Regularity Properties of Sectorial Operators: Extrapolation, Counterexamples and Generic Classes
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Universität Ulm Regularity Properties of Sectorial Operators: Extrapolation, Counterexamples and Generic Classes Vorgelegt von Stephan Fackler aus Ulm im Jahr 2014 Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät für Mathematik und Wirtschaftswissenschaften der Universität Ulm Amtierender Dekan: Prof. Dr. Dieter Rautenbach Gutacher: Prof. Dr. Wolfgang Arendt Prof. Dr. Rico Zacher Prof. Dr. Gilles Lancien Tag der Promotion: 5. November 2014 Contents Introduction iii I Regularity Properties of Sectorial Operators 1 1 Sectorial Operators – Regularity Properties 3 1.1 Sectorial Operators . 3 1.2 -Sectorial Operators and Maximal Regularity . 4 R 1.3 Bounded H1-Calculus for Sectorial Operators . 8 1.4 Sectorial Operators which Have a Dilation . 15 1.5 Bounded Imaginary Powers (BIP) . 16 2 Counterexamples 19 2.1 The Schauder Multiplier Method . 19 2.1.1 Schauder Multipliers . 20 2.1.2 Sectorial Operators without a Bounded H1-Calculus . 24 2.1.3 Sectorial Operators without BIP . 27 2.1.4 Sectorial Operators without Maximal Regularity: The Maximal Regularity Problem . 29 2.1.5 A First Application: Existence of Schauder Bases which are not -Bases . 59 R 2.1.6 A Second Application: Sectorial Operators whose Sum is not Closed . 60 2.2 Using Pisier’s Counterexample to the Halmos Problem . 64 2.3 Using Monniaux’s Theorem . 65 2.3.1 Some Results on Exotic Banach Spaces . 68 2.4 Notes & Open Problems . 69 3 Extrapolation of Regularity Properties 73 3.1 Extrapolation of Analyticity . 74 3.1.1 Via the Abstract Stein Interpolation Theorem . 75 3.1.2 Via a Kato–Beurling Type Characterization . 76 3.1.3 An Application: A Zero-Two Law for Cosine Families 86 3.2 Extrapolation of -Analyticity . 89 R 3.2.1 Via Abstract Stein Interpolation . 89 3.2.2 Via a Kato–Beurling Type Characterization . 90 3.2.3 A Counterexample to the Maximal Regularity Extrapo- lation Problem . 96 3.2.4 Exact Control of the Extrapolation Scale . 101 i Contents 3.3 Notes & Open Problems . 109 II Structural Characterizations of Sectorial Operators with a Bounded H 1-calculus 113 4 Semigroups with a Bounded H 1-Calculus 115 4.1 Contractive Semigroups on Hilbert Spaces . 115 4.2 Positive Contractive Semigroups on Lebesgue-Spaces . 116 4.2.1 Some Operator Theoretic Results for Lp-Spaces . 116 4.2.2 Fendler’s Dilation Theorem for Subspaces of Lp . 120 4.2.3 An Application to Ergodic Theory . 129 4.2.4 Bounded H1-Calculus for r-Contractive Semigroups . 131 4.3 Basic Persistence Properties of the H1-Calculus . 133 4.4 Notes & Open Problems . 135 5 Generic Classes for Bounded H 1-Calculus 139 5.1 p-Completely Bounded Maps and p-Matrix Normed Spaces . 140 5.2 The p-Matrix Normed Space Structure for H1(Σθ) . 140 5.3 A p-Completely Bounded H1-Calculus . 142 5.4 Pisier’s Factorization Theorem . 146 5.5 Regularization of the Constructed Semigroup . 148 5.6 Notes & Open Problems . 158 Appendices 161 A Banach Spaces and Lattices 163 A.1 Schauder Bases & Schauder Decompositions . 163 A.2 Geometry of Banach Spaces – General Methods . 168 A.3 Geometric Properties of Banach Spaces . 170 A.4 Banach Lattices . 175 A.5 Interpolation Theory . 176 B Operator Spaces 181 Bibliography 183 List of Mathematical Symbols 197 Index 199 Zusammenfassung in deutscher Sprache 203 ii Introduction The study of partial differential equations is a central topic in modern math- ematics. While Picard’s existence theorem and Peano’s existence theorem give very satisfying answers to the problem of existence and uniqueness of ordinary differential equations, such a complete theory does not exist for partial differential equations. In fact, for typical partial differential equations one has to deal with unbounded operators on infinite-dimensional spaces which causes fundamental new mathematical difficulties which are usually studied with sophisticated functional analytic methods. In this thesis we consider fundamental problems related to methods that are suited for the study of non-linear parabolic partial differential equations. Very prominent examples of this type are among many others the famous Navier–Stokes equations in fluid dynamics and in differential geometry the mean curvature flow for a surface given as the graph of a function u : Rn R ! which is described by the evolution equation 8 n > X @iu@j u > ∆ Rn >@tu u = 2 @i@j u in (0;T ) (MCF) < − − 1 + u × > i;j=1 jr j > :> u = u in Rn: jt=0 0 The existence of a unique local strong solution of such a non-linear parabolic equation can be established as follows. After linearizing the non-linear equa- tion one obtains an equation of the form 8 > <>u˙(t) + A(u(t)) = 0 > :> u(0) = u0; i.e. the abstract first order Cauchy problem for an (unbounded) closed opera- tor A on some function space X. For example, for (MCF) a natural choice is the realization of A = ∆ on L (Rn) for some p (1; ). Such abstract Cauchy − p 2 1 problems which are fundamental for the understanding of the behaviour of linear parabolic problems are naturally studied in the theory of strongly continuous one-parameter semigroups of linear operators on Banach spaces, or more shortly C0-semigroups. This theory is by now very well-understood. Hence, it is very natural to take this theory as a starting point for the under- standing of non-linear problems. Indeed, if one sees the right hand side of (MCF) in the abstract setting as a function u G(u) the non-linear equation 7! becomes 8 > <>u˙ + Au = G(u) > :> u(0) = u0: iii Introduction Suppose that the solution operator L: u (u˙ + Au;u ) is invertible on an 7! 0 appropriate X-valued function space Z. Then one can take its inverse in order to rewrite the equation into a fixed point problem which one may hope to solve with one of the well-known fixed point theorems, e.g. the Banach fixed point theorem or a variant of the Schauder fixed point theorem. One sees that for this at least the linear problem should be well-posed which is equivalent for A to be the generator of strongly continuous semigroup. − It is then most convenient to see the first component of the solution operator u as the sum of two closed operators + . Here is the derivation B A B operator on X-valued functions and is the pointwise application of A. For A L to be invertible one then needs that for each f Z there exists a unique 2 u D( ) D( ) with 2 A \ B u + u = u˙ + Au = f : B A In applications, this property together with some regularity of the given non-linearity G then guarantees the self-mapping property of the respective maps needed for the fixed point theorems. We are now interested in the case where Z = Lp([0;T ];X) is some vector-valued Lp-space which is a natural choice in view of Sobolev methods. In this case the above requirement then translates to the fact that for a given f L ([0;T ];X) and for the unique 2 p solution u of u˙ + Au = f together with the initial condition both u˙ and Au lie in Lp([0;T ];X). In other words, this means that both summands have the same regularity as the right hand side f . Since this is the best regularity one can hope for, one therefore speaks of maximal regularity or more precisely of maximal Lp-regularity. One can show that for this property to hold for the most general initial values u0 possible, namely the traces of functions in W 1([0;T ];X) L ([0;T ];D(A)) it suffices to verify it for the initial condition p \ p u0 = 0. We are therefore interested in the following property which we now formulate precisely. Let X be a Banach space and A the generator of a C -semigroup on X. − 0 We say that A has maximal L -regularity for p (1; ) and T > 0 if for all p 2 1 f L ([0;T ];X) the unique (mild) solution u of the abstract inhomogeneous 2 p Cauchy problem 8 > <>u˙ + Au = f > :> u(0) = 0 satisfies u W 1([0;T ];X) L ([0;T ];D(A)), where D(A) is the domain of the 2 p \ p closed operator A endowed with its graph norm. In fact, as a working example, one can show maximal regularity for the realization of ∆ on L (Rn) for all − p n N and all p (1; ). Together with some routine applications of the 2 2 1 Sobolev embedding theorems and a routine application of the Banach fixed point theorem this yields the existence of a unique local strong solution of iv (MCF) and many other important non-linear parabolic partial differential equations. In this thesis we answer some fundamental structural questions concern- ing maximal regularity and other closely related regularity properties, thereby making use of the geometric theory of Banach spaces and operator space the- ory. We now motivate the considered problems in more detail. The Maximal Regularity Problem Now that we have seen the importance of maximal regularity for non-linear equations, the most natural question to ask is which (negative) semigroup generators have maximal regularity. In fact, maximal regularity can be verified for a broad class of realizations of various differential operators. Moreover, it is a classical result that a necessary condition for A to have maximal regularity is that A generates an − analytic semigroup, i.e. the semigroup mapping t T (t) can be extended to 7! an analytic mapping on some sector around the positive real line.