On the lacunary spherical maximal function on the Heisenberg group

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

On the lacunary spherical maximal function on the Heisenberg group

Pritam Ganguly,Sundaram Thangavelu^∗

DepartmentofMathematics,IndianInstituteofScience,560012Bangalore,India

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received26December2019 Accepted15October2020 Availableonline27October2020 CommunicatedbyCamilMuscalu MSC:

primary43A80

secondary22E25,22E30,42B15, 42B25

Keywords:

Lacunarysphericalmeans Koranyisphere

Heisenberggroup L^p-improvingestimates Sparsedomination Weightedtheory

In this paper we investigate the L^p boundedness of the lacunarymaximalfunctionM_H^lacn associatedto thespherical means Arf taken over Koranyi spheres on the Heisenberg group.CloselyfollowinganapproachusedbyM.Laceyinthe Euclideancase, weobtain sparse boundsfor these maximal functionsleadingtonewunweightedandweightedestimates.

The key ingredients in the proof are the L^p improving propertyof theoperator Arf and a continuity propertyof thedifferenceArf−τyArf,whereτyf(x)=f(xy⁻¹) isthe righttranslationoperator.

©2020ElsevierInc.Allrightsreserved.

1. Introductionandthemain results

The study of spherical means has received considerable attention in the last few decades.In1976,SteinfirstconsideredthesphericalmaximalfunctiononRⁿ definedby

* Correspondingauthor.

E-mailaddresses:pritamg@iisc.ac.in(P. Ganguly),veluma@iisc.ac.in(S. Thangavelu).

https://doi.org/10.1016/j.jfa.2020.108832

0022-1236/©2020ElsevierInc. Allrightsreserved.

M f(x) = sup

r>0|f ∗μr(x)|= sup

r>0

|y|=r

f(x−y)dμr(y)

where μ_r is the normalised measure on the Euclidean sphere of radius r. In [18], for n≥3 heprovedthat

M fp≤Cfp if and only ifp > n n−1.

Later Cowling-Mauceri[7] revisitedthisand provedStein’sresultusing completelydif- ferent arguments. In 1986,Bourgain [3] settled thecase n= 2. On the other hand,as provedbyC.P.Calderon [4],thelacunary sphericalmaximalfunction

M_lacf(x) = sup

j∈Z|f∗μ₂^j(x)|

turned out tobe bounded on L^p(Rⁿ) for all1< p<∞.Recently, M.Lacey hascome up withanew ideato provethese two resultsandmuch more.In [13] hehas obtained sparse boundsforbothM and M_lac leadingto newweightednormestimates.

Our aiminthis paper isto prove sparse bounds forthelacunary sphericalmaximal functions ontheHeisenberg group.Recall thattheHeisenberg groupHⁿ :=Cⁿ×R is equipped withthegroupoperation

(z, t).(w, s) :=

z+w, t+s+1

2(z.w)¯

, ∀(z, t),(w, s)∈Hⁿ.

On this group,for every r > 0,we have afamily of non-isotropic dilations defined by δr(z,t) := (rz,r²t), ∀(z,t) ∈ Hⁿ These δr are automorphisms of the group Hⁿ. The Koranyi normof(z,t) inHⁿ isdefinedby

|(z, t)|:= (|z|⁴+ 16t²)¹⁴

and itis easyto seethat|δr(z,t)|=r|(z,t)|, i.e.,thenorm ishomogeneousof degree1 relativetothesenon-isotropicdilations.TheHaarmeasureonHⁿissimplytheLebesgue measure dzdt. Asinthe Euclideancase, thisHaar measure hasapolardecomposition.

Let SK := {(z,t) ∈ Hⁿ : |(z,t)| = 1}be the unitsphere with respect to the Koranyi norm. Then there is aunique Radon measure σ on SK such that for every integrable functiononHⁿ wehave

Hⁿ

f(z, t)dzdt= ∞

0

SK

f(δrω)r^Q−1dσ(ω)dr (1.1)

where Q= (2n+ 2) is knownas thehomogeneousdimensionofHⁿ.

Letusfixsomemorenotationsfirst.Givenf onHⁿ,itsdilationδrf issimplydefined by δrf(z,t) =f(δr(z,t)), ∀(z,t) ∈ Hⁿ. More generally, if U is a distribution on Hⁿ,

thenitsdilationδrU is definedbyδrU,φ:=U,δrφ, ∀φ∈C_c^∞(Hⁿ).Weletσr=δrσ anddefinethesphericalmeanvalueoperatorAr by

Arf(x) =f∗σr(x) =

|y|=1

f(x.δry⁻¹)dσ(y).

Theassociated sphericalmaximal functionMHⁿf(x)= sup_r>0|A_rf(x)|wasstudiedby M.Cowlingin[5] whereheestablishedthefollowingresult.

Theorem 1.1. The spherical maximal functionMHⁿ is bounded on L^p(Hⁿ)for all p>

2n+1 2n .

ThisistheHeisenberganalogueofStein’stheoremonRⁿbutCowlingusedanentirely different approach in proving this. He used a very clever idea of expressing σr as a weightedintegralofRiesz potentials whichare easyto handle,see Theorem3.2 below.

Otherproofsoftheabovetheoremarealsoavailableintheliterature.ByadaptingStein’s secondproofoftheEuclideancase[19] (Corollary 3inXI,section 3),OliverSchmidt,in hisstillunpublishedwork[17],studiedmaximalfunctionsonstratifiedgroupsassociated to hyper-surfaces with a non vanishing rotational curvature, giving another proof of theabovetheorem of Cowling. Alsoin[10], usinga squarefunction argument, Fischer provedtheboundednessofthesphericalmaximaloperatoronfunctionsonfreesteptwo nilpotentLiegroupswhichincludestheaboveresultofCowling.

Inthis paperweareinterestedinestablishingananalogueofCalderon’s theoremfor the Heisenberg group.Though the methodsused in [4] are adaptations of the original ideaofStein[18],itisunlikelythataproofalongsimilarlinesispossible.Thisismainly becausethegroupFouriertransformofthemeasureσrisnotknownexplicitly.Tostate ourmainresult,wefixδ∈(0,1/96) andlet

M_H^lacnf(x) = sup

k∈Z|A_δ^kf(x)|

standforthelacunarysphericalmaximalfunctiononHⁿ. Weprove:

Theorem 1.2.Let n≥2.The lacunary spherical maximal function M_H^lacn is bounded on L^p(Hⁿ)forall1< p<∞.

This is the analogue of Calderon’s theorem for the spherical averages on Koranyi spheres. Inview of the workby Fischer [10], it is natural to ask if the abovetheorem can be proved in the more general context of free step two nilpotent Lie groups. For reasonswe just mentioned before the statementof the theorem, itis going to be very difficulttoprovesucharesultusingspectralmethods.Inprovingtheabovetheoremfor theHeisenberg group,wecombinetheideaofCowlingalongwitharecentlyformulated strategy of Lacey [13] in the Euclidean context that has turned out to be fruitful in

handlingseveral maximalfunctions.Webelievethatourproof canbe suitablyadapted tohandleatleastthecaseofH-typegroups,werestrainourselvesfromdoingsoinorder to keep theproofselegant. Weplanto takeupthegeneralcaseoffreesteptwo groups inafuturework.

As mentioned above, weestablish Theorem 1.2 by closelyfollowing Lacey [13]. The main idea of Lacey in reproving Calderon’s theorem on Rⁿ is to establish a sparse dominationforthesphericalmeans.Inordertodoso,herequiredtwomainingredients:

(i) theL^p improvingproperty and (ii) thecontinuity property,of thespherical means.

These ingredients are then used to controlthe sphericalmaximal functionin termsof certain dyadicmaximalfunctions.

Once asparsedominationforthelacunarysphericalmaximalfunctionisobtained,it is then a routinematter to deduce weightedandunweighted L^p bounds. Theorem 1.2, aswellascertainweightedversionsthatarestatedinSection5areeasyconsequencesof thesparseboundstatedinTheorem1.3,whichisthemainresultofthispaper.Inorder to statetheresultweneedto setupsomemorenotation,see[13].

As inthe caseof Rⁿ, thereisanotionofdyadicgridsonHⁿ,the membersofwhich are called (dyadic) cubes.A collection ofcubes S inHⁿ is said to be η-sparse ifthere are sets {E_S ⊂S : S ∈ S} which are pairwise disjoint and satisfy |E_S| > η|S| for all S ∈ S.Forany cubeQand1< p<∞,wedefine

fQ,p:=

1

|Q|

Q

|f(x)|^pdx 1/p

, fQ := 1

|Q|

Q

|f(x)|dx.

Intheabove,x= (z,t)∈Hⁿanddx=dzdtistheLebesguemeasureonCⁿ×R,which, as wehavealreadymentioned,istheHaarmeasureontheHeisenberg group.Following Lacey [13],bytheterm(p,q)-sparseformwe meanthefollowing:

Λ_S,p,q(f, g) =

S∈S

|S|fS,pgS,q.

Theorem 1.3. Assume n≥2 andfix 0< δ < ₉₆¹. Let1< p,q < ∞ be suchthat (¹_p,¹_q) belong to the interior of the triangle joining the points (0,1),(1,0) and (_2n+1²ⁿ ,_2n+1²ⁿ ).

Then forany pairof compactly supportedbounded functions(f,g)there existsa (p,q)- sparse formsuchthat M_H^lacnf,g≤CΛ_S,p,q(f,g).

We remark thatas intheEuclidean case wecan takeany δ >0 inTheorem 1.3, in particularwecanchooseδ= 1/2.Inordertoprovethemainingredientsinourcase,we make use oftheideaof Cowlingalong withanalytic interpolation. Once wehavethese two ingredients in our hand, the method of Lacey can be adapted to the Heisenberg group settingto obtainsparse domination. This hasbeen alreadydone inajoint work of thesecondauthor [1] whereadifferentkindof sphericalmaximal functionisstudied ontheHeisenberg group.

Thesphericalmaximalfunctionmentionedinthepreviousparagraphis associatedto thesphericalmeanstakenovercomplexspheresSr,0={(z,0)∈Hⁿ :|z|=r}whichwere initiatedintheworkofNevoandThangavelu.In[16],theyshowedthatthecorresponding maximalfunctionisboundedonL^p(Hⁿ) forp>²ⁿ_2n⁻₋¹₂.LaterNarayanan-Thangaveluin [15] and independentlyMüller-Seeger in[14] improved thatresult andproved thatthe maximalfunctionisboundedonL^p(Hⁿ) ifandonlyifp> _2n−1²ⁿ .Recently,Bagchietal.

[1] haveprovedtheanalogueofCalderon’stheoremfortheassociatedlacunaryspherical maximalfunctionbyobtainingasparseboundas inLacey[13].

Letμr stand forthe normalised surfacemeasure on Sr ={z ∈Cⁿ : |z| =r}⊂Cⁿ whichcanbeconsideredasameasureonHⁿ.Theassociatedsphericalmaximalfunction isthengivenbysup_r>0|f∗μ_r(z,t)|.Eventhoughthemeasureμ_rismoresingularthanσ_r, itsgroupFouriertransformisexplicitlyknown,givenintermsofthesphericalfunctions associated to the action of the unitary group U(n) on Hⁿ. These spherical functions are given by Laguerre functions for which very precise estimates are available in the literature. Thus the spectral method turns out to be useful in studying the maximal functionsup_r>0|f∗μ_r(z,t)|ascanbeseenfromallthethreeworks[16],[15] and[1] (see also [21]). Having said this, we remark that it maybe interesting to see ifCalderon’s approach can be adapted to study the lacunary maximal function associated to the sphericalmeansf∗μr.In[1] theauthorswereinterestedinprovingsparseboundswhich immediately givesCalderon’s theorem as acorollary. Consequently, theydidnot make anyattempttoproveCalderon’stheoremusingspectralmethods.However,thespectral theoryofthesphericalmeanscannotbeavoidedcompletely.Eveninourcase,wewillsee thatinformation ontheFouriertransformofσ_rplays animportantroleinestablishing L^p improvingandcontinuitypropertiesofthesphericalmeansf∗σr.

Weconcludetheintroductionbybrieflydescribingtheplanofthepaper.Aftercollect- ingsomerelevantresultsfromtheharmonicanalysis onHeisenberggroupsinSection2 weestablishtheL^p improvingpropertiesofthesphericalaveragesinSection3.InSec- tion 4 we study the continuity properties of the same. Finally inSection 5 we sketch theproofofthesparsedomination(i.e.proofofTheorem1.3)anddeduceweightedand unweightedinequalities.

2. Preliminaries

2.1. FouriertransformonHⁿ

InthissectionwecollectsomebasicresultsfromtheharmonicanalysisonHeisenberg groups thatwill play important roles in the study of the sphericalmaximal function.

GeneralreferencesforthissectionareFolland[11] andThangavelu[23].TheHeisenberg group Hⁿ introduced in the previous section is a nilpotent Lie group which is non- commutativeand yet with avery simplerepresentation theory. For eachnon zero real numberλwehaveaninfinitedimensionalrepresentationπ_λrealisedontheHilbertspace L²(Rⁿ).These areexplicitlygivenby

πλ(z, t)ϕ(ξ) =e^iλte^{i(x·ξ+12x·y)}ϕ(ξ+y),

where z = x+iy and ϕ ∈ L²(Rⁿ). These representations are known to be unitary and irreducible. Moreover, upto unitary equivalence these account for all the infinite dimensionalirreducibleunitaryrepresentationsofHⁿ(see[11]).Asthefinitedimensional representations ofHⁿ donotcontributetothePlancherel measurewewill notdescribe them here.

The Fourier transform of a function f ∈ L¹(Hⁿ) is the operator valued function obtainedbyintegratingf againstπ_λ:

fˆ(λ) =

Hⁿ

f(z, t)π_λ(z, t)dzdt.

Note that fˆ(λ) is a bounded linear operator on L²(Rⁿ). It is known that when f ∈ L¹∩L²(Hⁿ) itsFouriertransformisactuallyaHilbert-Schmidtoperatorand onehas

Hⁿ

|f(z, t)|²dzdt= (2π)⁻ⁿ⁻¹ ∞

−∞

fˆ(λ)²HS|λ|ⁿdλ.

The above allows us to extend the Fourier transform as a unitary operator between L²(Hⁿ) andtheHilbertspaceofHilbert-SchmidtoperatorvaluedfunctionsonRwhich are squareintegrablewith respecttothePlancherelmeasure dμ(λ)= (2π)⁻ⁿ⁻¹|λ|ⁿdλ.

2.2. TheHeisenbergLie algebra

Welethn standfortheHeisenbergLiealgebraconsistingofleftinvariantvectorfields onHⁿ.Abasisforhn isprovidedbythe2n+ 1 vectorfields

Xj = ∂

∂xj

+1 2yj

∂

∂t, Yj = ∂

∂yj −1 2xj

∂

∂t, j= 1,2, ..., n

and T = _∂t^∂. These correspond to certain oneparameter subgroupsof Hⁿ. Recall that given suchasubgroupΓ={γ(s):s∈R}oneassociatestheleft invariantvectorfield

Xf(g) = d ds

s=0f(gγ(s)).

Associated toeachsuchX wealsohavearight invariantvectorfieldX definedby Xf(g) = d

ds

s=0f(γ(s)g).

Itthenfollowsthatany(x,y,t)∈Hⁿ canbewrittenas(x,y,t)= exp(x·X+y·Y+tT) whereX = (X1,....,Xn),Y = (Y1,....,Yn) andexp istheexponentialmaptakinghninto Hⁿ.Fromtheabovedefinitionright invariantvectorfieldscanbeexplicitlycalculated

X˜j= ∂

∂x_j −1 2yj

∂

∂t Y˜j= ∂

∂y_j +1 2xj

∂

∂t;

theyagreewiththeleft invariantonesat theorigin.Therepresentationsπλ of Hⁿ give riseto thederivedrepresentationsdπλoftheLie algebrahn.These aregivenby

dπλ(X)ϕ= d ds

s=0πλ(expsX)ϕ.

Forreasonablefunctionsf andanyright invariantvectorfieldX,itisknownthat π_λ(Xf) =dπ_λ( ˜X)π_λ(f)

wheredπλisthederivedrepresentationoftheHeisenbergliealgebracorrespondingtoπλ. Itisalsowell-knownthatdπλ(Xj)=iλξjanddπλ( ˜Yj)=_∂ξ^∂

j.NowwritingZ˜j = ˜Xj+iY˜j, Z˜j = ˜Xj−iY˜j andusingtheaboveresultswehave

πλ( ˜Zjf) =iAj(λ)πλ(f)and πλ( ˜Zjf) =iA^∗_j(λ)πλ(f) whereA_j(λ) andA^∗_j(λ) aretheannihilationandcreationoperatorsgiven by

A_j(λ) =

− ∂

∂ξ_j +iλξ_j

, A^∗_j(λ) = ∂

∂ξ_j +iλξ_j .

Wemake useof these relations intheproof of thecontinuityproperty of thespherical means,seeSection4.

2.3. The measureontheKoranyi sphere

Let SK = {(z,t)∈ Hⁿ : |(z,t)| = 1}be the Koranyi sphere of radius 1. Then it is wellknownthatthereisaRadonmeasureσonKwhichgivesrisetothefollowingpolar decompositionoftheHaarmeasureontheHeisenberggroup:

Hⁿ

f(x)dx= ∞ 0 SK

f(δry)dσ(y) r^Q−1dr

whereQ= 2n+2 isknownasthehomogeneousdimensionofHⁿ.Foranyr >0 wedefine themeasureσr=δrσandnote thatitis supportedonKr={(z,t)∈Hⁿ:|(z,t)|=r}. WealsohaveanotherpolardecompositionoftheHaarmeasure givenby

Hⁿ

f(z, t)dzdt= ∞

−∞

∞ 0 |ω|=1

f(rω, t)dμ(ω)

r²ⁿ⁻¹drdt

where μ is the surface measure on the unit sphere in Cⁿ. If we let μr stand for the surfacemeasureonthesphereSr={(z,0)∈Hⁿ:|z|=r}andδtfortheDiracmeasure on R supportedat the point t then themeasure μr,t =μr∗δt is supportedonthe set S_r,t={(z,t)∈Hⁿ:|z|=r}.Themeasureσ_rcanbeexpressedintermsofthemeasures μr,t asfollows(seeFaraut-Harzallah[9]):

σ_r= Γ_n+1

2

√πΓ_n

2

π/2

−π/2

μ_r^√_cosθ,1

4r²sinθ (cosθ)ⁿ⁻¹dθ.

2.4. Fouriertransformsofradial measures

The unitary groupU(n) hasa naturalaction onHⁿ given byk.(z,t)= (k.z,t),k∈ U(n) which inducesan actiononfunctionsand measures ontheHeisenberg group.We saythatafunctionf (measureμ)isradialifitisinvariantundertheactionofU(n).Itis wellknownthatthesubspaceofradialfunctionsinL¹(Hⁿ) formsacommutativeBanach algebraunderconvolution.Sois the spaceoffiniteradialmeasuresonHⁿ.TheFourier transformsofsuchmeasuresarefunctionsoftheHermiteoperatorH(λ)=−Δ+λ²|x|².

Infact,ifH(λ)=_∞

k=0(2k+n)|λ|Pk(λ) standsforthespectraldecompositionofthis operator,thenforaradialmeasure μwehave

ˆ μ(λ) =

∞ k=0

Rk(λ, μ)Pk(λ).

More explicitly, Pk(λ) stands for the orthogonal projection of L²(Rⁿ) onto the kth eigenspacespannedbyscaledHermitefunctionsΦ^λ_αfor|α|=k.ThecoefficientsRk(λ,μ) are explicitlygivenby

Rk(λ, μ) = k!(n−1)!

(k+n−1)!

Hⁿ

e^iλtϕ^λ_k(z)dμ(z, t).

Intheaboveformula,ϕ^λ_k are Laguerrefunctionsoftype(n−1):

ϕ^λ_k(z) =Lⁿ⁻¹_k (1

2|λ||z|²)e^{−14|λ||z|2}

whereLⁿ_k⁻¹ denotestheLaguerrepolynomialoftype(n−1).Wereferthereaderto[22]

forthedefinitionandproperties.Inparticular,forthemeasuresσrwe have

R_k(λ, σ_r) = Γ_n+1

2

√πΓ_n

2

k!(n−1)!

(k+n−1)!

π/2

−π/2

ϕ^λ_k(r√

cosθ)e^iλ14r2sinθ (cosθ)ⁿ⁻¹dθ.

Thoughtheaboveintegralcannotbeevaluatedinaclosedform,itcanbeusedtostudy themaximalfunctionassociated to thesphericalmeans f∗σr. Seetheworkof Fischer [10] wherethesphericalmaximalfunctioninaslightlygeneralcontexthasbeenstudied.

3. L^p -improving propertyofthesphericalmeans

In this section we prove certain L^p−L^q bounds for the spherical means operator A_r. In order to prove the required estimates we embed A_r into an analytic family of operators and then appeal to Stein’s analytic interpolation theorem. First we obtain the following representation of themeasure σr as asuperposition of certain operators which canbe handled easily. In whatfollows we let Pt(x) =cQ t (t²+|x|²)^−Q+12 and I_γ(x):=C(Q,γ)|x|^−Q+iγ, x∈Hⁿ,t>0,γ∈Rwhere c_Q is definedbythecondition

cQ

Hⁿ

(1 +|x|²)^−Q+12 dx= 1

and C(Q,γ) = cnΓ_Q−iγ

4

2

/Γ_iγ

2

. In the proof of the next theorem which gives a representationofσtintermsofPtandIγ wemakeuseofthefollowingsimplelemma.

Lemma3.1. LetF(t)=c_Q(1+e^2t)^−Q+12 e^QtandletF(γ)ˆ standsfortheEuclideanFourier transformofF.Then

√2πF(γ) =ˆ Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

.

Proof. BythedefinitionoftheFouriertransform Fˆ(γ) = cQ

√2π ∞

−∞

(1 +e^2t)^−Q+12 e^Qte^−itγdt

whichafterthechangeofvariablese^t=rleadsto

√2πFˆ(γ) =cQ

∞ 0

(1 +r²)^−Q+12 r^Q−1−iγdr.

Another change of variables (1−t) = (1+r²)⁻¹ converts the above integralinto the Betaintegral

1 2 1 0

(1−t)^{1+iγ2 −1}t^Q−2iγ−1dt= 1 2

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q+1

2

.

Consequently weobtain

√2πFˆ(γ) = 1 2cQ

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q+1

2

.

Observethat,bythedefinitionofcQ wehave 1 =√

2πFˆ(0) = 1 2c_QΓ_Q

2

Γ₁

2

Γ_Q+1

2

whichleadstotheconclusion√

2πFˆ(γ)= ^Γ

_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

completingtheproof.

Thefollowingresultistheanalogueofatheorem byCowlingandMauceri[7] proved inthecontextofRⁿ.WeprovidethedetailsintheHeisenbergsettingfortheconvenience of thereader. Wedefine anotherconstantd(Q,γ) bytherequirement

d(Q, γ)C(Q, γ) = 1−Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

. (3.1)

Theorem 3.2.Fort>0thefollowingrepresentationholds inthesense of distributions:

σt=Pt+ 1 2π

∞

−∞

d(Q, γ)t^−iγIγ dγ.

Proof. Letu∈C_c^∞(1−δ <|x|<1+δ).ThenbyusingpolardecompositionoftheHaar measure,

Hⁿ

u(x)|x|^−Q+iγdx= ∞ 0

SK

u(δ_rω)r^iγ−1dσ(ω)dr.

Bydefiningu(r)¯ =

SKu(δrω)dσ(ω) werewritetheaboveas

Hⁿ

u(x)|x|^−Q+iγdx= ∞ 0

¯

u(r)r^iγ−1dr

Butbyachangeofvariables wehave ∞

0

¯

u(r)r^iγ−1dr= ∞

−∞

¯

u(e^t)e^iγtdt.

HencebytheFourierinversionformulaweobtain

∞

−∞ Hⁿ

u(x)|x|^−Q+iγdx

dγ= 2πu(1) = 2π¯ σ, u. (3.2)

Wenowconsiderthefollowing equation:

∞

−∞

d(Q, γ)Iγdγ, u=

Hⁿ

^∞

−∞

d(Q, γ)C(Q, γ)|x|^−Q+iγdγ u(x)dx.

Nowchangingtheorderoftheintegrationand using(3.1) and(3.2) weget

∞

−∞

d(Q, γ)Iγdγ, u= 2πσ, u − ∞

−∞ Hⁿ

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

u(x)|x|^−Q+iγdx

dγ.(3.3)

Now wesimplifythe second integralintheaboveequation. Using polardecomposition wehave

∞

−∞

Hⁿ

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

u(x)|x|^−Q+iγdxdγ

= ∞

−∞

SK

∞ 0

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

r^iγ−1u(δrω)drdσ(ω)dγ

= ∞

−∞

SK

∞

−∞

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

e^iγtu(δ_e^tω)dtdσ(ω)dγ.

ByFubini’stheorem,changingtheorderoftheintegrationweobtain ∞

−∞

Hⁿ

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

u(x)|x|^−Q+iγdxdγ

= ∞

−∞

SK

^∞

−∞

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

e^iγtdγ

u(δ_e^tω)dσ(ω)dt.

WenowmakeuseofLemma3.1andobtain ∞

−∞

SK

^∞

−∞

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2) e^iγtdγ

u(δ_e^tω)dσ(ω)dt

=√ 2π

∞

−∞

SK

^∞

−∞

F(γ)eˆ ^itγdγ

u(δ_e^tω)dσ(ω)dt

= (2π) ∞

−∞

SK

F(t)u(δe^tω)dσ(ω)dt

= (2π)cQ

SK

∞ 0

(1 +r²)^−Q+12 u(δrω)r^Q−1drdσ(ω).

Asthelastintegralintheabovechainofequationsisnothingbut(2π)

Hⁿu(x)P1(x)dx we haveproved

∞

−∞

Hⁿ

Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

u(x)|x|^−Q+iγdxdγ= (2π)P₁, u. (3.4)

Combining (3.3) and (3.4) we obtainthefollowing equalitywhich holdsinthesense of distributions:

σ=P1+ (2π)⁻¹ ∞

−∞

d(Q, γ)Iγdγ.

As σ_t isobtainedfrom σbydilation,thetheoremisproved.

Wewouldliketo embedthesphericalmeansAr intoananalyticfamilyofoperators.

As in theEuclidean case, this isachieved by observing thatthe distributionsgiven by thefunctions

φr,α(x) = 2r^−Q Γ(α)

1−|x|²

r² α−1

+ , (α)>0

converge to σ_r as α → 0. In the Euclidean case the Fourier transform of φ_r,α(x) is known explicitly,given intermsof Bessel functions,which allowsimmediate extension asahomomorphicfamilyofdistributions.InthecaseoftheHeisenberggroupwedonot have ausefulformula for the(group) Fouriertransform ofφr,α.Hencewe make useof the following representation similar to the oneprovedfor σr inthe precedingtheorem inholomorphicallyextendingtheoperatorf∗φr,α.

Proposition 3.3. Letr > 0, Re(α) >0. Then for any Schwartz function f on Hⁿ,we have

f ∗φr,α(x) = 2r^−Q Γ(α)

r 0

1−t²

r² α−1

t^Q−1f∗Pt(x)dt

+ 1 2π

∞

−∞

d(Q, γ)r^−iγ Γ(^Q−iγ₂ )

Γ(α+^Q−iγ₂ )f∗Iγ(x)dγ.

Proof. Bydefinitionofconvolution onHeisenberggroupwehave f ∗φr,α(x) =

Hⁿ

f(x.y⁻¹)φr,α(y)dy.

Asφr,αis radial,integratinginpolarcoordinateswe get

f ∗φ_r,α(x) =2r^−Q Γ(α)

∞ 0 SK

f(x.δ_tω⁻¹) 1−t²

r² α−1

+ dσ(ω) t^Q−1dt

=2r^−Q Γ(α)

r 0

1− t² r²

α−1

t^Q−1f∗σ_t(x)dt.

Makinguseoftherepresentation

f ∗σt=f ∗Pt+ 1 2π

∞

−∞

d(Q, γ)t^−iγf ∗Iγdγ

provedintheprevioustheoremwegetf∗φ_r,α=S_r,αf+T_r,αf where

Sr,αf(x) = 2r^−Q Γ(α)

r 0

1−t²

r² α−1

t^Q−1f∗Pt(x)dt

and

Tr,αf(x) = 1 2π

∞

−∞

2r^−Q

Γ(α) r 0

1− t²

r² α−1

t^Q−1t^−iγdt

d(Q, γ)f∗Iγ(x)dγ.

Theinnerintegralcanbe explicitlycalculated:

r^−Q r 0

1−t² r²

α−1

t^Q−1t^−iγdt=r^−iγ 1 0

(1−t²)^α−1t^Q−1−iγdt

whichreducesto abetaintegraland yields

2r^−Q Γ(α)

r 0

1− t²

r² α−1

t^Q−1t^−iγdt= Γ(α)Γ(^Q−iγ₂ ) Γ(α+^Q−iγ₂ ). Consequently,weobtaintherepresentation

Tr,αf := 1 2π

∞

−∞

d(Q, γ)r^−iγ Γ(^Q−₂^iγ)

Γ(α+^Q−₂^iγ)f∗Iγ(x)dγ, proving thetheorem.

IfwedefineAr,αf =f∗φr,α,thenbytheabovepropositionwehaveAr,αf =Sr,αf+ Tr,αf where theaboveholds undertheassumption that(α) >0.But both Sr,α and T_r,α have analytic continuationto a largerdomain of α. Indeed, T_r,α canbe extended to thewholeofC asanentire functionandS_r,α extendsholomorphicallytotheregion (α) > −n. Thus Ar,α is an analytic family of operators and when α goes to 0 we recover f ∗σr. Weremarkthatthe L^p-improvingproperty ofthis generalisedspherical mean A_r,αf ontheEuclideanspacehasbeenstudiedbyStrichartz[20].

In order to study the L^p improving property of the spherical mean value operator f →f∗σrweuseanalyticinterpolation.ItisenoughtoproveanL^p-improvingproperty fortheoperatorTr,0 bystudyingthefamilyTr,α.Weshallshowthatforα= 1+iβ,the operatorT_r,αisboundedfromL^1+δ toL^∞foranyδ >0 and forsomenegativevalueof (α),itisboundedonL²(Hⁿ).Byadilationargument,wecanassumethatr= 1 and hence we deal with Tα :=T1,α. To handle theL² boundedness, we need the following Fouriertransformcomputation.

Proposition 3.4.For λ= 0, theHeisenberg group Fouriertransform of the distribution

|.|^−Q+iγ isgiven by

|.|^−Q+iγ(λ) = (2π)ⁿ⁺¹|λ|^−iγ/2 Γ(^iγ₂) Γ(^Q₄ −^iγ₄)²

∞ k=0

Γ_2k+n

2 +^2−iγ₄ Γ_2k+n

2 +^2+iγ₄ P_k(λ).

This has been proved in the work of Cowling and Haagerup [6]. From the above propositionitisnow easytoprovethefollowing:

Proposition 3.5.Assumethat n≥1.Thenforanyα∈C with Re(α)>−n+¹₂ wehave Tαf2≤C((α))f2

where C((α))has admissiblegrowth.

Proof. Notethatifwewrite a(Q,γ):=d(Q,γ)C(Q,γ) wehave

Tαf(x) = ∞

−∞

a(Q, γ)r^−iγ Γ(^Q−iγ₂ )

Γ(α+^Q−₂^iγ)f∗ |.|^−Q+iγ(x)dγ.

It isthereforeenoughtoshowthat

∞

−∞

|a(Q, γ)| |Γ(^Q−₂^iγ)|

|Γ(α+^Q−₂^iγ)|b(Q, γ)dγ≤C((α))

whereb(Q,γ) isthenormoftheoperatorf →f∗ |· |^−Q+iγ onL²(Hⁿ) so thatwehave theinequality

f ∗ | · |^−Q+iγ2≤b(Q, γ)f2.

InviewofPlanchereltheoremforthegroupFouriertransformonHⁿwehavetheestimate b(Q, γ)≤Csup

λ |.|^−Q+iγ(λ). Usingthecomputationinthepreviousproposition,we have

|.|^−Q+iγ(λ) ≤C |Γ(^iγ₂)|

|Γ(^Q₄ −^iγ₄)|²sup

k

|Γ_2k+n

2 +^2−iγ₄

|

|Γ_2k+n

2 +^2+iγ₄

| =C |Γ(^iγ₂)|

|Γ(^Q₄ −^iγ₄)|². Thusweonlyneedtoshowthat

∞

−∞

|a(Q, γ)| |Γ(^Q−₂^iγ)|

|Γ(α+^Q−iγ₂ )|

|Γ(^iγ₂)|

|Γ(^Q₄ −^iγ₄)|² dγ≤C((α)).

Inordertoprovetheabove,wefirstrecallthat a(Q, γ) =

1−Γ_Q−iγ

2

Γ_1+iγ

2

Γ_Q

2

Γ₁

2

andhencea(Q,γ) hasazeroat γ= 0.Consequently, 1

−1

|a(Q, γ)|

|γ|

|Γ(^Q−iγ₂ )|

|Γ(α+^Q−iγ₂ )|

|Γ(1 +^iγ₂)|

|Γ(^Q₄ −^iγ₄)|² dγ≤C₁((α))

aslongas(α)>−n−1.Toprovetheintegrabilityawayfromtheoriginwemakeuse ofthefollowingasymptoticformulaforthegammafunction:for|ν|large

Γ(μ+iν)∼√

2π|ν|^μ−1/2e^−12π|ν|. Sousingthisformula,asimplecalculationshows thatfor|γ|≥1

|a(Q, γ)|

|γ|

|Γ(^Q−iγ₂ )|

|Γ(α+^Q−iγ₂ )|

|Γ(1 +^iγ₂)|

|Γ(^Q₄ −^iγ₄)|² ≤C₂((α))|γ|^{−(α)−n−1/2}.

Therefore, itfollowsthat

|γ|≥1

|a(Q, γ)|

|γ|

|Γ(^Q−₂^iγ)|

|Γ(α+^Q−₂^iγ)|

|Γ(1 +^iγ₂)|

|Γ(^Q₄ −^iγ₄)|² dγ≤C2((α)) forall(α)>−n+ 1/2.Thiscompletes theproof oftheproposition.

Proposition 3.6. Foranyβ ∈R andp>1we have, T1+iβf∞≤C(β)fp.

Proof. For any p > 1, to prove L^p → L^∞ estimate, first note that in view of the Proposition3.3, foranyβ∈R wehave

T_1+iβf(x) =f ∗φ_1,1+iβ(x)− 1 Γ(1 +iβ)

1 0

1−t²iβ

t^Q−1f ∗P_t(x)dt. (3.5)

Soforany x∈Hⁿ wehave

|T1+iβf(x)| ≤ |f ∗φ1,1+iβ(x)|+ 1

|Γ(1 +iβ)| 1 0

t^Q−1|f∗Pt(x)|dt. (3.6)

Now seethat

|f∗φ1,1+iβ(x)| ≤ 1

|Γ(1 +iβ)||f| ∗χB1(x)≤C(β)fp. As wehavePt(x)=t^−QP1(δ_t⁻¹x) itfollows thatforanyp>1,

|f∗P_t(x)| ≤t^−Qfp Hⁿ

P₁(δ_t⁻¹x)^pdx1/p

≤Ct^−Q/pfp.

Consequently, the integral in (1.2) is bounded by fp. Finally, these two estimates together with(3.6) weget

T1+iβf∞≤C(β)fp.

Byusingtheabovetwopropositionsandananalyticinterpolationargument,wenow provethefollowingresult.

Proposition 3.7.For any 0 < δ < 1, we have T₀ : L^pδ(Hⁿ) → L^qδ(Hⁿ) where p_δ = (1+δ)^2n+1−δ_2n andqδ = (2n+ 1−δ).

Fig. 1.TriangleL_n,ontheleftside,showstheregionforL^p−L^qestimatesforA1.ThedualtriangleLnis ontheright.

Proof. Givenδ >0 weconsidertheholomorphicfamilyofoperatorsT_L(z),whereL(z)= (−n+^1+δ₂ )(1−z)+z.InviewofthePropositions3.5and3.6,Stein’sinterpolationtheorem gives

T_L(u):L^p(u)(Hⁿ)→L^q(u)(Hⁿ)

where _p(u)¹ = ^1−u₂ +_1+δ^u and _q(u)¹ = ¹₂(1−u). SolvingforL(u)= 0 weget u= ^2n−1−δ_2n+1−δ andsimplifyingwegetp(u)=p_δ = (1+δ)^2n+1−δ_2n andq(u)=q_δ = (2n+ 1−δ).

The following result which follows from the above end point estimates by means of analytic interpolation describes the L^p improving properties of the spherical mean operatorA_r.

Theorem3.8. Assumethat n≥1.Thenwehave

Arfq≤Cr^Q(1q−1p)fp

whenever (¹_p,¹_q) lies in the interior of the triangle joining the points (0,0),(1,1), and (_2n+1²ⁿ ,_2n+1¹ )aswellasthestraightlinejoining thepoints(0,0),(1,1).

Proof. An easy calculation shows that δ⁻¹_r A₁δ_r = A_r and hence we can assume that r= 1.Withpδ andqδ asintheproofofTheorem3.7,wefirstshowthat

Ar:L^pδ(Hⁿ)→L^qδ(Hⁿ). (3.7) RecallthatfromtheProposition3.3wehave