Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces

0.3 MB·

by  Michael Ruzhansky

HARDY-LITTLEWOOD, BESSEL-RIESZ, AND FRACTIONAL
INTEGRAL OPERATORS IN ANISOTROPIC MORREY AND
CAMPANATO SPACES
MICHAEL RUZHANSKY, DURVUDKHAN SURAGAN, AND NURGISSA YESSIRKEGENOV
Abstract. WeanalyseMorreyspaces,generalisedMorreyspacesandCampanato
spacesonhomogeneousgroups. TheboundednessoftheHardy-Littlewoodmaximal
7 operator,Bessel-Rieszoperators,generalisedBessel-Rieszoperatorsandgeneralised
1 fractional integral operators in generalised Morrey spaces on homogeneous groups
0
is shown. Moreover, we prove the boundedness of the modified version of the
2
generalised fractional integral operator and Olsen type inequalities in Campanato

n
spaces and generalised Morrey spaces on homogeneous groups, respectively. Our
a
results extend results known in the isotropic Euclidean settings, however, some of
J
them are new already in the standard Euclidean cases.
3

]
A
F
1. Introduction
.
h
t Consider the following Bessel-Riesz operators
a
m
|x−y|α−n
[ Iα,γf(x) = Kα,γ(x−y)f(y)dy = f(y)dy, (1.1)
1 ZRn ZRn (1+|x−y|)γ
v where f ∈ Lp (Rn),p ≥ 1,γ ≥ 0 and 0 < α < n. Here, I and K are called
loc α,γ α,γ
0 Bessel-Riesz operator and Bessel-Riesz kernel, respectively. The boundedness of the
5
Bessel-Riesz operators on Lebesgue spaces was shown by Hardy and Littlewood in
8
0 [HL27], [HL32] and Sobolev in [Sob38]. In the case of Rn, the Hardy-Littlewood
0
maximal operator, the Riesz potential I = I , the generalised fractional integral
. α,0 α
1
operators, which are a generalised form of the Riesz potential I = I , Bessel-Riesz
0 α,0 α
7 operatorsandOlsentypeinequalities arewidelyanalysedonLebesguespaces, Morrey
1 spaces and generalised Morrey spaces (see e.g. [Ada75], [CF87], [Nak94], [EGN04],
:
v [Eri02], [KNS99], [Nak01], [Nak02], [GE09], [SST12], [IGLE15] and [IGE16], as well
i
X as [Bur13] for a recent survey). For some of their functional analytic properties see
r also [BDN13, BNC14] and references therein.
a
In this paper we are interested in the boundedness of the Hardy-Littlewood maxi-
mal operator, Bessel-Riesz operators, generalised Bessel-Riesz operators, generalised
fractional integral operators and Olsen type inequalities in generalised Morrey spaces
on homogeneous Lie groups. The obtained results give new statements already in the
2010 Mathematics Subject Classification. 22E30, 43A80.
Key words and phrases. Fractional integral operator, generalised Morrey space, Campanato
space, Hardy-Littlewood maximal operator, Bessel-Riesz operator, Olsen type inequality, homo-
geneous Lie group.
TheauthorsweresupportedinpartsbytheEPSRCgrantEP/K039407/1andbytheLeverhulme
Grant RPG-2014-02, as well as by the MESRK grant 5127/GF4. No new data was collected or
generated during the course of research.
1
2 M. RUZHANSKY,D.SURAGAN,ANDN. YESSIRKEGENOV
Euclidean setting of Rn when we are working with anisotropic differential structure.
Furthermore, even in the isotropic situation in Rn, one novelty of all the obtained
results is also in the arbitrariness of the choice of any homogeneous quasi-norm, and
some estimates are also new in the usual isotropic structure of Rn with the Euclidean
norm, which we will be indicating at relevant places.
Thus, we could have worked directly in Rn with anisotropic structure, but since
the methods work equally well in the setting of Folland and Stein’s homogeneous
groups, we formulate all the results in such (greater) generality. In particular, it
follows the general strategy initiated by their work, of distilling results of harmonic
analysis depending only on the group and dilation structures: in this respect the
present paper shows that the harmonic analysis on Morrey spaces largely falls into
this category.
We refer to recent papers [RS16a], [RS16b], [RS16c], and [RS16d] for discussions
related to different functional inequalities with special as well as arbitrary homoge-
neous quasi-norms in different settings. Morrey spaces for non-Euclidean distances
find their applications in many problems, see e.g. [GS15a, GS15b] and [GS16].
For the convenience of the reader let us now shortly recapture the main results of
this paper.
For the definitions of the spaces appearing in the formulations below, see (3.1)
for Morrey spaces Lp,q(G), (3.2) for generalised Morrey spaces Lp,φ(G), and (7.1)
for generalised Camponato spaces Lp,φ(G), as well as (3.4) for the Hardy-Littlewood
maximaloperatorM, (2.4)forBessel-RieszoperatorsI ,(5.1)forgeneralisedBessel-
α,γ
Riesz operators I , and (6.1) for generalised fractional intergral operators T . Both
ρ,γ ρ
I and T generalise the Riesz transform and the Bessel-Riesz transform in different
ρ,γ ρ
directions.
Thus, in this paper we show that for a homogeneous group G of homogeneous
dimension Q and any homogeneous quasi-norm |·| we have the following properties:
• If 0 < α < Q and γ > 0, then K ∈ Lp1(G) for Q < p < Q , and
α,γ Q+γ−α 1 Q−α
1
(2kR)(α−Q)p1+Q p1
kKα,γkLp1(G) ∼ (1+2kR)γp1 !
k∈Z
X
for any R > 0, where K := |x|α−n .
α,γ (1+|x|)γ
• For any f ∈ Lp,φ(G) and 1 < p < ∞, we have
kMfk ≤ C kfk ,
Lp,φ(G) p Lp,φ(G)
where generalised Morrey space Lp,φ(G) and Hardy-Littlewood maximal op-
erator Mf are defined in (3.2) and (3.4), respestively.
• Let γ > 0 and 0 < α < Q. If φ(r) ≤ Crβ for every r > 0,β < −α,1 < p < ∞,
and Q < p < Q , then for all f ∈ Lp,φ(G) we have
Q+γ−α 1 Q−α
kI fk ≤ C kK k kfk ,
α,γ Lq,ψ(G) p,φ,Q α,γ Lp1(G) Lp,φ(G)
′
where q = βp1p and ψ(r) = φ(r)p/q. The Bessel-Riesz operator I on a
′ α,γ
βp +Q
1
homogenous group is defined in (2.4).
FRACTIONAL INTEGRAL OPERATORS ON HOMOGENEOUS GROUPS 3
• Let γ > 0 and 0 < α < Q. If φ(r) ≤ Crβ for every r > 0,β < −α,
Q < p ≤ p < Q and p ≥ 1, then for all f ∈ Lp,φ(G) we have
Q+γ−α 2 1 Q−α 2
kI fk ≤ C kK k kfk ,
α,γ Lq,ψ(G) p,φ,Q α,γ Lp2,p1(G) Lp,φ(G)
′
where 1 < p < ∞,q = βp1p ,ψ(r) = φ(r)p/q.
′
βp +Q
• Letω : R+ → R+ satisfyt1hedoublingconditionandassumethatω(r) ≤ Cr−α
for every r > 0, so that K ∈ Lp2,ω(G) for Q < p < Q and p ≥ 1,
α,γ Q+γ−α 2 Q−α 2
where 0 < α < Q and γ > 0. If φ(r) ≤ Crβ for every r > 0, where
β < −α < −Q−β, then for all f ∈ Lp,φ(G) we have
kIα,γfkLq,ψ(G) ≤ Cp,φ,QkKα,γkLp2,ω(G)kfkLp,φ(G),
where 1 < p < ∞,q = βp and ψ(r) = φ(r)p/q.
β+Q−α
• Let γ > 0 and let ρ and φ satisfy the doubling condition (3.3). Let 1 < p <
q < ∞. Let φ be surjective and satisfy
∞ φ(t)p
dt ≤ C φ(r)p,
1
t
Zr
and
r ρ(t) ∞ ρ(t)φ(t)
φ(r) dt+ dt ≤ C φ(r)p/q,
tγ−Q+1 tγ−Q+1 2
Z0 Zr
for all r > 0. Then we have
kI fk ≤ C kfk ,
ρ,γ Lq,φp/q(G) p,q,φ,Q Lp,φ(G)
where the generalised Bessel-Riesz operator I is defined in (5.1). This result
ρ,γ
is new already in the standard setting of Rn.
• Let ρ and φ satisfy the doubling condition (3.3). Let γ > 0, and assume that
φ is surjective and satisfies (5.3)-(5.4). Then we have
kW ·I fk ≤ C kWk kfk , 1 < p < p < ∞,
ρ,γ Lp,φ(G) p,φ,Q Lp2,φp/p2(G) Lp,φ(G) 2
provided that W ∈ Lp2,φp/p2(G). This result is new even in the Euclidean
cases.
• Let ρ and φ satisfy the doubling condition (3.3). Let 1 < p < q < ∞. Let φ
be surjective and satisfy
∞ φ(t)p
dt ≤ C φ(r)p,
1
t
Zr
and
r ρ(t) ∞ ρ(t)φ(t)
φ(r) dt+ dt ≤ C φ(r)p/q,
2
t t
Z0 Zr
for all r > 0. Then we have
kT fk ≤ C kfk ,
ρ Lq,φp/q(G) p,q,φ,Q Lp,φ(G)
where the generalised fractional integral operator T is defined in (6.1).
ρ
4 M. RUZHANSKY,D.SURAGAN,ANDN. YESSIRKEGENOV
• Let ρ and φ satisfy the doubling condition (3.3). Let φ be surjective and
satisfy (6.3)-(6.4). Then we have
kW ·T fk ≤ C kWk kfk , 1 < p < p < ∞,
ρ Lp,φ(G) p,φ,Q Lp2,φp/p2(G) Lp,φ(G) 2
provided that W ∈ Lp2,φp/p2(G).
• Letω : R+ → R+ satisfythedoublingconditionandassumethatω(r) ≤ Cr−α
for every r > 0, so that K ∈ Lp2,ω(G) for Q < p < Q and p ≥ 1,
α,γ Q+γ−α 2 Q−α 2
where 0 < α < Q, 1 < p < ∞,q = βp and γ > 0. If φ(r) ≤ Crβ for every
β+Q−α
r > 0, where β < −α < −Q−β, then we have
kW ·I fk ≤ C kWk kfk ,
α,γ Lp,φ(G) p,φ,Q Lp2,φp/p2(G) Lp,φ(G)
provided that W ∈ Lp2,φp/p2(G), where 1 = 1 − 1. This result is new already
in the Euclidean setting of Rn. p2 p q
• Let ρ satisfy (6.2), (3.3), (7.3), (7.4), and let φ satisfy the doubling condition
(3.3) and ∞ φ(t)dt < ∞. If
1 t
∞ φ(t)R r ρ(t) ∞ ρ(t)φ(t)
dt dt+r dt ≤ C ψ(r) for all r > 0,
t t t2 3
Zr Z0 Zr
then we have
kT fk ≤ C kfk , 1 < p < ∞,
ρ Lp,ψ(G) p,φ,Q Lp,φ(G)
where the generalised Campanato space Lp,ψ(G) and operator T are defined
e ρ
in (7.1) and (7.2), respectively.
This paper is structured as follows. In Section 2 we briefly recall the coencepts of ho-
mogeneous groups and fix the notation. The boundedness of the Hardy-Littlewood
maximal operator and Bessel-Riesz operators in generalised Morrey spaces on homo-
geneous groups is proved in Section 3 and in Section 4, respectively. In Section 5
we prove the boundedness of the generalised Bessel-Riesz operators and Olsen type
inequality for these operators in generalised Morrey spaces on homogeneous groups.
The boundedness of the generalised fractional integral operators and Olsen type in-
equality for these operators in generalised Morrey spaces on homogeneous groups
are proved in Section 6. Finally, in Section 7 we investigate the boundedness of the
modified version of the generalised fractional integral operator in Campanato spaces
on homogeneous groups.
2. Preliminaries
A connected simply connected Lie group G is called a homogeneous group if its Lie
algebra g is equipped with a family of dilations:
∞
1
D = Exp(Alnλ) = (ln(λ)A)k,
λ
k!
k=0
X
where A is a diagonalisable positive linear operator on g, and each D is a morphism
λ
of g, that is,
∀X,Y ∈ g, λ > 0, [D X,D Y] = D [X,Y].
λ λ λ
FRACTIONAL INTEGRAL OPERATORS ON HOMOGENEOUS GROUPS 5
The exponential mapping exp : g → G is a global diffeomorphism and gives the
G
dilation structure, which is denoted by D x or just by λx, on G.
λ
Then we have
|D (S)| = λQ|S| and f(λx)dx = λ−Q f(x)dx, (2.1)
λ
G G
Z Z
where dx is the Haar measure on G, |S| is the volume of a measurable set S ⊂ G
and Q := TrA is the homogeneous dimension of G. Recall that the Haar measure on
a homogeneous group G is the standard Lebesgue measure for Rn (see e.g. [FR16,
Proposition 1.6.6]).
Let |·| be a homogeneous quasi-norm on G. We will denote the quasi-ball centred
at x ∈ G with radius R > 0 by
B(x,R) := {y ∈ G : |x−1y| < R}
and we will also use the notation
Bc(x,R) := {y ∈ G : |x−1y| ≥ R}.
Theproofofthefollowingimportant polardecompositiononhomogeneousLiegroups
was given by Folland and Stein [FS82], which can be also found in [FR16, Section
3.1.7]: there is a (unique) positive Borel measure σ on the unit sphere
S := {x ∈ G : |x| = 1}, (2.2)
so that for any f ∈ L1(G), one has
∞
f(x)dx = f(ry)rQ−1dσ(y)dr. (2.3)
ZG Z0 ZS
Now, for any f ∈ Lp (G), p ≥ 1 and γ ≥ 0, 0 < α < Q, we shall define the
loc
Bessel-Riesz operators on homogeneous groups by
|xy−1|α−Q
I f(x) := K (xy−1)f(y)dy = f(y)dy, (2.4)
α,γ α,γ
(1+|xy−1|)γ
G G
Z Z
where|·|isanyhomogeneousquasi-norm. Here, K istheBessel-Riesz kernel. Here-
α,γ
after, C, C , C , C and C are positive constants, which are not necessarily
i p p,φ,Q p,q,φ,Q
the same from line to line.
Let us recall the following result, which will be used in the sequel.
Lemma 2.1 ([IGLE15]). If b > a > 0 then (ukR)a < ∞, for every u > 1 and
(1+ukR)b
k∈Z
R > 0. P
We now calculate the Lp-norms of the Bessel-Riesz kernel.
Theorem 2.2. Let G be a homogeneous group of homogeneous dimension Q. Let |·|
be a homogeneous quasi-norm. Let K (x) = |x|α−Q . If 0 < α < Q and γ > 0 then
α,γ (1+|x|)γ
K ∈ Lp1(G) and
α,γ
1
(2kR)(α−Q)p1+Q p1
kK k ∼ ,
α,γ Lp1(G) (1+2kR)γp1 !
k∈Z
X
for Q < p < Q .
Q+γ−α 1 Q−α
6 M. RUZHANSKY,D.SURAGAN,ANDN. YESSIRKEGENOV
Proof of Theorem 2.2. Introducing polar coordinates (r,y) = (|x|, x ) ∈ (0,∞)×S
|x|
on G, where S is the sphere as in (2.2), and using (2.3) for any R > 0, we have
|x|(α−Q)p1
|K (x)|p1dx = dx
α,γ
G G (1+|x|)γp1
Z Z
∞ r(α−Q)p1+Q−1 r(α−Q)p1+Q−1
= dσ(y)dr = |σ| dr,
Z0 ZS (1+r)γp1 k∈ZZ2kR≤r<2k+1R (1+r)γp1
X
where |σ| is the Q−1 dimensional surface measure of the unit sphere.
Then it follows that
1
|K (x)|p1dx ≤ |σ| r(α−Q)p1+Q−1dr
α,γ
ZG k∈Z (1+2kR)γp1 Z2kR≤r<2k+1R
X
|σ|(2(α−Q)p1+Q −1) (2kR)(α−Q)p1+Q
= .
(α−Q)p1 +Q k∈Z (1+2kR)γp1
X
On the other hand, we obtain
|σ| 1
|K (x)|p1dx ≥ r(α−Q)p1+Q−1dr
ZG α,γ 2γp1 k∈Z (1+2kR)γp1 Z2kR≤r<2k+1R
X
|σ|(2(α−Q)p1+Q −1) (2kR)(α−Q)p1+Q
= .
2γp1((α−Q)p1 +Q) k∈Z (1+2kR)γp1
X
Therefore, for every R > 0 we arrive at
(2kR)(α−Q)p1+Q
|K (x)|p1dx ∼ .
G α,γ (1+2kR)γp1
Z k∈Z
X
For p ∈ Q , Q using Lemma 2.1 with u = 2,a = (α−Q)p +Q,b = γp , we
1 Q+γ−α Q−α 1 1
obtain (cid:16) (2kR)(α−Q)(cid:17)p1+Q < ∞ which implies K ∈ Lp1(G). (cid:3)
k∈Z (1+2kR)γp1 α,γ
The fPollowing is well-known on homogeneous groups, see e.g. [FR16, Proposition
1.5.2].
Proposition 2.3 (Young’s inequality). Let G be a homogeneous group. Suppose
1 ≤ p,q,p ≤ ∞ and 1 +1 = 1 + 1 . If f ∈ Lp(G) and g ∈ Lp1(G) then
1 q p p1
kg ∗fk ≤ kfk kgk .
Lq(G) Lp(G) Lp1(G)
In view of Proposition 2.3 and taking into account the definition of Bessel-Riesz
operator 2.4, we immediately get the following Corollary 2.4:
Corollary 2.4. Let G be a homogeneous group of homogeneous dimension Q. Let
|·| be a homogeneous quasi-norm. Then for 0 < α < Q,γ > 0, we have
kIα,γfkLq(G) ≤ kKα,γkLp1(G)kfkLp(G)
for every f ∈ Lp(G) where 1 ≤ p,q,p ≤ ∞, 1 +1 = 1 + 1 and Q < p < Q .
1 q p p1 Q+γ−α 1 Q−α
Corollary 2.4 shows that the I is bounded from Lp(G) to Lq(G) and
α,γ
kIα,γkLp(G)→Lq(G) ≤ kKα,γkLp1(G).
FRACTIONAL INTEGRAL OPERATORS ON HOMOGENEOUS GROUPS 7
3. The boundedness of Hardy-Littlewood maximal operator in
generalised Morrey spaces
In this section we define Morrey and generalised Morrey spaces on homogeneous
groups. Then we prove that the Hardy-Littlewood maximal operator is bounded in
these spaces. Notethat in theisotropic Abelian case the result was obtainedby Nakai
[Nak94]. Let G be a homogeneous group of homogeneous dimension Q.
Let us define the Morrey spaces Lp,q(G) by
Lp,q(G) := {f ∈ Lploc(G) : kfkLp,q(G) < ∞}, 1 ≤ p ≤ q, (3.1)
where
1/p
kfk := suprQ(1/q−1/p) |f(x)|pdx .
Lp,q(G)
r>0 (cid:18)ZB(0,r) (cid:19)
Next, for a function φ : R+ → R+ and 1 ≤ p < ∞, we define the generalised Morrey
space Lp,φ(G) by
Lp,φ(G) := {f ∈ Lp (G) : kfk < ∞}, (3.2)
loc Lp,φ(G)
where
1/p
1 1
kfk := sup |f(x)|pdx .
Lp,φ(G) φ(r) rQ
r>0 (cid:18) ZB(0,r) (cid:19)
Here we assume that φ is nonincreasing and tQ/pφ(t) is nondecreasing, so that φ
satisfies the doubling condition, i.e. there exists a constant C > 0 such that
1
1 r 1 ρ(r)
≤ ≤ 2 =⇒ ≤ ≤ C . (3.3)
1
2 s C ρ(s)
1
Now, for every f ∈ Lp (G), we define the Hardy-Littlewood maximal operator M by
loc
1
Mf(x) := sup |f(y)|dy, x ∈ G, (3.4)
|B(0,r)|
x∈B ZB(0,r)
where |B(0,r)| denotes the Haar measure of the ball B = B(0,r).
Using the definition of Morrey spaces (3.1), one can readily obtain the following
Lemma 3.1:
Lemma 3.1. Let G be a homogeneous group of homogeneous dimension Q. Let |·|
be a homogeneous quasi-norm. Then
kKα,γkLp2,p1(G) ≤ kKα,γkLp1,p1(G) = kKα,γkLp1(G), (3.5)
where 1 ≤ p ≤ p and Q < p < Q .
2 1 Q+γ−α 1 Q−α
We now prove the boundedness of the Hardy-Littlewood maximal operator on
generalised Morrey spaces.
Theorem 3.2. Let G be a homogeneous group. For any f ∈ Lp,φ(G) and 1 < p < ∞,
we have
kMfk ≤ C kfk . (3.6)
Lp,φ(G) p Lp,φ(G)
8 M. RUZHANSKY,D.SURAGAN,ANDN. YESSIRKEGENOV
Proof of Theorem 3.6. By the definition of the norm of the generalised Morrey space
(3.2), we have
1/p
1 1
kfk = sup |f(x)|pdx .
Lp,φ(G) φ(r) rQ
r>0 (cid:18) ZB(0,r) (cid:19)
This implies that
1/p
|f(x)|pdx ≤ φ(r)rQpkfkLp,φ(G), (3.7)
(cid:18)ZB(0,r) (cid:19)
for any r > 0.
On the other hand, using Corollary 2.5 (b) from Folland and Stein [FS82] we have
1/p 1/p
|Mf(x)|pdx ≤ C |f(x)|pdx . (3.8)
p
(cid:18)ZB(0,r) (cid:19) (cid:18)ZB(0,r) (cid:19)
Combining (3.7) and (3.8) we arrive at
1/p
1 1
|Mf(x)|pdx ≤ C kfk ,
φ(r) rQ p Lp,φ(G)
(cid:18) ZB(0,r) (cid:19)
for all r > 0. Thus
kMfk ≤ C kfk ,
Lp,φ(G) p Lp,φ(G)
completing the proof. (cid:3)
4. Inequalities for Bessel-Riesz operators on generalised Morrey
spaces
In this section, we prove the boundedness of the Bessel-Riesz operators on gener-
alised Morrey spaces (3.2).
Theorem 4.1. Let G be a homogeneous group of homogeneous dimension Q. Let |·|
be a homogeneous quasi-norm. Let γ > 0 and 0 < α < Q. If φ(r) ≤ Crβ for every
r > 0,β < −α,1 < p < ∞, and Q < p < Q , then for all f ∈ Lp,φ(G) we have
Q+γ−α 1 Q−α
kI fk ≤ C kK k kfk , (4.1)
α,γ Lq,ψ(G) p,φ,Q α,γ Lp1(G) Lp,φ(G)
′
where q = βp1p and ψ(r) = φ(r)p/q.
′
βp +Q
1
Proof of Theorem 4.1. For every f ∈ Lp,φ(G), let us write I f(x) in the form
α,γ
I f(x) := I (x)+I (x),
α,γ 1 2
where I (x) := |xy−1|α−Qf(y)dy and I (x) := |xy−1|α−Qf(y)dy, for some
1 B(x,R) (1+|xy−1|)γ 2 Bc(x,R) (1+|xy−1|)γ
R > 0.
R R
FRACTIONAL INTEGRAL OPERATORS ON HOMOGENEOUS GROUPS 9
By using dyadic decomposition for I , we obtain
1
−1 |xy−1|α−Q|f(y)|
|I (x)| ≤ dy
1 (1+|xy−1|)γ
k=−∞Z2kR≤|xy−1|<2k+1R
X
−1 (2kR)α−Q
≤ |f(y)|dy
(1+2kR)γ
k=−∞ Z2kR≤|xy−1|<2k+1R
X
−1 (2kR)α−Q+Q/p1(2kR)Q/p′1
≤ CMf(x) .
(1+2kR)γ
k=−∞
X
From this using H¨older inequality for 1 + 1 = 1 we get
′
p1 p1
′
−1 (2kR)(α−Q)p1+Q 1/p1 −1 1/p1
|I (x)| ≤ CMf(x) (2kR)Q .
1 (1+2kR)γp1 ! !
k=−∞ k=−∞
X X
Since
−1 (2kR)(α−Q)p1+Q 1/p1 (2kR)(α−Q)p1+Q 1/p1
≤ ∼ kK k , (4.2)
(1+2kR)γp1 ! (1+2kR)γp1 ! α,γ Lp1(G)
k=−∞ k∈Z
X X
we arrive at
′
|I1(x)| ≤ CkKα,γkLp1(G)Mf(x)RQ/p1. (4.3)
For the second term I , by using H¨older inequality for 1 + 1 = 1 we obtain that
2 p p′
∞ (2kR)α−Q
|I (x)| ≤ |f(y)|dy
2 (1+2kR)γ
k=0 Z2kR≤|xy−1|<2k+1R
X
′
∞ (2kR)α−Q 1/p 1/p
≤ dy |f(y)|pdy
(1+2kR)γ
k=0 (cid:18)Z2kR≤|xy−1|<2k+1R (cid:19) (cid:18)Z2kR≤|xy−1|<2k+1R (cid:19)
X
′
∞ (2kR)α−Q 2k+1R 1/p 1/p
= rQ−1dσ(y)dr |f(y)|pdy
(1+2kR)γ
k=0 Z2kR ZS ! (cid:18)Z2kR≤|xy−1|<2k+1R (cid:19)
X
∞ (2kR)α−Q ′ 1/p
≤ C (2kR)Q/p |f(y)|pdy .
(1+2kR)γ
k=0 (cid:18)Z2kR≤|xy−1|<2k+1R (cid:19)
X
This implies that
∞ (2kR)α−Q+Q/p1 ′
|I2(x)| ≤ CkfkLp,φ(G) (1+2kR)γ φ(2kR)(2kR)Q/p1.
k=0
X
Since φ(r) ≤ Crβ, we write
∞ (2kR)α−Q+Q/p1 ′
|I2(x)| ≤ CkfkLp,φ(G) (1+2kR)γ (2kR)β+Q/p1.
k=0
X
10 M. RUZHANSKY,D.SURAGAN,ANDN. YESSIRKEGENOV
Applying H¨older inequality again, we get
′
∞ (2kR)(α−Q)p1+Q 1/p1 ∞ ′ 1/p1
|I2(x)| ≤ CkfkLp,φ(G) (1+2kR)γp1 ! (2kR)βp1+Q! .
k=0 k=0
X X
From the conditions p < Q and β < −α, we have βp′ +Q < 0. By Theorem 2.2,
1 Q−α 1
we also have
∞ (2kR)(α−Q)p1+Q 1/p1 (2kR)(α−Q)p1+Q 1/p1
(1+2kR)γp1 ! ≤ (1+2kR)γp1 ! ∼ kKα,γkLp1(G).
k=0 k∈Z
X X
Using these, we arrive at
′
|I2(x)| ≤ CkKα,γkLp1(G)kfkLp,φ(G)RQ/p1+β. (4.4)
Summing up the estimates (4.3) and (4.4), we obtain
′ ′
|Iα,γf(x)| ≤ CkKα,γkLp1(G)(Mf(x)RQ/p1 +kfkLp,φ(G)RQ/p1+β).
Assuming that f is not identically 0 and that Mf is finite everywhere, we can choose
R > 0 such that Rβ = Mf(x) , that is,
kfkLp,φ(G)
|Iα,γf(x)| ≤ CkKα,γkLp1(G)kfkL−pβ,Qφp′1(G)(Mf(x))1+βQp′1,
′
for every x ∈ G. Setting q = βp1p , for any r > 0 we get
′
βp +Q
1
1 1/q
q
|Iα,γf(x)|qdx ≤ CkKα,γkLp1(G)kfk1L−p,pφ/(qG) |Mf(x)|pdx .
(cid:18)Z|x|<r (cid:19) (cid:18)Z|x|<r (cid:19)
Then we divide both sides by φ(r)p/qrQ/q to get
1 1/q
|I f(x)|qdx q |Mf(x)|pdx
|x|<r α,γ 1−p/q |x|<r
≤ CkK k kfk ,
(cid:16)R ψ(r)rQ/q (cid:17) α,γ Lp1(G) Lp,φ(G)(cid:16)R φ(r)p/qrQ/q (cid:17)
where ψ(r) = φ(r)p/q. Now by taking the supremum over r > 0, we obtain that
1−p/q p/q
kI fk ≤ CkK k kfk kMfk ,
α,γ Lq,ψ(G) α,γ Lp1(G) Lp,φ(G) Lp,φ(G)
which gives (4.1), after applying estimate (3.6). (cid:3)
Lemma 3.1 gives the property that the Bessel-Riesz kernel belongs to Morrey
spaces, which will be used in the next theorem.
Theorem 4.2. Let G be a homogeneous group of homogeneous dimension Q. Let |·|
be a homogeneous quasi-norm. Let γ > 0 and 0 < α < Q. If φ(r) ≤ Crβ for every
r > 0,β < −α, Q < p ≤ p < Q and p ≥ 1, then for all f ∈ Lp,φ(G) we have
Q+γ−α 2 1 Q−α 2
kIα,γfkLq,ψ(G) ≤ Cp,φ,QkKα,γkLp2,p1(G)kfkLp,φ(G), (4.5)
′
where 1 < p < ∞,q = βp1p ,ψ(r) = φ(r)p/q.
′
βp +Q
1