(
sup
r>0, u∈Hn
[tr]−λ
1
)1/p
∫
p
|f (v)| dV (v)
B(u,tr)
λ
p
∥f ∥Lep,λ ,
= t− p [t]1,+
and
Iα ft (u) = t−α Iα f (δt u),
(
∥Iα ft ∥Leq,λ = t−α
=t
sup
u∈Hn , r>0
−α− Q
q
(
sup
r>0
Q
λ
[r]−λ
1
Iα ft (u) = t−α Iα f (δt u),
)1/q
∫
q
|Iα f (δt v)| dV (v)
B(u,r)
[tr]1
[r]1
)λ/q
sup
r>0, u∈Hn
q
= t−α− q [t]1,+
∥Iα f ∥Leq,λ .
(
[tr]−λ
1
)1/q
∫
q
|Iα f (v)| dV (v)
B(δt u,tr)
204
Vagif S. Guliyev and Yagub Y. Mammadov
Also
Q
λ
q
∥Iα ft ∥Leq,λ = t−α− q [t]1,+
∥Iα f ∥Leq,λ .
e p,λ (Hn ) to L
e q,λ (Hn )
By the boundedness of Iα from L
Q
−λ
Q
−λ
∥Iα f ∥Leq,λ = tα+ q [t]1,+q ∥Iα ft ∥Leq,λ
≤ tα+ q [t]1,+q ∥ft ∥Lep,λ
Q
Q
λ
−λ
p
q
∥f ∥Lep,λ ,
≤ Cp,q,λ tα+ q − p [t]1,+
where Cp,q,λ depends only on p, q, λ and n.
α
If p1 < 1q + Q
, then in the case t → 0 we have ∥Iα f ∥Leq,λ = 0 for all
e
f ∈ Lp,λ (Hn ).
α
, then at t → ∞ we obtain ∥Iα f ∥Leq,λ = 0 for all
As well as if p1 > 1q + Q−λ
e
f ∈ Lp,λ (Hn ).
α
α
Therefore Q
≤ p1 − 1q ≤ Q−λ
. Analogously we get the last equality for the
Iα .
e 1,λ (Hn ). We have
2) Sufficiency. Let f ∈ L
|{v ∈ B(u, t) : |Iα f (v)| > 2β}|
≤ |{v ∈ B(u, t) : |A(v, t)| > β}| + |{v ∈ B(u, t) : |C(v, t)| > β}| .
Then
C(v, t) =
∞ ∫
∑
j=0
|f (z)||vw−1 |α−Q dV (w)
B(v,2j+1 t)\B(v,2j t)
≤ tα−Q ∥f ∥Le1,λ
∞
∑
2−(Q−α)j [2j+1 t]λ1
j=0
=t
α−Q
∥f ∥Le1,λ

1
[log2 2t

∑ ] (λ−Q+α)j


2
+
 2λ tλ




{
j=0
∞
∑
∞
∑
1
j=[log2 2t
]+1
2−(Q−α)j , 0 < t < 21 ,
2−(Q−α)j ,
j=0
tλ + C15 tQ−α , 0 < t < 12 ,
≈ tα−Q ∥f ∥Le1,λ
1,
t ≥ 21
{ λ+α−Q
t
, 0 < t < 12 ,
≈ ∥f ∥Le1,λ
= [2t]λ1 tα−Q ∥f ∥Le1,λ .
α−Q
t
,
t ≥ 12
t≥
1
2
Riesz potential on the Heisenberg group and modified Morrey spaces
205
Taking into account inequality (9) and Theorem 2.1, we have
{
|{v ∈ B(u, t) : |A(v, t)| > β}| ≤ v ∈ B(u, t) : M f (v) >
≤
}
β
α
C1 t C2 tα
· [t]λ1 ∥f ∥Le1,λ ,
β
where C2 = C1 · C1,λ and thus if C2 [2t]λ1 tα−Q ∥f ∥Le1,λ = β, then |C(v, t)| ≤ β
and consequently, | {v ∈ B(u, t) : |C(v, t)| > β} | = 0.
Then
1 λ α
[t] t ∥f ∥Le1,λ
β 1
(
) Q−λ
∥f ∥Le1,λ Q−λ−α
λ
, if 2t < 1
. [t]1
β
|{v ∈ B(u, t) : |Iα f (v)| > 2β}| .
and
1 λ α
[t] t ∥f ∥Le1,λ
β 1
(
) Q
∥f ∥Le1,λ Q−α
λ
. [t]1
, if 2t ≥ 1.
β
|{v ∈ B(u, t) : |Iα f (v)| > 2β}| .
Finally we have
|{v ∈ B(u, t) : |Iα f (v)| > 2β}|
(
Q−λ
Q 
) Q−λ−α
(
) Q−α
(
)q
 ∥f ∥ e

∥f
∥
e
1
L1,λ
L1,λ
. [t]λ1 min
,
. [t]λ1
∥f ∥Le1,λ .


β
β
β
e 1,λ (Hn ) to
Necessity. Let the operators Iα and Iα are bounded from L
206
Vagif S. Guliyev and Yagub Y. Mammadov
e q,λ (Hn ). We have
WL
(
∥Iα ft ∥W Leq,λ = sup r
r>0
sup
u∈Hn , τ >0
= sup r
sup
r>0
u∈Hn , τ >0
Q
= t−α− q sup
τ >0
(
×
sup
dV (v)
{v∈B(u,τ ) : |Iα ft (v)|>r}
(
[tτ ]1
[τ ]1
)1/q
∫
[τ ]−λ
1
[tτ ]−λ
1
u∈Hn , τ >0
Q
(
)1/q
∫
[τ ]−λ
1
dV (v)
{v∈B(δt u,τ ) : |Iα f (δt v)|>rtα }
)λ/q
sup rtα
r>0
)1/q
∫
dV (v)
{v∈B(u,tτ ) : |Iα f (v)|>rtα }
λ
q
= t−α− q [t]1,+
∥Iα f ∥W Leq,λ .
e 1,λ (Hn ) to W L
e q,λ (Hn )
By the boundedness of Iα from L
Q
−λ
Q
−λ
∥Iα f ∥W Leq,λ = tα+ q [t]1,+q ∥Iα ft ∥W Leq,λ
. tα+ q [t]1,+q ∥ft ∥Le1,λ
λ− λ
Q
= tα+ q −Q [t]1,+q ∥f ∥Le1,λ .
α
If 1 < 1q + Q
, then in the case t → 0 we have ∥Iα f ∥W Leq,λ = 0 for all
e
f ∈ L1,λ (Hn ).
α
Similarly, if 1 > 1q + Q−λ
, then for t → ∞ we obtain ∥Iα f ∥W Leq,λ = 0 for
e
all f ∈ L1,λ (Hn ).
α
α
Therefore Q
≤ 1 − 1q ≤ Q−λ
. Analogously we get the last equality for the
Iα .
Thus Theorem 2.1 is proved.
Proof of Corollary 2.1.
Sufficiency of Corollary 2.1 follows from Theorem 2.1 and inequality (3).
e p,λ (Hn ) to L
e q,λ (Hn ) for 1 < p <
Necessity. (1) Let Mα be bounded from L
Q−λ
α . Then we have
Mα ft (u) = t−α Mα f (δt u),
and
Q
λ
q
∥Mα ft ∥Leq,λ = t−α− q [t]1,+
∥Mα f ∥Leq,λ .
207
Riesz potential on the Heisenberg group and modified Morrey spaces
α
By the same argument in Theorem 2.1 we obtain Q
≤ p1 − 1q ≤
e 1,λ (Hn ) to W L
e q,λ (Hn ). Then
(2) Let Mα be bounded from L
α
Q−λ .
λ
Q
q
∥Mα ft ∥W Leq,λ = t−α− q [t]1,+
∥Mα f ∥W Leq,λ .
Hence we obtain
(3) Let
α
Q
≤p≤
Q−λ
α
∥Mα f ∥L∞ = 2−Q
≤1−
1
q
≤
α
Q−λ .
Q
α.
Then by the Hölder’s inequality we have
∫
|f (v)|dV (v)
sup tα−Q
u∈Hn , t>0
−Q
p
≤2
sup
B(u,t)
(
t
α− Q
p
u∈Hn , t>0
λ
p
[t]−λ
1
[t]1
)1/p
∫
|f (v)| dV (v)
p
B(u,t)
Q
λ
Q
≤ 2− p ∥f ∥Lep,λ sup tα− p [t]1p
t>0
{
}
Q−λ
Q
Q
−Q
p
=2
∥f ∥Lep,λ max sup tα− p , sup tα− p = 2− p ∥f ∥Lep,λ .
t>1
0<t≤1
Thus Corollary 2.1 is proved.
Proof of Theorem 2.2.
Let 0 < α < Q, 0 ≤ λ < Q − α,
given t > 0 we denote
Q−λ
α
f1 (u) = f (u)χB(e,2t) (u),
Then
≤p≤
Q
α
e p,λ (Hn ). For
and f ∈ L
f2 (u) = f (u) − f1 (u).
Ieα f (u) = Ieα f1 (u) + Ieα f2 (u) = F1 (u) + F2 (u),
where
∫
(
F1 (u) =
B(e,2t)
∫
F2 (u) =
)
|v −1 u|α−Q − |v|α−Q χ {B(e,1) (v) dV (v),
(
{
B(e,2t)
)
|v −1 u|α−Q − |v|α−Q χ {B(e,1) (v) dV (v)
Note that the function f1 has compact (bounded) support and thus
∫
a1 = −
|v|α−Q dV (v)
B(e,2t)\B(e,min{1,2t})
is finite.
(13)
208
Vagif S. Guliyev and Yagub Y. Mammadov
Note also that
∫
∫
−1 α−Q
F1 (u) − a1 =
|v u|
f (v)dV (v) −
|v|α−Q f (v)dV (v)
B(e,2t)
B(e,2t)\B(e,min{1,2t})
∫
∫
+
|v|α−Q f (v)dV (v) =
|v −1 u|α−Q f1 (v)dV (v) = Iα f1 (u).
Hn
B(e,2t)\B(e,min{1,2t})
Therefore
∫
|F1 (u) − a1 | ≤
Hn
|v|α−Q f1 (v −1 u) dV (v) =
∫
|v|α−Q f (v −1 u) dV (v).
B(u,2t)
Further, for u ∈ B(e, t), v ∈ B(u, 2t) we have
|v| ≤ |u| + |v −1 u| < 3t.
Consequently for u ∈ B(e, t)
∫
|F1 (u) − a1 | ≤
|v|α−Q f (v −1 u) dV (v).
(14)
B(e,3t)
Then
|B(e, t)|−1
.t
−Q
∫
∫
F1 (w−1 u) − a1 dV (w)
B(e,t)
(∫
|v|
B(e,t)
.t
−Q
=t
|f (v
−1
−1
−1
w
−1
u)|dV (v) dV (w)
B(e,3t)
(∫
∫
)
|f (v
B(e,3t)
−Q
)
α−Q
∫
(∫
w
B(e,t)
u)|dV (w) |v|α−Q dV (v)
)
|f (wv)|dV (w) |v|α−Q dV (v)
B(e,3t)
B(u,t)
. t−Q tQ−α ∥f ∥Le1,Q−α
∫
|v|α−Q dV (v)
B(e,3t)
≈ ∥f ∥L1,Q−α .
(15)
∫
Denote
|v|α−Q f (v)dV (v).
a2 =
B(e,max{1,2t})\B(e,2t)
and estimate |F2 (u) − a2 | for u ∈ B(e, t):
∫
|f (v)| |v −1 u|α−Q − |v|α−Q dV (v).
|F2 (u) − a2 | ≤
{
B(e,2t)
209
Riesz potential on the Heisenberg group and modified Morrey spaces
Applying Lemma 4.3 we have
∫
|F2 (u) − a2 | ≤ 2
Q−α+1
|u|
= 2Q−α+1 |u|
|f (v)||v|α−Q−1 dV (v)
{
B(e,2t)
∞ ∫
∑
j=1
. |u|
∞
∑
(
|f (v)||v|α−Q−1 dV (v)
B(e,2j+1 t)\B(e,2j t)
)α−Q−1 ( j+1 )Q−α
2j t
2 t
∥f ∥L1,Q−α
j=1
≈ |u| t−1 ∥f ∥L1,Q−α .
Denote
(16)
∫
|v|α−Q dV (v).
af = a1 + a2 =
B(e,max{1,2t})
Finally, from (15) and (16) we have
∫
1
e
sup
Iα f (v −1 u) − af dV (v) . ∥f ∥L1,Q−α .
u,t |B(e, t)| B(e,t)
(17)
By the Hölder’s inequality we have
∫
∥f ∥L1,Q−α = sup tα−Q
|f (v)|dV (v)
u∈Hn , t>0
≤2
n
p′
B(u,t)
sup
t
α− Q
p
u∈Hn , t>0
(
λ
p
[t]1
[t]−λ
1
)1/p
∫
|f (v)| dV (v)
p
B(u,t)
n
λ
Q
≤ 2 p′ ∥f ∥Lep,λ sup tα− p [t]1p
t>0
{
}
n
Q−λ
Q
= 2 p′ ∥f ∥Lep,λ max sup tα− p , sup tα− p
0<t≤1
=2
Finally for
e Iα f BM O
Q−λ
α
n
p′
(18)
t>1
∥f ∥Lep,λ .
≤p≤
≤ 2 sup
u,t
Q
α
e p,λ (Hn ), from (17) and (18) we get
and f ∈ L
1
|B(e, t)|
∫
Thus Theorem 2.2 is proved.
B(e,t)
e
Iα f (v −1 u) − af dV (v) . ∥f ∥Lep,λ .
210
Vagif S. Guliyev and Yagub Y. Mammadov
Acknowledgment
The research of V. Guliyev was partially supported by the grant of Science
Development Foundation under the President of the Republic of Azerbaijan
project EIF-2010-1(1)-40/06-1, by the Scientific and Technological Research
Council of Turkey (TUBITAK Project No: 110T695) and by the grant of Ahi
Evran University Scientific Research Projects (BAP FBA-10-05).
References
[1] D.R. Adams, A note on Riesz potentials . Duke Math., 42 (1975), 765778.
[2] D.R. Adams, Choquet integrals in potential theory, Publ. Mat. 42 (1998),
3-66.
[3] L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano 58 (1990), 253-284.
[4] L. Capogna, D. Danielli, S. Pauls and J. Tyson, An introduction to the
Heisenberg group and the sub-Riemannian isoperimetric problem, Progr.
Math., 259, Birkhauser, Basel, 2007.
[5] F. Chiarenza, M. Frasca, Morrey spaces and Hardy–Littlewood maximal
function. Rend. Math. 7 (1987), 273-279.
[6] G. Di Fazio, D.K. Palagachev and M.A. Ragusa, Global Morrey regularity
of strong solutions to the Dirichlet problem for elliptic equations with
discontinuous coefficients, J. Funct. Anal. 166 (1999), 179-196.
[7] G.B. Folland, A fundamental solution for a subelliptic operator. Bull.
Amer. Math. Soc. 79 (1973), No.2, 373-376.
[8] G.B. Folland, Subelliptic estimates and function spaces on nilpotent Lie
groups. Ark. Mat. 13 (1975), 161-207.
[9] G.B. Folland and E.M. Stein, Hardy Spaces on Homogeneous Groups.
Mathematical Notes Vol. 28, Princeton Univ. Press, Princeton, 1982.
[10] G. Furioli, C. Melzi, and A. Veneruso, Littlewood-Paley decompositions
and Besov spaces on Lie groups of polynomial growth. Math. Nachr. 279
(2006), no. 9-10, 1028-1040.
[11] N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Atin. Inst.
Fourier, Grenoble, 40 (1990), 313-356.
Riesz potential on the Heisenberg group and modified Morrey spaces
211
[12] V.S. Guliyev, Integral operators on function spaces on the homogeneous
groups and on domains in Rn , Doctor of Sciencies, Moscow, Mat. Inst.
Steklova, (1994, Russian), 1-329.
[13] V. Guliev, Integral operators, function spaces and questions of approximation on Heisenberg groups. Baku- ELM, 1996.
[14] V.S. Guliyev, Function spaces, integral operators and two weighted inequalities on homogeneous groups. Some applications. Baku, (1999, Russian), 1-332.
[15] V.S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl., 2009, Art. ID
503948, 20 pp.
[16] S. Giulini, Approximation and Besov spaces on stratified groups. Proc.
Amer. Math. Soc. 96 (1986), No. 4, 569-578.
[17] D.S Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I., J. Funct. Anal., 43 (1981), 97142.
[18] D.S Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II., J. Funct. Anal., 43 (1981), 224257.
[19] A. Kufner, O. John and S. Fucik, Function Spaces, Noordhoff, Leyden,
and Academia, Prague, 1977.
[20] A.L. Mazzucato, Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc. 355 (2003), 12971364.
[21] A. Meskhi, Maximal functions, potentials and singular integrals in grand
Morrey spaces, Complex Var. Elliptic Eqns. (2011), 1-18.
[22] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential
equations. Trans. Amer. Math. Soc. 43 (1938), 126-166.
[23] C. Perez, Two weighted norm inequalities for Riesz potentials and uniform Lp -weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990),
3144.
[24] A. Ruiz and L. Vega, Unique continuation for Schrödinger operators with
potential in Morrey spaces, Publ. Mat. 35 (1991), 291-298.
[25] A. Ruiz and L. Vega, On local regularity of Schrödinger equations, Int.
Math. Res. Notices 1993:1 (1993), 13-27.
212
Vagif S. Guliyev and Yagub Y. Mammadov
[26] K. Saka, Besov spaces and Sobolev spaces on a nilpotent Lie group. Tohoku Math. J. 31 (1979), 383-437.
[27] Z. Shen, The periodic Schrödinger operators with potentials in the Morrey
class, J. Funct. Anal. 193 (2002), 314-345.
[28] Z. Shen, Boundary value problems in Morrey spaces for elliptic systems
on Lipschitz domains, Amer. J. Math. 125 (2003), 1079-1115.
[29] S. Spanne, Sur l’interpolation entre les espaces Lp,Φ
k , Ann. Schola Norm.
Sup. Pisa, 20 (1966), 625-648.
[30] E.M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ,
1993.
[31] L. Skrzypczak, Besov spaces and Hausdorff dimension for some CarnotCaratheodory metric spaces. Canad. J. Math. 54 (2002), 1280-1304.
[32] M.E. Taylor, Analysis on Morrey spaces and applications to NavierStokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407-1456.
[33] Jinsen Xiao and Jianxun He, Riesz potential on the Heisenberg group
Journal of Inequalities and Applications, Volume 2011 (2011), Article ID
498638, 13 pagesdoi:10.1155/2011/498638
Vagif S. Guliyev,
Ahi Evran University, Kirsehir, Turkey
and
Institute of Mathematics and Mechanics,
Baku, Azerbaijan
Email: vagif@guliyev.com
Yagub Y. Mammadov,
Institute of Mathematics and Mechanics,
Baku, Azerbaijan
and
Nakhchivan Teacher-Training Institute,
Nakhchivan, Azerbaijan
Email: yagubmammadov@yahoo.com