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Progress 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 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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) ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1 J.J.Marínetal.,SingularIntegralOperators,QuantitativeFlatness,andBoundary Problems,ProgressinMathematics344, https://doi.org/10.1007/978-3-031-08234-4_1