functions of general L functions on domains Weak differentiability for fractional maximal Advances in Mathematics

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Advances in Mathematics

www.elsevier.com/locate/aim

Weak differentiability for fractional maximal functions of general L

functions on domains

JoãoP.G. Ramos^a, Olli Saari^a,∗, Julian Weigt^b

aMathematicalInstitute,UniversityofBonn,EndenicherAllee60,53115,Bonn, Germany

bDepartmentofMathematicsandSystemsAnalysis,AaltoUniversity,Finland

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

Articlehistory:

Received24November2019 Receivedinrevisedform26March 2020

Accepted1April2020 Availableonlinexxxx CommunicatedbyLarsOlsen

MSC:

primary42B25,46E35

Keywords:

Maximalfunction Sobolevspace Sphericalmeans Domains

LetΩ⊂Rⁿ beboundedadomain.Weproveundercertain structuralassumptionsthatthefractionalmaximaloperator relativetoΩ mapsL^p(Ω)→W^1,p(Ω) forallp>1,whenthe smoothnessindexα≥1.Inparticular,theresultsarevalidin therangep∈(1,n/(n−1)] thatwaspreviouslyunknown.As anapplication,weproveanendpointregularityresultinthe domainsetting.

©2020ElsevierInc.Allrightsreserved.

1. Introduction

RegularityoftheHardy–LittlewoodmaximalfunctionofaSobolevfunctionwasfirst studiedin[13].It wasshownthatthemaximaloperatorpreserves W^1,p(Rⁿ) regularity for p > 1. This continues to hold true at the derivative level when p = 1 and n = 1

* Correspondingauthor.

E-mailaddresses:jpgramos@math.uni-bonn.de(J.P.G. Ramos),saari@math.uni-bonn.de(O. Saari), julian.weigt@aalto.fi(J. Weigt).

https://doi.org/10.1016/j.aim.2020.107144 0001-8708/©2020ElsevierInc. Allrightsreserved.

[21,16] andforradialfunctions[18].ExtendingsuchastatementtomoregeneralSobolev functionsofseveralvariablesisadifficultopenproblem,whichhasinspiredmanyresults in relatedtopics. Forinstance, slightlystrongerbounds havebeen proved formaximal operators withmorespecialconvolution kernels(see[7],[3],[4] and[20]),thecontinuity of the mappinghas been studiedin[17] and [6], and apartof thetechniques used for continuity,alsorelevantforthecurrentpaper,havebeenextendedto p= 1 in[11].

Another aspect of the problem is the fractional endpoint question proposed by CarneiroandMadrid[5].Thefractionalmaximal functionisgiven by

Mαf(x) = sup

r>0

r^α

|B(x, r)| ˆ

B(x,r)

|f(y)|dy,

anditdefinesaboundedoperatorL^p(Rⁿ)→L^q(Rⁿ) whenq=np/(n−αp),0< α < n/p and p > 1. This boundedness fails at the endpoint p = 1, but the question about boundednessof∇Mα fromW^1,1(Rⁿ) toL^n/(n−α)(Rⁿ) hasnotbeenansweredso farfor α < 1 (see [19], [1] and [2] for related research and partial results). The case α ≥ 1 turned outtobeverysimple,andthereasoncanbetracedbacktotheinequality

|∇Mαf(x)| ≤cα,nMα−1f(x) (1.1) of Kinnunenand Saksman[15].CarneiroandMadrid[5] notedthat(1.1) togetherwith theGagliardo–Sobolev–NirenberginequalityandtheL^p→L^q boundsforthefractional maximal functionimplytheexpected endpointbound whenα≥1.

In thepresent paper,we studythese problems ingeneralopen subsetsof Rⁿ,which is anaturalcontext for analysis from thepoint of viewof potential theoryand partial differential equations. Regularity ofthe local Hardy–Littlewood maximal functionof a SobolevfunctiononanopenΩ⊂Rⁿ wasfirststudiedbyKinnunenandLindqvist[14], andalocalvariantoftheinequalityforthederivativeofthefractionalmaximalfunction (1.1) wasprovedin[12]. Thisisourstartingpoint,and formorethoroughdiscussionof whatwasprovedandwhatisunknown, weintroducesomemorenotation.

IfΩ⊂Rⁿ isanopenset,thelocal fractionalmaximalfunctionisdefinedas M_α^Ωf(x) = sup

0<r<dist(x,Ω^c)

r^α

|B(x, r)| ˆ

B(x,r)

f(y)dy.

As theboundaryofΩ restrictsthechoiceofrinthedefinition,onecannotexpect (1.1) to triviallycarryovertothelocalsetting. Indeed,suchapointwiseinequalityisfalsein general(Example4.1in[12]).Ontheotherhand,ifoneaddsacorrectionterminvolving thesurfacemeasureofthespheretotheright handsideof(1.1),oneobtains

|∇M_α^Ωf(x)| ≤c_α,n

M_α−1f(x) + sup

r>0|r^α−1σ_r∗f(x)|

, (1.2)

which is valid in all domains. This was used in [12] to prove that L^p functions with p> n/(n−1) large enoughhaveM_α^Ωf inafirstorder Sobolev class.The lower bound onprules outfunctionstoo singularforanapplicationofasphericalmaximalfunction argument.

Our main theorem shows that under suitable assumptions on the domain Ω, the maximalfunctionM_α^Ωmaps L^p(Ω) intoafirstorderSobolevspaceforallp>1.

Theorem1.1. Let Ω⊂Rⁿ be open,n≥2, p>1and f ∈L^p(Ω). ThenM_α^Ωf is weakly differentiableand

∇M_α^Ωf L^p(Ω)≤C f L^p(Ω)

ifany one ofthefollowingholds:

(1) α >1andΩisbounded.

(2) α= 1 andΩ^c is convex.

(3) α = 1 and Ω is bounded and satisfies a uniform curvature bound in the sense of Section 2.2.

(4) α= 1 andp>1+_n¹.

Theconstant C dependsonthedimension, andin(1)and(3)italso dependsonαand thedomain.

Unlike[12],weare notabletoproveanL^p→L^q smoothingeffectontopofwinning onederivative.However,ourmethoddoesapplytosingularfunctionsinL^p spaceswith 1≤p≤n/(n−1) wherethe argumentin[12] failstogiveany result.Inparticular, we havethefollowing endpointregularity result,whichwaspreviouslyoutofreach.

Corollary1.2. LetΩ⊂Rⁿ be aLipschitzdomain. Thenforallf ∈W^1,1(Ω) ∇M₁^Ωf _Ln/(n−1)(Ω)≤C f _W^1,1_(Ω)

wheretheconstant C only dependsonΩandthedimension.

Webrieflyoutlinetheproofofthemaintheorem.Themaximalfunctiononadomain behavesdifferently depending onwhether theball attaining themaximum touchesthe boundaryornot.Incaseitdoesnot,thelocalmaximalfunctionbehavesliketheglobal one, and the analysis is very similar. Otherwise it coincides with a linear averaging operator(2.3),whichdependsonthedomain.These twoparts areanalyzedseparately, andthemainpartoftheproofistoestablishL^pboundsforthederivativeof(2.3).This leadsto studyingadomaindependentweightedsphericalaveragingoperator(5.1).

InsteadofresortingtomaximalaveragesandtheBourgain–Steintheorem,anangular decompositionoftheoperatoriscarriedout.Theadditionalgeometricinformationallows

Fig. 1.A setP(y) and a tangent line.

instead to establishgood L¹ bounds that canbe interpolated with trivialL^∞ bounds inorder toobtainadomination of(5.1) byaconvergingsumofL^pbounded operators.

ImprovingtheL¹boundoverwhatfollowsfromthebehavior ofgenericsphericalmeans is crucial when aiming atL^p bounds forall p>1.Such aconclusioncannot be drawn from merepolynomialdecayoftheFouriertransformoftheweightedsphericalmeasure inquestion,ifnoadditionalL¹informationistakenintoaccount.Turningthefocusfrom the Littlewood–PaleydecompositionandL²methodsto anangulardecompositionand geometric estimatesinL¹istheleadinginsightoftheproof.

Thekey ideaintheL¹ estimatescanbedescribed asfollows. Each domainΩ comes endowedwith afamilyofsets(Fig. 1)

{P(y) :y∈Ω}, P(y) ={x∈Ω :y∈∂B(x,dist(x,Ω^c))},

which can morallybe used to dualize the spherical averaging operators (5.1) through Fubini’s theorem.The L¹ boundsfor the constituentsinthe angular decomposition of thesphericalaveragingoperatorcorrespondtoweightedintegralsoverthepiecesofP(y).

IfΩ isaball,thenthesetsP(y) areellipsoidswithfociatthecenteroftheballandaty.

Inthecasesofthecomplementofaballandahalf-space,theP(y) takethesimpleforms ofhyperboloidsandparaboloids.Onecannothopeforas explicitdescriptionsasthatin more general domains, butall P(y) are boundaries of convex sets. This observationis used extensivelyintheproof.

The structure of the paper is as follows. In the first section, we introduce notation and sometools thatwill be helpfulthroughout theproof. The first sections are about differentiating themaximalfunctionon so-calledunconstrainedpoints andproving the

weakdifferentiabilityofthemaximalfunctionconditionallytotheL^pboundednessofthe averagingoperator(2.3).TherestofthepaperisdevotedtoprovingthoseL^pboundsby firstcomputingaformulaforthederivativeandthencarryingoutthestrategysketched above.Finally,thereisaconcludingsectionwithremarksonopenproblemsandcertain observationsabouttheproofwhichmightbeofindependentinterestforfutureresearch.

2. Preliminaries

2.1. Notation

We letn ≥1 denote the dimension. For ameasurable set E, we let |E| denote the n-dimensionalLebesguemeasure.Thek-dimensionalHausdorffmeasuresaredenotedby H^k. An Euclidean ball with centerx ∈Rⁿ and radius r > 0 is denoted byB(x,r).A finiteconstantonlydependingonquantitiesthatarenotbeing kepttrackofisdenoted by C. If A ≤ CB for such constant, we denote A B or write A is B. We write A∼B ifbothAB andBAhold.

2.2. Domains

WealwaysassumeΩ⊂Rⁿ to beanopen set,whichwe interchangeablycalldomain asthedistinctionobviouslyplaysnoroleinthispaper.Weassumeittohavenon-empty complement. The distance function is denoted by δ(x) = dist(x,Ω^c). As Ω^c is closed, thereexists atleast onebx∈Ω^c so that|x−bx|=δ(x).Wereserve thenotationbx for suchapoint,whichneednotbeuniqueunlessΩ^cisconvex.Thedistancefunctionδ: Ω→ [0,∞) is always 1-Lipschitz. The gradient exists almost everywhere by Rademacher’s theorem,and itholdsthat

∇δ(x) =x−bx

δ(x) . (2.1)

This is because clearly the one sided directional derivative of δ(x) in the direction of bx−x always exists and is −1. Where the gradientexists, we canuse |∇δ(x)|≤ 1 to concludethat the directional derivativein all directions orthogonal to x−bx must be zero.

AdomainissaidtosatisfyauniformcurvatureboundifthereisanR >0 sothatfor everypoint x∈Ω itholds

B

bx+Rx−bx

δ(x) , R

⊂Ω.

AllboundedC² domains satisfythis condition,butadomainsatisfying auniform cur- vature bound might be non-smooth and have inwards-pointing cusps. A domain with aninteriorballconditionneednotsatisfytheuniformcurvature bound.Anexampleof suchadomain isB(0,1)\({0}∪ {(0,. . . ,0,2^k):k∈Z,k <0}).

2.3. Functionspaceson domains

Functionsf ∈L^p(Ω) areapriorionlydefinedinthedomainΩ,butwealwaysextend them by zeroto Rⁿ withoutadditional comments.TheSobolev classW^1,p(Ω) consists of functionsf ∈L^p(Ω) suchthat|∇f|∈L^p(Ω).Theweakderivativesaredefinedusing test functionsinC_c^∞(Ω).

Fortheapplicationofthemaintheoremtotheendpointregularityproblem,weneed aSobolevembedding theoremfor domains.Oneconcrete casewecandealwith isthat of aLipschitzdomain.

Proposition 2.1 (Section 4.4 in [9]).Let Ω ⊂ Rⁿ be a bounded open set so that ∂Ω is Lipschitz. Thenforevery1≤p<∞ thereexistsaboundedextension operator

E:W^1,p(Ω)→W^1,p(Rⁿ)

such thatsupp(Ef)⊂B(x0,2diam(Ω))forsomex0∈Ωandallf ∈W^1,p(Ω).

BytheboundarybeingLipschitz,wemeanthatitcanbecoveredbyafinitenumber of open balls B_i so that for each i the domain B_i∩Ω is the epigraph of a Lipschitz function.

The propositiontogether with the Gagliardo–Nirenberg–Sobolevinequality(see e.g.

Section4.5.1in[9])impliesarudimentarylocal Sobolevembedding

f _Lpn/(n−p)(Ω)≤CΩ,p,n f W^1,p(Ω) (2.2)

valid for all f ∈ W^1,p(Ω) whenever Ω is abounded open set with Lipschitz boundary.

This issufficientforourpurposes.

2.4. Maximal function

Forα∈[1,n),definethelocalfractionalmaximal functionrelative toΩ as M_α^Ωf(x) = sup

0<r<δ(x)

r^α − ˆ

B(x,r)

f(y)dy

wheneverf ∈L¹_loc(Ω).WeomitthesuperscriptwhenΩ isthewholeRⁿ.Inaddition,we define forα∈Rtheauxiliarylinearoperator

A_αf(x) =δ(x)^α − ˆ

B(z,δ(x))

f(y)dy. (2.3)

2.5. Constrainedpoints

Letfbecontinuous.Fixx∈Ω.BecausethecomplementofΩ isnon-empty,δ(x)<∞ andthereexistsaconvergentsequencerj∈(0,δ(x)) withlimitr= lim_j→∞rj∈[0,δ(x)]

suchthat

M_α^Ωf(x) = lim

j→∞r^α_j ˆ−

B(x,rj)

f(y)dy=r^α ˆ−

B(x,r)

f(y)dy

ifr >0.If

M_α^Ωf(x)> δ(x)^α − ˆ

B(x,δ(x))

f(y)dy,

thesequence rj mustbe chosen so thatr < δ(x), and thepoint xissaid to be uncon- strained.Allotherpointsarecalled constrained.

3. Theunconstrainedpart

Thelocalmaximal functionbehavessimilarlyto theglobaloneintheunconstrained set,and we reduce the differentiability questionof the unconstrained partaccordingly tothatoftheglobal maximalfunction.Thisisthecontentofthefollowingproposition.

Proposition 3.1. Let p > 1, α ≥ 1 and f ∈ L^p_loc(Ω) be continuous. The set U of the unconstrained pointsis open,the maximal function M_α^Ωf is weaklydifferentiable in U, andthepointwisebound

|∇M_α^Ωf(x)| ≤cM_α−1f(x)

holdsforaconstant c onlydepending on thedimensionandαwheneverx∈U. Proof. Considerthefractionalaveragefunction

A(z, r) :=r^α ˆ−

B(z,r)

f(y)dy.

Itiscontinuousin(z,r)∈Ω×R+.Fixnowanunconstrainedpointx.Bydefinition,there exists ε> 0 sothatM_α^Ωf(x)−A(x,δ(x)) > . Moreover, there exists γ >0 so thatif

|(z,r)−(x,δ(x))|< γ,thenM_α^Ωf(x)−A(z,r)> ε/2.SinceM_α^Ωf islowersemicontinuous, onecanfindforeveryz closeenoughtoxasequencer_z,j→r_z< δ(x)−γ/2 sothat

M_α^Ωf(z) = lim

j→∞r^α_z,j ˆ−

B(z,rz,j)

f(y)dy.

Inparticular, thereisanopenneighborhood Uxofxsothatforallz∈Ux

M_α^Ωf(z) =Mα(1_B(x,δ(x))f)(z).

ByTheorem3.1 in[15],

|∇M_α^Ωf(x)| ≤CMα−1f(x) follows.

4. Thefull maximalfunction

Next we provethe differentiability of the local maximal function conditional to L^p boundsforthederivativeoftheaveragingoperator(2.3).Thisstepmorallyfollowsfrom thelatticeproperty ofSobolevfunctions,butasweonlyknowtheweakdifferentiability of M_α^Ωf intheunconstrainedset,someextraworkisneeded.

Lemma 4.1.Let p >1, α≥ 1 and Ω⊂ Rⁿ be such that ∇Aα and Mα−1 are bounded L^p(Ω) → L^p(Ω). If f ∈ L^p(Ω), then the local fractional maximal function is weakly differentiable and

∇M_α^Ωf _L^p_(Ω) f _L^p_(Ω).

Proof. Assumefirstthatf iscontinuous andcompactlysupported.Following theargu- ments in[15], we inferthat M_α^Ωf canbe seenas supremum over radii betweenafixed upperandlowerbound.ThefractionalaveragesareLipschitzcontinuouswithconstants only dependingontheradii,and hencetheirsupremumisalso Lipschitz.Inparticular, we knowthatM_α^Ωf iscontinuous.

Denote byg⁺ = max(g,0) thepositivepartofafunctiong andwrite M_α^Ωf = (M_α^Ωf−A_αf)⁺+A_αf.

By assumption,the second termadmits the desired Sobolevbounds. To deal with the other term,let>0 anddefine

F(t) =

((t−)²+²)^1/2−, t >

0, t≤.

These functions are of class C¹(R) and converge pointwise to t → (t)⁺ as → 0.

Moreover, as M_α^Ωf and A_αf arecontinuous,E ={x∈Ω:F(M_α^Ωf(x)−A_αf(x))>0} hasitsclosurecontainedintheopensetofunconstrainedpointsU.ByProposition3.1, theassumptiononAαandthechainruleforSobolevderivatives(4.2.2in[9]),weobtain forallpartial derivatives∂i

∂iF(M_α^Ωf−Aαf) = (∂iM_α^Ωf−∂iAα)F(M_α^Ωf −Aαf).

Takingatest functionϕandcomputing ˆ

Ω

F(M_α^Ωf−Aαf)∂iϕ dx= ˆ

Ω

(∂iM_α^Ωf−∂iAαf)F(M_α^Ωf−Aαf)ϕ dx,

we see that taking the limit → 0 proves the claim for continuous and compactly supportedf.

Todealwiththegeneralcase,letf ∈L^p(Ω) andletfj becontinuous andcompactly supported functions converging to f in L^p norm. By L^p continuity of the fractional maximaloperator,M_α^Ωfj→M_α^Ωf inL^p.As wehaveprovedthefollowinginequality

∇M_α^Ωf_{j L}^p_(Ω) f_{j L}^p_(Ω),

forcontinuousfunctionsfj,thesequenceM_α^Ωfj isboundedinW^1,p(Ω).Wecanextract a weaklyconvergent subsequence. By taking limits along this sequence and using the uniquenessof distributionallimit,weconcludetheproofforgeneralf ∈L^p(Ω).

As the main theorem is a direct consequence of the previous lemma, it remains to investigatetheboundedness oftheoperator∇Aα onL^p(Ω).The followingsections are devotedtoestablishingtherequiredL^p boundswhenΩ issufficientlywell-behaved.

5. Constrainedpart

Byachangeofvariables, wecanwrite theaveragingoperator(2.3) as

A_αf(x) =δ(x)^α ˆ−

B(0,1)

f(x+yδ(x))dy.

This operatoris linear, and as we areaiming for L^p bounds, there is no loss of gener- ality inrestricting theattention to smooth functions.If xis aconstrained point, then M_α^Ωf(x) =A_αf(x), which justifies our reference to A_α as the constrained part. Also, Lemma4.1showedthatL^p boundsforthederivativeof Aαf areenoughtoimplyweak differentiabilityofthefullmaximaloperator,sothemaximalfunctiondoesnotplayany role inwhatfollows. A versionof thefollowing proposition wasalready provedin[12], butasweneedaformulamoreprecisethanwhattheystated,weincludetheshortproof forclarity.

Proposition 5.1. Letf ∈C^∞(Ω).Thenforalmosteveryx∈Ω

|∇A_αf(x)| ≤c_n,α|A_α−1f(x)|+c_nδ(x)^α−1 − ˆ

∂B(x,δ(x))

|y−b_x|

δ(x) f(y)dHⁿ⁻¹(y) where b_x∈∂Ωisapointsuchthat |b_x−x|=δ(x).

Proof. Fixapointx.AsAαf(x)=δ(x)^αA0f(x),itholdsthat

∇A_αf(x) =αδ(x)^α−1A₀f(x)∇δ(x) +δ(x)^α(∇A₀f)(x).

Since|∇δ(x)|≤1 (cf. (2.1)),thefirstsummandaboveisboundedbyA_α−1f(x).Thusit sufficesto analyzethegradientofA₀f.Take theunitvector

e=∇A0f(x)/|∇A0f(x)|. Then

|∇(A₀f)(x)|= (e· ∇)A₀f(x)

= −

ˆ

B(0,1)

e+y(e· ∇δ(x))

· ∇f(x+δ(x)y)dy

= 1

δ(x) − ˆ

B(0,1)

div_y

(e+y(e· ∇δ(x)))f(x+δ(x)y) dy

−ne· ∇δ(x) δ(x) ˆ−

B(0,1)

f(x+δ(x)y)dy=: I + II.

Since |∇δ(x)| ≤ 1, the contribution δ(x)^α·II is pointwise bounded by nAα−1f. To estimate theother term,weapplyGauss’stheoremtoobtain

I = cn

δ(x) ˆ

∂B(0,1)

y·(e+y(e· ∇δ(x)))f(x+δ(x)y)dHⁿ⁻¹(y)

= cn

δ(x) ˆ−

∂B(x,δ(x))

(y−bx)·e δ(x) f(y)dy.

Sowereachtheinequality

|∇Aαf(x)| ≤αn|Aα−1(x)|+cnδ(x)^α−1 ˆ−

∂B(x,δ(x))

|y−bx|

δ(x) f(y)dHⁿ⁻¹(y), whichproves theclaim.

Because A_α−1f(x) ≤M_α^Ω₋₁f(x),and M_α^Ω₋₁ satisfies the right L^p →L^q bounds, we havereducedthematterto understandingtheweightedsphericalaverage

B_αf(x) :=δ(x)^α−1 − ˆ

B(x,δ(x))

|y−b_x|

δ(x) f(y)dHⁿ⁻¹(y) (5.1) on the right hand side of the conclusion of the previousproposition. The weight|y− bx|/δ(x) measurestheangle betweenbx−xand y−xwhen |y−bx|/δ(x) is small. We decomposetheweightedsphericalaveragingoperatoraccordingtotheangleandlocation inthedomain asfollows.Fork∈Z,let

Ω_k={x∈Ω : 2^k ≤δ(x)<2^k+1} andforeverypointx∈Ω andintegerj≥0

ωj(x) = y ∈∂B(x, δ(x)) : 2^−j < |y−b_x|

δ(x) ≤2^−j+1

.

Define

S_j^kf(x) = 1Ωk(x) ˆ

ωj(x)

f(y)dHⁿ⁻¹(y).

Then

Bαf(x)

k∈Z

∞ j=1

2^{k(α−n)−j}S_j^kf(x) (5.2)

anditremains toproveboundsfor S_j^k so thattherighthandsidesumsupinL^p.This isdonebyinterpolatingboundsonL^∞ andL¹.

Proposition 5.2. Let Ω be any domain. It holds that S_j^k L^∞→L^∞ 2^{(n−1)(k−j)}, and consequently

k2^k(1−n)S_j^k L^∞→L^∞ 2^−(n−1)j. Proof. Thisfollows fromHⁿ⁻¹(ω_j(x))2^{(n−1)(k−j)}. 6. L¹ bounds

To proveL¹ bounds, we introducesome morenotation. For eachinteger j ≥0 and eachpointy∈Ω,define

Pj(y) ={x∈Ω :y∈ωj(x)}, P(y) = ∞ j=0

Pj(y). (6.1)

Inaddition, let

A^k_j =

x∈Ωk

ωj(x). (6.2)

Formally, certain weighted integrals over P(y) give the adjoint operator of B_α. A naive changeoforderofintegrationisnotjustifiedinthis case,butusingthedecompo- sitionofBα,wecanmaketheideaprecise.Thefollowingtwo propositionsgiveeffective descriptionofP(y) andprovideasubstituteforFubini’stheorem.

Proposition 6.1. LetΩ bean opensetandlety∈Ω.Then E(y) ={x∈Ω :|x−y| ≤δ(x)} is closedandconvex setsuchthat

P(y) =∂E(y).

For each x∈P(y), thesupporting hyperplaneat xbisects theangle betweeny−x and bx−xandisnormal tobx−y.

Proof. RecallthatP(y) consistsofthepointswith{x∈Ω:|y−x|=δ(x)}.Forx∈P(y), itholds that

x+ b_x−x

|bx−x| ∈E(y)^c, and itiseasytosee∂E(y)=P(y).Considerthehyperplane

{z∈Rⁿ :|z−b_x|=|z−y|}.

It dividesthe space into two half spaces H₁ ={z : |z−b_x| <|z−y|} and H₂ ={z :

|z−b_x|≥ |z−y|}.Ifx∈P(y),then E(y)⊂H₂and x∈H₂.Thus∂H₂ isasupporting hyperplaneforE(y) atx.AseveryboundarypointofE(y) hasasupportinghyperplane, E(y) isconvex.Theremainingassertionsreadilyfollowfromthedefinitionof∂H2. Proposition6.2.LetΩbeadomain,j≥0andkintegersandf ≥0aboundedcontinuous function onΩ.Then

ˆ

Ω

S_j^kf(x)dx2^j

|j−j|≤1,|k−k|≤1

ˆ

A^k_j

f(y)Hⁿ⁻¹(P_j^k(y))dy,

where weletP_j^k(y)=Pj(y)∩Ωk.

Fig. 2.The construction to findx1.

Notethaty∈A^k_j ifandonlyifP_j^k(y)=∅.

Proof. The parameterk plays norole inthe following computation,butis includedin thestatementforfuturereference.Letϕ≥0 beasmoothfunctionofonevariable with compactsupportin(0,1) and ϕ _L¹_(R)= 1.Denotethe-dilationbyϕ(t)=⁻¹ϕ(t⁻¹).

Foranyfixedx,wedefine theset ofrelevantdirections

ω_j^dir(x) =δ(x)⁻¹(ωj(x)−x)⊂∂B(0,1).

Asf ispositive,theweakconvergence S_j^kf(x) =

ˆ

ωj(x)

f(y)dHⁿ⁻¹(y) = lim

→0

ˆ

x+Rω^dir_j (x)

f(y)ϕ(δ(x)− |x−y|)dy

holds.Integratingoverx,applyingthedominatedconvergencetheorem(thisisjustified, seetheremarkat theend oftheargument),andusingFubini’stheorem,weobtain

ˆ

Ωk

S_jf(x)dx ˆ

A^k_j

f(y)

→0lim

|{x∈Ω_k :y∈ω⁻_j (x)}|

dy (6.3)

wheretheone-sidedneighborhoodisdefinedas(see Fig.2) ω_j⁻(x) =x+ω_j^dir(x)(δ(x)−, δ(x)).

Nextweestimatethelimitexpressionin(6.3).Asj andkarefixed,wecanassume tobe verysmallrelativetothem.Letx∈Ωk.Assumethaty∈ω_j⁻(x).Then

− <|y−x| −δ(x)<0 (6.4) and bydefinitionxbelongstotheinteriorofE(y).

Set

e= b_x−x

|bx−x|

and letr ∈(0,δ(x)) be such thatx+re ∈P(y).Next we give anupper bound for r.

Because y∈ω_j⁻(x),italso holdsthat y−x

|y−x| ∈ω_j^dir(x).

Themapping

g(ρ) :=|y−(x+ρe)| −δ(x+ρe) =|y−x−ρe| −δ(x) +ρ is Lipschitzandhenceabsolutely continuous.

Forallρ≥0 wehavethelowerbound

g(ρ) =∂ρ[|y−(x+ρe)| −δ(x+ρe)] =−e· y−x−ρe

|y−x−ρe| + 1

= 1−cos(b_x−x, y−x−ρe)

≥1−cos(bx−x, y−x) 2^−2j

Thelast inequalityis duetoy∈ω_j⁻(x).Recallthatg(0)≥ −andg(r)= 0.Since gis absolutely continuous,weconclude

2^−2jr ˆr

0

g(s)ds=g(r)−g(0)≤,

and

r2^2j.

Denote x0 = x+re ∈ P(y). Consider the 2-planecontaining x, y, bx (and x0). Its intersection with theconvex body E(y) provided by Proposition6.1 is again aconvex set E in the plane. Let be its supporting line at x₀. Then x₀ ∈ P_j^k(y) for some j∈ {j,j−1}, k∈ {k,k−1}because

(bx−x0, y−x0)≥(bx−x, y−x)≥2^−j sin(bx−x0, y−x0)≤ |bx−y|

δ(x)−C2^2j = |bx−y|δ(x)⁻¹

1−Cδ(x)⁻¹2^2j ≤sin 2^−j+2

forsmallenough.ByProposition6.1thisalsomeansthaty−x₀makesanangle∼2^−j with , and hence so does x−x₀. Let e be the unit vector perpendicular to and e·(y−x)<0.Thenthere is

s|x−x0|sin 2^−j2^j

sothatx+se∈.Sincex∈E(y) andintersectsE(y) onlyin∂E(y),thereiss < s withx₁=x+se ∈∂E(y),whichmeans

dist(x, P(y))2^j. (6.5)

Recallthat

dist(x, P_j^k(y))≤ |x−x₀|2^2j. Let

N() =

j∈{j−1,j},k∈{k−1,k}

{x∈P(y) : dist(x, P_j^k(y))≤}.

Then

lim→0

|{x∈Ωk :y∈ω_j⁻(x)≤}|

≤ lim

→0lim

→0

|{x∈Ω_k : dist(x, P(y))≤c_n2^j} ∩N()|

lim

→02^jHⁿ⁻¹(P(y)∩N())

≤

|j−j|≤1,|k−k|≤1

2^jHⁿ⁻¹(P_j^k(y)),

where the second inequality follows, for instance, by Theorem 3.2.39 in [10]. The in- tegrable majorant of the sequence above that was needed for the application of the dominatedconvergencetheorembefore canbeobtainedbyanapplicationofthecoarea formula.Thiscompletes theproof.

These two propositions are enough to conclude a general L¹ bound for the pieces S_j^k. This bound can be refined further,when additional regularity onthe domain Ω is assumed.

Proposition 6.3. LetΩ bean openset.Then S_j^k L¹→L¹2^k(n−1)+j.

Proof. Ifx∈P(y)∩Ωk,then|x−y|= dist(x,Ω^c)≤2^k+1.HenceP(y)∩Ωk ⊂B(y,2^k+1).

RecallthatP(y)=∂E(y) andthatE(y) isconvex.ThusP(y)∩Ωk⊂∂(B(y,2^k+1)∩E(y)) where B(y,2^k+1)∩E(y) is convex. Since the perimeter of B(y,2^k+1) dominates the perimetersof allconvex setswithnon-emptyinteriorcontainedinit,wecanconclude Hⁿ⁻¹(P_j^k(y))≤ Hⁿ⁻¹(P(y)∩B(y,2^k+1))≤ Hⁿ⁻¹(∂(B(y,2^k+1)∩E(y)))

≤ Hⁿ⁻¹(∂B(y,2^k+1))2^k(n−1). Now theclaimfollows fromProposition6.2.

Remark6.4.IncaseΩ isboundedand∂Ω isC² smooth,theestimateforHⁿ⁻¹(P_j^k(y)) can be refined as follows. If x ∈ P_j^k(y), then |y −bx| ≤ δ(x)2^−j+1. This implies dist(y,∂Ω)≤δ(x)·2^−j+1 andfurther

|b_y−b_x| ≤ |b_y−y|+|y−b_x| ≤4δ(x)·2^−j.

As theinward-pointing unitnormalN_Ωattheboundaryiswell-definedandLipschitz,

|NΩ(by)−NΩ(bx)|diam(Ω)2^−j. Because N(bz)= (z−bz)/|z−bz|,thisimplies

NΩ(by)−(x−y) δ(x)

≤ |NΩ(by)−NΩ(bx)|+|y−bx|

δ(x) diam(Ω)·2^−j.

Therefore, allvectorsx−y withy ∈ωj(x) are withinanangle∼˜c(Ω)·2^−j of NΩ(by).

Hence theset P_j^k(y) iscontained inacylinder of height ∼2^k and basis ∼˜c(Ω)·2^k−j. Bytheinequalityforperimetersofconvex setsasintheproofofProposition6.3

Hⁿ⁻¹(P_j^k(y))c(Ω)2^k·2^{(k−j)(n−2)}.

Thisdependencyonj issharpevenforveryflatdomainsascanbe seenbylettingΩ be asmoothedoutB(0,10)∩ {x1≥0}andy= 2^−je1 andk≤0.

However, asthe estimateon Hⁿ⁻¹(P_j^k(y)) isnotthe narrowgap ofthe proofof our main theorem,wedonotpursuethisaspectfurther.

TheestimateHⁿ⁻¹(P_j^k(y))2^k(n−1)cannotbeimprovedingeneral.Iftheboundary of the domain is a single point, the equality is achieved up to a constant. However, focusing on the whole P_j(y) instead of single pieces P_j^k(y), onecan obtain adifferent estimate at costof worseningthe dependencyonj. Thefollowing proposition isuseful forsmall valuesofj, anditholdsinverygeneraldomains.

Proposition6.5. LetΩbean openset andy∈Ω.Then ˆ

Pj(y)

1

dist(x, y)ⁿ⁻¹dHⁿ⁻¹(x)2^j withtheconstant independentof y.Inparticular,

k

2^k(1−n)S_j^k _L¹_→L¹ 2^2j Proof. Wehave

ˆ

Pj(y)

1

dist(x, y)ⁿ⁻¹dHⁿ⁻¹(x)lim inf

→0

1

ˆ

{x∈E(y)^c:dist(x,Pj(y))≤}

1

dist(x, y)ⁿ⁻¹dx.

Given any point x ∈ P_j(y) and a line l_x = {y +t(x−y) : t ∈ R}, we see that by Proposition6.1thelinemakesanangle∼2^−j withPj(y),andhence

H¹(lx∩ {z∈E(y)^c : dist(z, Pj(y))≤})2^j.

Thefirstclaimedboundfortheintegralfollowsimmediatelyfrompassingtopolarcoor- dinateswithoriginat y.

Toprovethesecond claim,notethatbyProposition6.2 ˆ

Ω

k

2^k(1−n)S^k_jf(x)dx2^j

|j−j|≤1

ˆ

Ω

f(y)

k

2^k(1−n)Hⁿ⁻¹(P_j^k(y))

dy

2^j

|j−j|≤1

ˆ

Ω

f(y)

⎛

⎜⎝ ˆ

Pj(y)

1

dist(x, y)ⁿ⁻¹dHⁿ⁻¹(x)

⎞

⎟⎠dy

2^2j f L¹,

wherethelaststepwasanapplicationof thefirstclaim.

7. L^p boundsandgeometry

To conclude bounds for the operator Bα, we have to sum up all the pieces in the decomposition.Inordertomakethiswork,onehastoensurethatthereisenoughdecay inj andk.Although the L¹ bounds do notsumup, interpolationwith the betterL^∞ bounds provides us with enough decay in the angle parameter j. If Ω is bounded, we can take advantage of the L^p(Ω) spaces being nested and use the decay in the scale parameterkneartheboundarytocomplete theproof withnosmoothnessassumptions

ontheboundaryofthedomain.Thisispossibleonlywhen wedonotattempttoprove scalableestimatesthatwouldcaptureL^p→L^qsmoothingbeyondonederivativegain.

Theorem 7.1.LetΩbe aboundedopenset,p,α >1.Then B_{α L}^p_(Ω)→L^p_(Ω)diam(Ω)^α−1

where theimplicit constantonly dependson p,αandthedimension.

Proof. LetSj=

k2^k(α−n)S_j^k so thatBα=

j2^−jSj. ThenbyProposition6.3

Sj L¹(Ω)→L¹(Ω)≤

log diam(Ω)+1

k=−∞

2^k(α−n) S_j^k _L¹_(Ω)→L¹_(Ω)

log diam(Ω)+1

k=−∞

2^k(α−1)2^j2^jdiam(Ω)^α−1.

ByProposition5.2

2^−jS_{j L}∞(Ω)→L^∞(Ω)2^−njdiam(Ω)^α−1 and byinterpolationweobtain

2^−jSj L^p(Ω)→L^p(Ω)2^{−(p−1)np j}diam(Ω)^α−1.

As theexponentisnegative, wecansumupinj toconcludetheproof.

To dealwith thecriticalcaseα= 1 whereourestimateshavethecorrectscaling,we havetotakeintoaccountfinerpropertiesoftheboundary,astheestimationasroughas aboveleadstoalogarithmic blow-upofthek-sumattheboundary.

Proposition 7.2. LetΩ bean openset.

• If Ωsatisfies theuniform curvature boundwith R,thenforall y∈Ωandx∈P(y) with δ(x)≤R,itholdsthat

δ(x)(1−δ(x)

R )(1−cosβ)≤dist(y, ∂Ω) (7.1) where β=(b_x−x,y−x).

• If Ω^c isconvex,then

δ(x)(1−cosβ)≤dist(y, ∂Ω). (7.2)

Fig. 3.The balls and points appearing in the proof of Proposition7.2.

Proof. Take x ∈ Ω and y ∈ ∂B(x,δ(x)) and let β be the angle between b_x−x and y−x.BecauseΩ satisfiesauniformcurvaturebound, thereisanR >0 independentof xand y so thatwe canfind aball B(z,R)⊂Ω withz =b_x+ (x−b_x)R/δ(x) sothat B(z, R)∩∂Ω={b_x}(seeFig.3). ThePythagoreanidentityreads

|z−y|²= (δ(x) sinβ)²+ (R−δ(x)(1−cosβ))²

=R²(1−2δ(x)

R (1−δ(x)

R )(1−cosβ))

≤R²(1−δ(x)

R (1−δ(x)

R )(1−cosβ))².

Letw be theclosestpoint toy in∂B(z,R).Since z,y andw areonthe sameline,we get

dist(y, ∂Ω)≥ |y−w|=|z−w| − |z−y|

≥R−R(1−δ(x)

R (1−δ(x)

R )(1−cosβ))

=δ(x)(1−δ(x)

R )(1−cosβ)

asclaimed. IfΩ^c isconvex, thenthe uniformcurvaturebound issatisfiedwith R=∞, whencethesecond claimfollows.

Theorem7.3. LetΩbean openset.Letα= 1andp>1.Then

• If Ωisbounded andsatisfiestheuniformcurvature bound,then B1 L^p(Ω)→L^p(Ω)log

diam(Ω) R + 1

where Ristheradiusfrom theuniformcurvature bound.

• If Ω^c isconvex,then

B1 L^p(Ω)→L^p(Ω)1

andtheoperatornormonly dependsonthedimensionandp.

• If Ωismerelyopen,then

B1 L^p(Ω)→L^p(Ω)1 under therestrictionp>1+_n¹.

Proof. Proposition6.3implies ˆ

Ω

2^k(1−n)−jS_j^kf(x)dxˆ

f(y)1_A^k

j(y)dy. (7.3)

Recallthedefinition(6.2).Thereareonly∼log(diam(Ω)/R+1) valuesofksothatR/8≤ 2^k ≤2diam(Ω).Forksuchthat2^k+3≤R,wecanusethefirstiteminProposition7.2to seethatforfixedy,thesetP_j^k(y) isnon-emptyonlyforksuchthat2^−2j+kdist(y,∂Ω).

Ontheother hand,theupperbound

dist(y, ∂Ω)≤ |y−bx|2^k−j

is always valid, so P_j^k(y) isnon-empty only for 2^−2j+k dist(y,∂Ω) 2^−j+k. Conse- quently,

A^k_j ⊂ {y∈Ω : 2^−2j+k dist(y, ∂Ω)2^−j+k}.

For anyy,there areonlyj valuesksuchthattheset aboveisnon-empty,and hence by(7.3)

k

2^k(1−n)−jS_j^k _L¹_→L¹ log

diam(Ω) R + 1

+j.

InterpolationasintheproofofTheorem7.1 impliestheclaim.

Toprovetheseconditem,justnotethattheconvexityassumptiononthecomplement meanssendingR→ ∞sothat2^k+3≤Ralwaysholds.Toprovethethirditem,westudy Sj as inthe proof of Theorem 7.1 and replace the L¹ bound from Proposition 6.3 by thatfromProposition6.5.

Corollary 7.4.LetΩbe adomain,p>1andf ∈L^p.ThenA_αf(x)from(2.3) isweakly differentiable and

∇Aαf L^p(Ω) f L^p(Ω)

ifany one ofthefollowingholds:

• α >1andΩisbounded.

• α= 1 andΩisbounded andsatisfiesauniformcurvature bound.

• α= 1 andΩ^c isconvex.

• α= 1 andp>1+_n¹

The constantdependson thedomain,αandthedimension.

Proof. By linearity, it suffices to prove the norm inequality for smooth functions. By Proposition5.1,it sufficestobound Bα from (5.1).Thisfollows from Theorem7.1 and Theorem7.3

Theorem1.1follows fromCorollary7.4 andLemma4.1.

8. Remarks

8.1. Roleof thedomain

It isnotclear ifthe conditionsonthe domain inthehypothesis ofTheorem 1.1 are necessary.Onemayask if

∇M₁^Ω _L^p_(Ω)→L^p_(Ω)1

holdsforalldomains Ω andallp>1.Wearenotawareofany counterexamplesso far.

SinceM₀^ΩdoessatisfyanL^p(Ω) boundindependentofthedomain,thequestionisabout thebehavior ofB1 (seeTheorem7.3)ingeneraldomains.Wepointoutthatoneavenue forimprovingtheL^p boundsforB₁ couldbe toreplacethestrongL¹ boundsforS_j^k by weaktypeboundsinordertoimprove theoperatornormboundwith respecttoj.

8.2. Endpointregularity indomains

Corollary1.2follows fromTheorem1.1,since

∇M₁^Ωf _L^n/(n−1)(Ω) f _L^n/(n−1)(Ω) f _W^1,1_(Ω).

Hereweusedthemaintheoremand(2.2).Thesameobservationwasdoneby[5] tonotice thatthe fractional endpoint regularity problem follows from inequality (1.1) asα ≥1 in the full space Rⁿ. The domain case was not known before as the inequality (1.1) should have been replaced by (1.2). This amounts to changing the Hardy–Littlewood maximal function to thespherical maximal function inthe display above.Thatone is