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Sobolev-Type Spaces (Mainly Based on the Lp Norm) on Metric Spaces, and Newto- Nian Spaces in Particular, Have Been Under Intensive Study Since the Mid-"##$S

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Sobolev-Type Spaces (Mainly Based on the Lp Norm) on Metric Spaces, and Newto- Nian Spaces in Particular, Have Been Under Intensive Study Since the Mid- - ) . / 0 &/ . ! " # $" ! %&' '( ! ) *+, !"## $ "% ' ( &)*+) ,##(-./*01*2/ ,#3(+140+)01*)+0.*/0- %"52-)/ 6 7 %80$%6!"6#62-)/ i Abstract !is thesis consists of four papers and focuses on function spaces related to !rst- order analysis in abstract metric measure spaces. !e classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose de!nition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behav- ior along (almost) all recti!able curves, which gives rise to the so-called Newtonian spaces. !e summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newto- nian spaces in particular, have been under intensive study since the mid-"##$s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newto- nian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, New- tonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s di%erentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev func- tions in analysis of partial di%erential equations. In fact, Lipschitz continuity, which is (unlike C!-smoothness) well-de!ned even for functions on metric spaces, of- ten su&ces as a regularity condition. !us, Paper III concentrates on the question when Lipschitz functions provide good approximations of Newtonian functions. As shown in the paper, it su&ces that the function lattice quasi-norm is absolutely continuous and a fractional sharp maximal operator satis!es a weak norm estimate, which it does, e.g., in doubling Poincaré spaces if a non-centered maximal operator of Hardy–Littlewood type is locally weakly bounded. !erefore, such a local weak boundedness on rearrangement-invariant spaces is explored as well. Finer qualitative properties of Newtonian functions and the Sobolev capacity get into focus in Paper IV. Under certain hypotheses, Newtonian functions are proven to be quasi-continuous, which yields that the capacity is an outer capacity. Various su&cient conditions for local boundedness and continuity of Newtonian functions are established. Finally, quasi-continuity is applied to discuss density of locally Lip- schitz functions in Newtonian spaces on open subsets of doubling Poincaré spaces. ii D !"#!$%& #'()*) !+ L,-./ M%&0 iii Popul ärvetenskaplig sammanfattning Avhandlingen best år av fyra artiklar som behandlar s å kallade newtonrum baserade på vissa kvasinormer . L åt oss f örs öka belysa det som d öljs bakom dessa begrepp. Många f öreteelser i naturvetenskap kan beskrivas med hj älp av di%erentialekva- tioner. Till exempel kan temperaturf ördelningen i en kropp uttryckas som en funk- tion beroende p å tid och position och den l öser den s å kallade v ärmeledningsekva- tionen. Vanligtvis kan man visa att det !nns en entydig l ösning, men det är s ällan möjligt att !nna den explicit. Å andra sidan kan man med hj älp av datorer f örs öka ber äkna en approximation av den s ökta l ösningen. Ett problem med detta angrepps- sätt är att man inte i f örv äg vet om det numeriska resultatet n ärmar sig den verkliga lösningen. I allm änhet kan l ösningarna vara ganska ”vilda” funktioner. Emellertid går det o/a att visa 0era trevliga egenskaper hos dem. Dessa egenskaper kan t.ex. vara uppskattningar av hur stora v ärden funktionen antar och hur snabbt v ärdena sv änger. Med hj älp av ett l ämpligt m ätningss ätt kan man avg öra om en approxima- tion till di%erentialekvationens l ösning representerar verkligheten tillr äckligt v äl. Det !nns olika s ätt att m äta storleken av funktioner samt deras variationer (allts å i princip derivator), och denna avhandling behandlar olika s ådana s ätt samt den rika teorin omkring dessa. Teorin f ör m ätning av icke-sl äta funktioner p å våra vanliga ( åsk ådliga) rum stu- derades noggrant under st örre delen av "#$$-talet. Under de senaste "1–2$ åren har man f örs ökt utvidga detta till allm änna s å kallade metriska rum, d är man endast känner till avst åndet mellan tv å godtyckliga punkter. Metriska rum kan vara v äl- digt oregelbundna (t.ex. fraktaler) s å att det i sj älva verket inte beh över !nnas n ågra räta linjer mellan olika punkter, vilket f örhindrar till ämpning av den klassiska teo- rin. Änd å kan man o/a f örbinda punkterna med kurvor. Om man n öjer sig med att mäta funktionen l ängs kurvor, s å förenklas problemet. Finns det tillr äckligt m ånga kurvor, s å kan man änd å dra viktiga slutsatser om beteendet av en funktion med hj älp av m ätningar l ängs alla m öjliga kurvor. Detta är ungef är den princip som st år bakom teorin om newtonrum. Newtonrum har redan studerats f ör vissa typer av storleksbegrepp. Teorin i av- handlingen byggs, mer allm änt än tidigare, baserad p å kvasinormer i banachfunk- tionslattice , vilket ger m ånga generaliseringar och kvalitativa f ör!ningar av hittills kända resultat. Låt oss nu betrakta ett 0ertal fr ågor som man o/a vill f å besvarade och som besvaras i avhandlingen. Antag att man vill unders öka en funktion som beskriver någon fysikalisk storhet och att funktionens variation redan har m ätts (eller sna- rare uppskattats) l ängs alla kurvor. Man kan fr åga sig huruvida det verkligen var nödv ändigt att kolla p å funktionens beteende p å alla kurvor. Det visar sig att man iv D !"#!$%& #'()*) !+ L,-./ M%&0 mycket v äl kunde ha struntat i ett litet antal kurvor (i en v äldigt precis mening) utan att f örlora v äsentliga data. Dessutom inser man att funktionen egentligen är gans- ka sn äll ( absolutkontinuerlig ) l ängs alla kurvor med undantag av ett litet antal (d är ”litet antal” har samma betydelse som ovan). Man kan beh öva ta reda p å om det !nns st ällen (m ängder) d är funktionsv är- dena inte styrs av m ätningar l ängs kurvor och hur stora dessa m ängder egentligen är. I avhandlingen visas att det exakt är de m ängder vars s å kallade sobolevkapacitet är lika med noll. Sobolevkapaciteten kan i princip tolkas som ett volymm ått med en !nare uppl ösning än det vanliga volymm åttet. Str ängt taget inneb är det att m äng- derna d är man saknar kontroll över funktionsv ärden är ännu mindre än n är man anv änder sig av den klassiska sobolevteorin om icke-sl äta funktioner i det vanliga rummet Rn. Trots att man i de 0esta fall endast överuppskattar funktionernas variationer, s å känns det bara naturligt att f örv änta sig att det !nns en b ästa (allts å minsta) upp- skattning. Fr ågan om huruvida den existerar är i sj älva verket mer komplicerad med tanke p å att funktionernas beteende kan vara of örutsebart l ängs ett visst litet antal kurvor. Dessutom beror den egentliga inneb örden av frasen ”litet antal” p å storleks- begreppet som best äms av funktionslatticets kvasinorm . F ör vissa storleksbegrepp var den h är existensfr ågan ett öppet problem inom newtonteori tills den besvarades jakande i en av avhandlingens artiklar. När partiella di%erentialekvationer studeras, s å kr äver m ånga resonemang att sl äta funktioner utg ör bra approximationer till alla funktioner som kommer i fr åga som svaga l ösningar till ekvationen. Om omr ådet har goda geometriska egenskaper, så beror approximerbarheten p å storleken av s å kallade maximalfunktioner . D ärf ör unders öks maximalfunktioner samt deras uppskattningar noggrant i avhandlingen. Följaktligen f år man fram n ågra enkla villkor som r äcker f ör att kunna anv ända sig av sl äta approximationer. D ärtill ges typexempel d är s ådana approximationer inte är m öjliga. Till sist studeras det om uppskattningar av storleken av en funktions variationer längs kurvor leder till att sj älva funktionen är begr änsad eller till och med kontinu- erlig. S å ä r fallet om det storleksbegrepp som anv änds vid uppskattningar (d.v.s. kvasinormen av banachfunktionslatticet ) är tillr äckligt restriktivt i j ämf örelse med dimensionen av volymm åttet. Man inser n ämligen att det d å inte kan !nnas n ågra icke-tomma m ängder som har sobolevkapacitet noll. Det inneb är i sin tur att funk- tionen är kontrollerad överallt av sina variationer som enbart m ätts eller uppskattats längs n ästan alla kurvor. v Acknowledgements It has been "" years since I set out on a journey through the wondrous lands ruled by the mighty Queen of the Sciences; and some journey it has been! Not always did I know where the path I decided to take was about to lead. I have had the chance to experience the marvelous beauty of mathematics, but I have also come across places I personally deem somewhat gloomy.
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