Table Of ContentProgress in Mathematics
344
Juan José Marín
José María Martell
Dorina Mitrea
Irina Mitrea
Marius Mitrea
Singular Integral
Operators,
Quantitative
Flatness, and
Boundary Problems
Progress in Mathematics
Volume 344
SeriesEditors
AntoineChambert-Loir ,UniversitéParis-Diderot,Paris,France
Jiang-HuaLu,TheUniversityofHongKong,HongKongSAR,China
MichaelRuzhansky,GhentUniversity,Belgium,QueenMaryUniversityofLondon,
London,UK
YuriTschinkel,CourantInstituteofMathematicalSciences,NewYork,USA
Juan José Marín (cid:129) José María Martell (cid:129)
Dorina Mitrea (cid:129) Irina Mitrea (cid:129) Marius Mitrea
Singular Integral Operators,
Quantitative Flatness, and
Boundary Problems
JuanJoséMarín JoséMaríaMartell
InstituteofMathematicalSciences InstituteofMathematicalSciences
Madrid,Spain UniversidadAutónomadeMadrid
Madrid,Spain
DorinaMitrea IrinaMitrea
DepartmentofMathematics DepartmentofMathematics
BaylorUniversity TempleUniversity
Waco,TX,USA Philadelphia,PA,USA
MariusMitrea
DepartmentofMathematics
BaylorUniversity
Waco,TX,USA
ISSN0743-1643 ISSN2296-505X (electronic)
ProgressinMathematics
ISBN978-3-031-08233-7 ISBN978-3-031-08234-4 (eBook)
https://doi.org/10.1007/978-3-031-08234-4
MathematicsSubjectClassification:31B10,35B65,35C15,35J25,35J57,35J67,42B20,42B37
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Preface
We develop the theory of layer potentials in the context of δ-AR domains in Rn
(aka δ-flat Ahlfors regular domains) where the parameter δ > 0, regulating the
size of the BMO semi-norm of the outward unit normal ν to (cid:4), is assumed to be
small.Thisisasub-categoryoftheclassoftwo-sidedNTAdomainswithAhlfors
regular boundaries, and our results complement work carried out [61] in regular
SKT domains (withSKT acronym for Semmes-Kenig-Toro). The latter brand was
designed to work well when the domains in question have compact boundaries.
By way of contrast, the fact that we are now demanding ||ν||[BMO(∂(cid:4),σ)]n is small
enough (where σ is the “surface measure” Hn−1(cid:2)∂(cid:4)) has topological and metric
implicationsfor(cid:4),namely(cid:4)isaconnectedunboundedopenset,withaconnected
unboundedboundaryandanunboundedconnectedcomplement.Forexample,inthe
two-dimensional setting, we show that the class of δ-AR domains with δ ∈ (0,1)
smallagreeswiththecategoryofchord-arcdomainswithsmallconstant.
Assuming (cid:4) ⊆ Rn to be a δ-AR domain with δ ∈ (0,1) sufficiently small
(relativetothedimensionnandtheAhlforsregularityconstantof∂(cid:4)),weprovethat
theoperatornormofCalderón-Zygmundsingularintegralswhosekernelsexhibita
certainalgebraicstructure(specifically,theyconta(cid:2)intheinne(cid:3)rproductofthenormal
ν(y) with the “chord” x − y as a factor) is O δln(1/δ) as δ → 0+. This is
true in the context of Muckenhoupt weighted Lebesgue spaces, Lorentz spaces,
Morreyspaces,vanishingMorreyspaces,blockspaces,(weighted)Banachfunction
spaces,aswellasforthebrandsofSobolevspacesnaturallyassociatedwiththese
scales. Simply put, the problem that we solve here is that of determining when
(andhow)singularintegraloperatorsofdouble-layertypehavesmalloperatornorm
on domains which are relatively “flat.” We also establish estimates in the opposite
direction, quantifying the flatness of a “surface” by estimating the BMO semi-
normofitsunitnormalintermsoftheoperatornormsofcertainsingularintegrals
canonically associated with the given surface (such as the harmonic double layer,
the family of Riesz transforms, and commutators between Riesz transforms and
pointwise multiplication by the components of the unit normal). Ultimately, this
goes to show that the two-way bridge between geometry and analysis constructed
hereisinthenatureofbestpossible.
v
vi Preface
Significantly, the operator norm estimates described in the previous paragraph
permit us to invert the boundary double-layer potentials associated with certain
classes of second-order PDE (such as the Laplacian, any scalar homogeneous
constant complex coefficient second-order operator which is weakly elliptic when
n ≥ 3 or strongly elliptic in any dimension, the Lamé system of elasticity, and,
most generally, any weakly elliptic homogeneous constant complex coefficient
second-order system having a certain distinguished coefficient tensor), acting on
a large variety of function spaces considered on the boundary of a sufficiently flat
domain(specifically,aδ-ARdomainwithδ ∈(0,1)suitablysmallrelativetoother
geometriccharacteristicsofsaiddomain).Inparticular,thisportionofourworkgoes
inthedirectionofansweringthequestionposedbyC.Kenigin[71,Problem3.2.2,
p.117] asking to invert layer potentials in appropriate spaces on certain uniformly
rectifiablesets.
In turn, these invertibility results allow us to establish solvability results for
boundary value problems in the class of weakly elliptic second-order systems
mentionedabove,inasufficientlyflatAhlforsregulardomain,withboundarydata
from Muckenhoupt weighted Lebesgue spaces, Lorentz spaces, Morrey spaces,
vanishingMorreyspaces,blockspaces,Banachfunctionspaces,andfromSobolev
spacesnaturallyassociatedwiththesescales.
In summary, a central theme in Geometric Measure Theory is understanding
how geometric properties translate into analytical ones, and here we explore the
implications of demanding that Gauss’ map ∂(cid:4) (cid:7) x (cid:8)→ ν(x) ∈ Sn−1 has small
BMO semi-norm in the realm of singular integral operators and boundary value
problems. The theory developed here complements the results of S. Hofmann,
M. Mitrea, and M. Taylor obtained in [61] and extends previously known well-
posednessresultsforellipticPDEintheupperhalf-spacetotheconsiderablymore
inclusiverealmofδ-ARdomainswithδ ∈(0,1).
Acknowledgments
Portions of this work were completed at Baylor University, Waco, USA, Temple
University,Philadelphia,USA,andICMAT,Madrid,Spain.Theauthorsthankthese
institutionsfortheirhospitality.Theauthorsalsoacknowledgepartialsupportfrom
the Simons Foundation (through grants #426669, # 958374, #318658, #616050,
#637481), National Science Foundation (grant #1900938), and MCIN/AEI/
10.13039/501100011033 (grants CEX2019-000904-S and PID2019-107914GB-
I00).
Madrid,Spain JuanJoséMarín
Madrid,Spain JoséMaríaMartell
Waco,TX,USA DorinaMitrea
Philadelphia,PA,USA IrinaMitrea
Waco,TX,USA MariusMitrea
March2022
Contents
1 Introduction .................................................................. 1
2 GeometricMeasureTheory................................................. 27
2.1 ClassesofEuclideanSetsofLocallyFinitePerimeter ............... 28
2.2 ReifenbergFlatDomains .............................................. 53
2.3 Chord-ArcCurvesinthePlane ........................................ 66
2.4 TheClassofDelta-FlatAhlforsRegularDomains ................... 85
2.5 TheDecompositionTheorem .......................................... 100
2.6 Chord-ArcDomainsinthePlane ...................................... 120
2.7 Dyadic Grids and Muckenhoupt Weights on Ahlfors
RegularSets ............................................................ 128
2.8 SobolevSpacesonAhlforsRegularSets .............................. 145
3 Calderón–ZygmundTheoryforBoundaryLayersinURDomains.... 163
3.1 BoundaryLayerPotentials:TheSetup ................................ 163
3.2 SIOsonMuckenhouptWeightedLebesgueandSobolev
Spaces ................................................................... 179
3.3 DistinguishedCoefficientTensors ..................................... 200
4 BoundednessandInvertibilityofLayerPotentialOperators........... 241
4.1 EstimatesforEuclideanSingularIntegralOperators ................. 241
4.2 EstimatesforCertainClassesofSingularIntegralsonUR
Sets ...................................................................... 259
4.3 NormEstimatesandInvertibilityResultsforDoubleLayers ........ 294
4.4 Invertibility on Muckenhoupt Weighted Homogeneous
SobolevSpaces ......................................................... 318
4.5 AnotherLookatDoubleLayersfortheTwo-Dimensional
LaméSystem ........................................................... 330
5 ControllingtheBMOSemi-NormoftheUnitNormal................... 339
5.1 CliffordAlgebrasandCauchy–CliffordOperators ................... 340
5.2 EstimatingtheBMOSemi-NormoftheUnitNormal ................ 344
vii
viii Contents
5.3 UsingRieszTransformstoQuantifyFlatness ........................ 352
5.4 Using RieszTransforms toCharacterize Muckenhoupt
Weights ................................................................. 355
6 BoundaryValueProblemsinMuckenhouptWeightedSpaces ......... 365
6.1 TheDirichletProbleminWeightedLebesgueSpaces ................ 367
6.2 TheRegularityProbleminWeightedSobolevSpaces ................ 379
6.3 TheNeumannProbleminWeightedLebesgueSpaces ............... 396
6.4 TheTransmissionProbleminWeightedLebesgueSpaces ........... 411
7 SingularIntegralsandBoundaryProblemsinMorreyand
BlockSpaces.................................................................. 433
7.1 BoundaryLayerPotentialsonMorreyandBlockSpaces ............ 433
7.2 InvertingDoubleLayerOperatorsonMorreyandBlock
Spaces ................................................................... 460
7.3 InvertibilityonMorrey/Block-BasedHomogeneous
SobolevSpaces ......................................................... 467
7.4 CharacterizingFlatnessinTermsofMorreyandBlock
Spaces ................................................................... 475
7.5 BoundaryValueProblemsinMorreyandBlockSpaces ............. 481
8 Singular Integrals and Boundary Problems in Weighted
BanachFunctionSpaces .................................................... 497
8.1 BasicPropertiesandExtrapolationinBanachFunction
Spaces ................................................................... 497
8.2 BoundaryLayerPotentialsonWeightedBanachFunction
Spaces ................................................................... 511
8.3 InvertingDoubleLayerOperatorsonWeightedBanach
FunctionSpaces ........................................................ 529
8.4 Invertibility on Homogeneous Weighted Banach
Function-BasedSobolevSpaces ....................................... 533
8.5 Characterizing Flatness in Terms of Weighted Banach
FunctionsSpaces ....................................................... 540
8.6 BoundaryValueProblemsinWeightedBanachFunction
Spaces ................................................................... 546
8.7 ExamplesofWeightedBanachFunctionSpaces ..................... 564
8.7.1 UnweightedBanachFunctionSpaces ......................... 564
8.7.2 RearrangementInvariantBanachFunctionSpaces ........... 566
References......................................................................... 587
SubjectIndex..................................................................... 595
SymbolIndex..................................................................... 599
Chapter 1
Introduction
Morethan25yearsago,in[71,Problem3.2.2,p.117],C.Kenigaskedto“Provethat
thelayerpotentialsareinvertibleinappropriate[...]spacesin[suitablesubclasses
of uniformly rectifiable] domains.” Kenig’s main motivation in this regard stems
from the desire of establishing solvability results for boundary value problems
formulated in a rather inclusive geometric setting. In the buildup to this open
questionon[71,p.116],itisremarkedthatthereexistsomerathergeneralclasses
ofopensets(cid:4) ⊆ Rn withthepropertythatifσ := Hn−1(cid:2)∂(cid:4)(whereHn−1 stands
forthe(n−1)-dimensionalHausdorffmeasureinRn)thensaidlayerpotentialsare
boundedoperatorsonLp(∂(cid:4),σ)foreachexponentp ∈ (1,∞).Remarkably,this
isthecasewhenever(cid:4) ⊆ Rn isanopensetwithauniformlyrectifiableboundary
(cf.[40]).
Tofurtherelaborateonthisissue,weneedsomenotation.Fixn∈Nwithn≥2,
alongwithM ∈ N,andconsiderasecond-order,homogeneous, constantcomplex
coefficient,weaklyelliptic,M ×M systeminRn
(cid:2) (cid:3)
L= aαβ∂ ∂ , (1.1)
jk j k 1≤α,β≤M
where the summation convention over repeated indices is in effect (here and
elsewhere in the manuscript). The weak ellipticity of the system L amounts to
demandingthat
(cid:2) (cid:3)
thecharacteristicmatrixL(ξ) := −aαβξ ξ is
jk j k 1≤α,β≤M (1.2)
invertibleforeachvectorξ =(ξ ,...,ξ )∈Rn\{0}.
1 n
This should be contrasted with the more stringent Legendre–Hadamard (strong)
ellipticityconditionwhichasksfortheexistenceofsomec>0suchthat
(cid:4) (cid:5)
Re −L(ξ)ζ,ζ ≥c|ξ|2|ζ|2 forall ξ ∈Rn and ζ ∈CM. (1.3)
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J.J.Marínetal.,SingularIntegralOperators,QuantitativeFlatness,andBoundary
Problems,ProgressinMathematics344,
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