Volume 2009, Article ID 864191,13pages doi:10.1155/2009/864191
Research Article
Meda Inequality for Rearrangements of
the Convolution on the Heisenberg Group and
Some Applications
V. S. Guliyev,
1A. Serbetci,
2E. G ¨uner,
2and S. Balcı
31Department of Mathematical Analysis, Institute of Mathematics and Mechanics,
AZ1145 Baku, Azerbaijan
2Department of Mathematics, Ankara University, 06100 Ankara, Turkey 3Department of Mathematics, Istanbul Aydin University, 34295 Istanbul, Turkey
Correspondence should be addressed to A. Serbetci,serbetci@science.ankara.edu.tr
Received 13 May 2008; Revised 6 January 2009; Accepted 24 February 2009
Recommended by Yeol Je Cho
The Meda inequality for rearrangements of the convolution operator on the Heisenberg groupHn is proved. By using the Meda inequality, an O’Neil-type inequality for the convolution is obtained. As applications of these results, some sufficient and necessary conditions for the boundedness of the fractional maximal operatorMΩ,αand fractional integral operatorIΩ,αwith rough kernels in
the spacesLpHnare found. Finally, we give some comments on the extension of our results to the case of homogeneous groups.
Copyrightq2009 V. S. Guliyev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
LetHnCn×Rbe the 2n1-dimensional Heisenberg group. We define the Lebesgue space onHnby
LpHn
⎧ ⎨
⎩f :fp≡
Hn fup
du 1/p
<∞ ⎫ ⎬
⎭, 1≤p <∞, 1.1
and ifp ∞, thenL∞Hn {f : fL
∞ ess supu∈Hn|fu|< ∞}.The convolution of the functionsf, g∈L1Hnis defined by
f∗gu
Hn
fvgv−1udv
Hn
LetΩ∈LsSH,s≥1, andΩbe homogeneous of degree zero onHn, and let 0< α < Q, whereSH Se,1is the unite sphere centered at the identity ofHnwith radius 1, andQ 2n2 is the homogeneous dimension ofHn. We define the fractional maximal function with rough kernel by
MΩ,αfu sup r>0
1 rQ−α
Be,r|Ωv|
fv−1udv, 1.3
and the fractional integral with rough kernel by
IΩ,αfu
Hn Ωv
|v|Q−αf
v−1udv. 1.4
It is clear that, whenΩ≡1,MΩ,αandIΩ,αare the usual fractional maximal operatorMαand
the Riesz potentialIαsee, e.g.,1–5 , respectively.
In this paper, we obtain the Meda inequality for rearrangements of the convolution defined on the Heisenberg group Hn. By using this inequality we get an O’Neil-type inequality for the convolution on Hn. As applications of these inequalities, we find the necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and fractional integral operator with rough kernels from the spacesLpHn
toLqHn, 1< p < q <∞, and from the spacesL1Hnto the weak spacesWLqHn, 1< q <∞.
We also show that the conditions on the parameters ensuring the boundedness cannot be weakened for the fractional maximal operator and fractional integral operator with rough kernels. Finally, we extend our results to the case of homogeneous groups.
2. Preliminaries
LetHnbe the 2n1-dimensional Heisenberg group. That is,HnCn×R, with multiplication
z, t·w, s zw, ts2Imz·w, 2.1
wherez·w nj1zjwj. The inverse element ofu z, tisu−1 −z,−t, and we write the
identity ofHnase 0,0. The Heisenberg group is a connected, simply connected nilpotent Lie group. We define one-parameter dilations onHn, forr >0, byδrz, t rz, r2t. These
dilations are group automorphisms, and the Jacobian determinant isrQ, whereQ2n2 is
the homogeneous dimension ofHn. A homogeneous norm onHnis given by
|z, t| |z|2|t|1/2. 2.2
With this norm, we define the Heisenberg ball centered atu z, twith radiusrbyBu, r
Using coordinatesu z, t xiy, tfor points inHn, the left-invariant vector fields Xj,Yj, andTonHnequal to∂/∂xj,∂/∂yj, and∂/∂tat the origin are given by
Xj
∂ ∂xj
2yj
∂
∂t, Yj ∂ ∂yj −
2xj
∂
∂t, T ∂
∂t, 2.3
respectively. These 2n1 vector fields form a basis for the Lie algebra ofHnwith commutation relations
Yj, Xj
4T 2.4
forj1, . . . , n, and all other commutators equal to 0. It is easy to check that Lebesgue measure dudzdtis a Haar measure onHnsee, e.g.,6 .
Letf :Hn → Rbe a measurable function. We define rearrangement offin decreasing order by
f∗t infs >0 :f∗s≤t, ∀t∈0,∞, 2.5
wheref∗tdenotes the distribution function offgiven by
f∗t u∈Hn:fu> t, 2.6
and|A|denotes the Haar measure of any measurable subsetA⊂Hn.
We note the following properties of rearrangement of functionssee1,7,8 :
1if 0< p <∞, then
Hn
fupdu ∞
0
f∗tp dt; 2.7
2for anyt >0,
sup
|E|t
E
fudu t
0
f∗sds; 2.8
3
Hn
fugudu≤∞
0
f∗tg∗tdt. 2.9
If 1< p <∞, the following inequality is valid: f∗∗L
p0,∞≤p
f∗L
p0,∞, 2.10
wherep p/p−1.
We denote byWLpHn the weakLpHn space of all measurable functionsf with
finite norm
f
WLp sup
t>0
tf∗t1/p, 1≤p <∞. 2.11
3. Meda Inequality for Rearrangements of the Convolution on the
Heisenberg Group
H
nFor the convolution
Kα∗f
x
Rn Kα
x−yfydy, 3.1
Meda9 proved the following pointwise rearrangement estimate:
Kα∗f∗∗t≤C
δαf∗∗t δα−n/pf p
, δ >0, 3.2
and gave a new proof of the Hardy-Littlewood-Sobolev theorem forKα∗f by using this
inequality, whereKα∈WLn/n−αRn, 0< α < n, andf∈LpRn, 1< p < n/α.
In this section we use the same notation as in the previous sections. We prove the Meda inequality for rearrangements of the convolution on the Heisenberg groupHn.
Theorem 3.1. Letg∈WLrHn,1< r <∞,f∈LpHn,1≤p < r
. Then for anyδ >0
f∗g∗∗t≤C1δQ/r
f∗∗t C2δQ/r
−Q/pf
p, 3.3
where C1 B1grWLr, B1 2r2r−1−1−1, C2 B2gr/p
WLr, B2 1 forp 1, and B2 22p−r −1−1/p
for1< p < r.
Proof. SupposeFδ{v∈Hn:|gv| ≥δ−Q/r},then
f∗g u≤
Fδ
Hn\Fδ
fuv−1gvdvD1u D2u. 3.4
Suppose thatFδ
∞
j1F
δ,j,where
Then from equality2.7we get
1
|E|
E
D1udu 1
|E|
E
Fδ
gvfuv−1dv du
≤∞
j1 2jδ−Q/r
Fδ,j
1
|E|
E
fuv−1du dv
≤δ−Q/rf∗∗t∞j12j
Fδ,j
dv
≤2rδ−Q/rf∗∗tgrWL r
∞
j1
2j2jδ−Q/r−r
C1δQ/r
f∗∗t,
3.6
whereC12r2r−1−1−1grWLr. Thus
1
|E|
E
D1udu≤C1δQ/r
f∗∗t. 3.7
Letp1. We have
|D2u| ≤fu·1sup Hn\Fδ
gv≤f
1δ−Q/r. 3.8
Let now 1< p < r. By using H ¨older inequality we get
|D2u| ≤fu·p
Hn\Fδ
gvp
dv 1/p
≤fp
Hn\Fδ
gvp
dv 1/p
.
3.9
Sinceg ∈WLrHnandHn\Fδ
∞
j1Bδ,j,where
Bδ,j
then
Hn\Fδ
gvp
dv
∞
j1
Bδ,j
gvp
dv
≤∞
j1
2−j1δ−Q/rp
{v∈Hn:|gv|≥2−jδ−Q/r} dv
≤2pδ−Qp/rgrWL r
∞
j1
2−jp2−jδ−Q/r−r
2pδQ−Qp/rgrWL r
∞
j1 2−jp−r
Cp
2 δ
Q−Q/rp,
3.11
whereC222p
−r−1−1/p
PgPr/p
WLr. Hence
|D2u| ≤C2fpδQ/r
−Q/p. 3.12
Thus
1
|E|
E
f∗gudu≤C1δQ/r
f∗∗t C2δQ/r
−Q/p
fp. 3.13
Hence from equality2.8we get3.3, and thereforeTheorem 3.1is proved.
Corollary 3.2. LetΩbe homogeneous of degree zero onHn,Ω ∈LQ/Q−αSH,0 < α < Q, and let f∈LpHn,1≤p < Q/α. Then for anyδ >0the following inequality holds:
IΩ,αf
∗∗
t≤C3δαf∗∗t C4δα−Q/pfp, 3.14
where C3 B3A/Q, C4 B4A/Q1/p
, B3 2Q/Q−α2α/Q−α−1− 1
, B4
22p−Q/Q−α−1−1/p for1< p < Q/α, andB
41forp1and
AΩQ/L Q−α
Q/Q−α , ΩLQ/Q−α
SH
|Ωu|Q/Q−αdu
Q−α/Q
. 3.15
Proof. If we takegu Ωu/|u|Q−α, 0< α < Q, andr Q/Q−αinTheorem 3.1, then the proof ofCorollary 3.2is straightforward, whereΩis homogeneous of degree zero onHnand Ω∈LQ/Q−αSH. In this case
g∗t A/Qt−Q/Q−α, g∗t A/Qt1−α/Q. 3.16
Thereforeg∈WLQ/Q−αHnandgWLQ/Q−α A/Q
Corollary 3.3. Letf ∈LpHn,1≤p < Q/α,0< α < Q,then for the Riesz potential
Iαfu
Hn
v−1uα−Qfvdv 3.17
for all0< t <∞the following inequality holds:
Iαf
∗∗
t≤C5δαf∗∗t C6δα−Q/pfp, 3.18
whereC5CQB3andC6 C1/p
Q B4for1≤p < Q/α.
Proof. By the same argument inCorollary 3.2if we take
gu |u|α−Q∈WLQ/Q−αHn, 0< α < Q, 3.19
inTheorem 3.1, we easily get the proof of the Corollary. In this case
g∗t CQt−Q/Q−α, g∗t
CQt−1
1−α/Q
, gWL
Q/Q−α C
1−α/Q Q ,
3.20
whereCQis the volume of the unit ballB1.
Note that, the following estimate
MΩ,αfu≤I|Ω|,αfu 3.21
is valid. Indeed, for allr >0 we have
I|Ω|,αfu≥
Be,r
|Ωv| yQ−α
fv−1udv
≥ 1
rQ−α
Be,r|Ωv|
fv−1udv.
3.22
Taking supremum over allr >0, we get3.21.
FromCorollary 3.2and inequality3.21we get the following
Corollary 3.4. Suppose thatΩis homogeneous of degree zero onHn,Ω∈LQ/Q−αSH,0< α < Q and letf∈LpHn, 1≤p < Q/α. Then for allδ >0the following inequality holds:
MΩ,αf
∗∗t≤C
4. O’Neil-Type Inequality for the Convolution on
H
nand
Some Applications
In this section we prove an O’Neil-type inequality see 10 for the convolution on the Heisenberg group Hn. With the help of this inequality, we obtain some sufficient and necessary conditions for the boundedness of the fractional maximal operator MΩ,α and
fractional integral operatorIΩ,αwith rough kernels in the spacesLpHn.
Theorem 4.1. 1Letg ∈WLrHn,1 < r <∞,f ∈LpHn,1< p < r
, and1/p−1/q1/r. Thenf∗g ∈LqHnand
f∗g
q ≤2B
1−p/r
1 B
p/r
2 p
p/qgWL
rfp. 4.1
2Letf∈L1Hn,g∈WLqHn, and1< q <∞. Thenf∗g∈WLqHnand
f∗gWL q≤2B
1/q
1 gWLqf1. 4.2
Proof. Step 1.g∈WLrHn, 1< r <∞,f ∈LpHn, 1< p < r
, and 1/p−1/q1/r. If we take
δ
C1f∗∗t C2fp
−p/Q
4.3
in3.3, then we get
f∗g∗∗t≤2C1f∗∗t1−p/r
C2fpp/r
2C1−p/r
1 C
p/r
2 f∗∗t
p/qf1−p/q p
2B1−p/r
1 B
p/r
2 gWLrf
1−p/q p f∗∗t
p/q.
4.4
Thus
f∗g
q≤f∗g∗∗tLq0,∞
≤2B1−p/r
1 B
p/r
2 gWLrf
1−p/q p
f∗∗p/q
Lq0,∞
2B1−p/r
1 B
p/r
2 gWLrf
1−p/q p f∗∗
p/q Lp0,∞
≤2B1−p/r
1 B
p/r
2 p
p/q gWL rfp.
4.5
We take
δ
C1f∗∗t C2fp
−1/Q
4.6
in3.3, then we get
f∗g∗∗t≤2B11/qgWL qf
1/q
1 f∗∗t
1/q. 4.7
Thus
f∗g
WLqsup
t>0
t1/qf∗g∗t
≤2B11/qgWL qf
1/q
1 sup
t>0
t1/qf∗∗t1/q
2B11/qgWL qf
1/q
1 sup
t>0
t
0
f∗sds 1/q
≤2B11/qgWL qf
1/q
1 f∗ 1/q L10,∞
2B11/qgWL qf1,
4.8
and therefore the proof ofTheorem 4.1is completed.
Corollary 4.2. LetΩbe homogeneous of degree zero onHnandΩ∈LQ/Q−αSH,0< α < Q. 1If1< p < Q/α,f∈LpHnand1/p−1/qα/Q, thenMΩ,αf,IΩ,αf∈LqHnand
MΩ,αf
q≤IΩ,αfq≤2A/Q
1−α/QB1−αp/Q
3 B
αp/Q
4 p
p/qfp. 4.9
2Iff∈L1Hnand1−1/qα/Q, thenMΩ,αf,IΩ,αf ∈WLqHnand
MΩ,αfWLq ≤IΩ,αfWLq ≤2A/Q1−α/QB31/qf1. 4.10
Corollary 4.31 . Let0< α < Q.
1If1< p < Q/α,f∈LpHnand1/p−1/qα/Q, thenMαf, Iαf∈LqHnand
Mαf
q≤Iαfq ≤2C
1−α/Q
Q B
1−αp/Q
3 B
αp/Q
4 p
p/qfp. 4.11
2Iff∈L1Hnand1−1/qα/Q, thenMαf, Iαf∈WLqHnand
In the following we give necessary and sufficient conditions for the Lp, Lq
-bound-edness of the fractional integral operatorIΩ,αwith rough kernel onHn.
Theorem 4.4. LetΩbe homogeneous of degree zero onHnandΩ∈LQ/Q−αSH,0< α < Q.
1If1 < p < Q/α, then the condition1/p−1/q α/Qis necessary and sufficient for the boundedness ofIΩ,αfromLpHntoLqHn.
2Ifp1, then the condition1−1/qα/Qis necessary and sufficient for the boundedness ofIΩ,αfromL1HntoWLqHn.
Proof. Sufficiency ofTheorem 4.4follows fromCorollary 4.2.
Necessity
1Suppose that the operatorIΩ,αis bounded fromLpHntoLqHnand 1< p < Q/α.
Defineftu :fδtufort >0. Then it can be easily shown that
ft
pt−Q/pfp,
IΩ,αft
u t−αIΩ,αfδtu,
IΩ,αftq t−α−Q/qIΩ,αfq.
4.13
Since the operatorIΩ,αis bounded fromLpHntoLqHn, we have
IΩ,αf
q ≤Cfp, 4.14
whereCis independent off. Then we get
IΩ,αf
qtαQ/qIΩ,αftq≤CtαQ/qftpCtαQ/q−Q/pfp. 4.15
If 1/p <1/qα/Q,then for allf ∈LpHnwe haveIΩ,αfLq0 ast → 0.
As well as if 1/p >1/qα/Q, then for allf ∈LpHnwe haveIΩ,αfq 0 ast → ∞.
Therefore we get 1/p1/qα/Q.
2Suppose that the operatorIΩ,αis bounded fromL1HntoWLqHn. It is easy to show that
ft1t−Qf1,
IΩ,αft
u t−αIΩ,αf
δtu,
IΩ,αftWLq t−α−Q/qIΩ,αfWLq.
4.16
By the boundedness ofIΩ,αfromL1HntoWLqHn, we have
IΩ,αf
whereCis independent off. Then we have
IΩ,αft∗τ t−QIΩ,αf∗tατ,
IΩ,αftWLq t−α−Q/qIΩ,αfWLq,
IΩ,αf
WLqt
αQ/qI
Ω,αftWLq≤CtαQ/qft1CtαQ/q−Qf1.
4.18
If 1<1/qα/Q,then for allf∈L1Hnwe haveIΩ,αfWLq 0 ast → 0. If 1>1/qα/Q,then for allf∈L1Hnwe haveIΩ,αfWLq 0 ast → ∞. Therefore we get the equality 11/qα/Q.
Corollary 4.5. Let0< α < Q,Ωbe homogeneous of degree zero onHnandΩ∈LQ/Q−αSH.
1If1 < p < Q/α, then the condition1/p−1/q α/Qis necessary and sufficient for the boundedness ofMΩ,αfromLpHntoLqHn.
2Ifp1, then the condition1−1/qα/Qis necessary and sufficient for the boundedness ofMΩ,αfromL1HntoWLqHn.
5. Conclusions
In this section we make some comments on the extension of above results to the case of homogeneous groups. Homogeneous groups can be exemplified by an n-dimensional Euclidean space, Heisenberg groups, and so on. We begin by reviewing some definitionssee 1–3,5 .
Let G be a homogeneous group, that is, G is a connected and simply connected nilpotent Lie group which is endowed with a family of dilations {δr}r>0. We recall that a family of dilations on the Lie algebragofGis a one parameter family of automorphisms ofg of the form{expAlogr:r >0},whereAis diagonalizable linear operator ongwith positive eigenvalues 1 d1 ≤ d2 ≤ · · · ≤ dn,Q dimG. Then the exponential map fromg toG
defines the corresponding family of dilations{δr}r>0onG. We will often use the abbreviated notationδrxrxforx∈Gandr >0.
We fix a homogeneous norm onG, that is, a continuous map| · | :G → 0,∞that is C∞onG\ {e}and satisfies
x−1|x| ∀x∈G,
|δrx|r|x| ∀x∈G, r >0,
|x|0⇐⇒xe.
5.1
The ballBx, rof radiusr >0 and centeredx∈Gis defined as
LetQ d1d2 · · ·dn be the homogeneous dimension of G, and letγ be the minimal
constant such that
xy≤γ|x|y ∀x, y∈G. 5.3
Onto the group G we fixed the normed Haar measure in such a manner that the measure of the unit ballBe,1is equal to 1. The Haar measure of any measurable setE⊂G will be denoted by|E|, and the integral onEwith respect to this measure byEfxdx.
It readily follows that
|δtE|tQ|E|, dδtx tQdx. 5.4
In particular, for arbitraryx∈Gandr >0 we have|Be, r|rQ.
The spacesLpGandWLpGare defined the same as in the caseHn. For example,
LpGis defined as the set of all measurable functionsfonGwith finite norm
fL pG
G
fxpdx 1/p
, 1≤p <∞. 5.5
The convolution of the functionsf, g∈L1Gis defined by
f∗gx
Gf
ygy−1xdy
Gf
xy−1gydy. 5.6
By using the methods of Theorem 3.1 it is not hard to show the Meda inequality for rearrangements of the convolution on the homogeneous groupG.
For a homogeneous groupG,it is well known thatsee1, page 14 there exists a unique countably additive measureσdefined onSGsuch that the equality
Gfxdx ∞
0 tQ−1
SG
fδtξdσξ dt 5.7
holds for anyf∈L1G,whereSG{x∈G:|x|1}. The equality5.7is an important tool for proving the following two theorems.
The fractional integralRΩ,αwith rough kernel onGis defined by
RΩ,αfu
G Ωy yQ−α f
y−1xdy, 0< α < Q. 5.8
Theorem 5.1. 1Letg ∈WLrG,1< r <∞,f∈LpG,1< p < r, and1/p−1/q1/r. Then
f∗g∈LqGand
f∗gL
qG≤2B
1−p/r
1 B
p/r
2 p
p/qgWL
rGfLpG, 5.9
whereB1andB2are defined inTheorem 3.1.
2Letf∈L1G,g ∈WLqG, and1< q <∞. Thenf∗g∈WLqGand
f∗g
WLqG≤2B
1/q
1 gWLqG fL1G. 5.10
In Theorem 5.2we give necessary and sufficient conditions for the Lp, Lq
-bound-edness of the fractional integral operatorRΩ,αwith rough kernel onG.
Theorem 5.2. LetΩbe homogeneous of degree zero onGandΩ∈LQ/Q−αSG,0< α < Q. 1If1 < p < Q/α, then the condition1/p−1/q α/Qis necessary and sufficient for the
boundedness ofRΩ,αfromLpGtoLqG.
2Ifp1, then the condition1−1/qα/Qis necessary and sufficient for the boundedness ofRΩ,αfromL1GtoWLqG.
Acknowledgments
Guliyev and Serbetci were partially supported by The Turkish Scientific and Technological Research CouncilTUBITAK, programme 2221, no. 220.01-619-4889. Guliyev was partially supported by BGP II project ANSF/AZM1-3110-BA-08. The authors are grateful to the referee for his/her suggestions.
References
1 G. B. Folland and E. M. Stein,Hardy Spaces on Homogeneous Groups, vol. 28 ofMathematical Notes, Princeton University Press, Princeton, NJ, USA, 1982.
2 I. Genebashvili, A. Gogatishvili, V. Kokilashvili, and M. Krbec,Weight Theory for Integral Transforms on Spaces of Homogeneous Type, vol. 92 ofPitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Harlow, UK, 1998.
3 V. S. Guliyev,Function Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups. Some Applications, Baku, Casioglu, Russia, 1999.
4 Y. S. Jiang, M. J. Liu, and L. Tang, “Some estimates for convolution operators with kernels of typel, r on homogeneous groups,”Acta Mathematica Sinica, vol. 21, no. 3, pp. 577–584, 2005.
5 E. M. Stein,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of
Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993.
6 L. Capogna, D. Danielli, S. D. Pauls, and J. T. Tyson,An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, vol. 259 ofProgress in Mathematics, Birkh¨auser, Basel, Switzerland, 2007.
7 D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer Monographs in Mathematics, Springer, Berlin, Germany, 2004.
8 E. M. Stein and G. Weiss,Introduction to Fourier analysis on Euclidean Spaces, Princeton Mathematical Series, no. 3, Princeton University Press, Princeton, NJ, USA, 1971.
9 S. Meda, “A note on fractional integration,” Rendiconti del Seminario Matematico della Universit`a di Padova, vol. 81, pp. 31–35, 1989.
