Lecture 2: Some basic principles of the b-calculus

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1 Lecture 2: Some basic principles of the b-calculus Daniel Grieser (Carl von Ossietzky Universität Oldenburg) September 20, 2012 Summer School Singular Analysis Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

2 General principles for studying singular problems Preliminary step: Put problem in the form (X, V). (This may involve blow-ups, e.g. cone X ) General principles for studying (X, V) Split into geometric and analytic aspects: Geometry encodes singular structure Analysis: conormal distributions ( hide Fourier transform) This helps when studying complicated singularities. Separate different types of singular behavior by blow-ups Describe operators via their Schwartz kernels Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

3 Motivation: Significance of Schwartz kernel Schwartz kernel K A D (M 2 ) of a linear operator A on C (M): (Af )(p) = K A (p, p ) f (p ) dp M Knowing K P 1 yields info. on solutions of Pu = f, e.g. (supp f compact): P = on R n, n 3: K P 1(p, p ) = c n p p 2 n. Order -2 singularity at diagonal p = p regularity of u Polynomial decay as p p polynomial decay of u at P = + 1: K P 1(p, p ) = ĥ(p p ), h(ξ) = 1 ξ Order -2 singularity at diagonal p = p regularity of u Exponential decay as p p rapid decay of u at Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

4 Basics: Geometry Spaces: Manifolds with corners, p-submanifolds Operation on spaces: Blow-up Definition/Lemma: Blow-up Let Z be a manifold, p Z. Then there is a unique manifold with boundary, denoted [Z, p], and smooth map β : [Z, p] Z with [Z, p] \ β 1 (p) Z \ {p} is a diffeomorphism locally near p, β = polar coordinates R + S n 1 R n Generalizations: Replace p by a closed submanifold Y, get [Z, Y ] Z = manifold with corners, Y a p-submanifold Iterate Useful coordinates on [Z, Y ]: projective coordinates Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

5 Basics: Analysis Definition: Conormal distribution Z manifold, D Z closed submanifold. A distribution u D (Z) is conormal with respect to D if u is smooth on Z \ D, and in any local coordinates with D = {z = 0} u(y, z) = e izζ a(y, ζ) dζ R N k where a is a symbol. (N = dim Z, k = dim D) (principal) symbol of u: a (mod lower order terms), well-defined as function on N D Warning: no relation to conormal symbol of cone calculus Examples: δ, δ, p.v. 1 x, Schwartz kernels of (pseudo)-differential operators Generalization: Z manifold with corners, D p-submanifold Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

6 Basics: Classical ΨDO calculus X = compact smooth manifold Operators Symbols (on T X ) Schwartz kernels (in D (X 2 )) Diff (X ) homog. polynomials in ξ δ-type at D X Ψ (X ) homog. functions in ξ Conormal w.r.t. D X Composition Theorem: Ψ (X ) is closed under products and the Symbol map σ : Ψ (X ) S (T X ) preserves products There is a short exact symbol sequence 0 Ψ m 1 (X ) Ψ m (X ) S [m] (T X ) 0 Theorem Asymptotic completeness These properties give parametrix construction: P Ψ m (X ) elliptic Q Ψ m (X ) with PQ I, QP I Ψ (X ). Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

7 Basics: Classical ΨDO calculus Functional analysis: P Ψ m (X ) bounded H s (X ) H s m (X ) R Ψ (X ) K R smooth R compact operator Corollary P Ψ m (X ) elliptic, then elliptic regularity: Pu = f, f H s m (X ) u H s (X ) P Fredholm Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

8 What we ll get: V ΨDO calculus X = compact manifold with corners, V Lie algebra of vector fields Operators Symbols (on V T X ) Schwartz kernels (in D (X 2 V )) Diff V (X ) homog. polynomials in ξ δ-type at D X,V Ψ V (X ) homog. functions in ξ Conormal w.r.t. D X,V Composition Theorem: Ψ V (X ) is closed under products and the Symbol map V σ : Ψ V (X ) S ( V T X ) preserves products There is a short exact symbol sequence 0 Ψ m 1 V (X ) Ψ m V (X ) S [m] ( V T X ) 0 Asymptotic completeness Theorem These properties give parametrix construction: P Ψ m V (X ) elliptic Q Ψ m V (X ) with PQ I, QP I Ψ V (X ). Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

9 What we ll get: V ΨDO calculus Functional analysis: P Ψ m V (X ) bounded Hs V (X ) Hs m V (X ) R Ψ V (X ) K R smooth (but R compact operator) Corollary P Ψ m V (X ) V-elliptic, then small elliptic regularity: Pu = f, f H s m V (X ) u HV s (X ) To get compact errors (hence Fredholm P), need larger calculus or stronger ellipticity condition. Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

10 Main steps in building a V ΨDO calculus 1 Construct double space X 2 V. Requirements: Diagonal D X lifts to p-submanifold D X,V For any V V, the vector field V 0 on X 2 lifts smoothly to X 2 V These lifts span the normal space to D X,V 2 Define small V-calculus Ψ V (X ) via Schwartz kernels on X 2 V : conormal w.r.t. D X,V (uniformly to the boundary) vanish to all orders at all faces except those intersecting D X,V symbols are functions on V T X = N D X,V can invert V-elliptic operators up to smoothing errors. 3 Identify obstruction to compactness of smoothing operators. normal, indicial operator 4 If needed, enlarge calculus by including inverses of normal operator get compact errors Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11

11 Example: b-calculus The problem: X = cpct manifold with boundary, V b = {vector fields tangent to X } (spanned by x x, yi near boundary) The solution: 1 Double space: X 2 b := [X 2, ( X ) 2 ] (check lifts of x x, yi ) 2 Small b-calculus: Ψ b (X ) Motivating example P = x x + c on X = R + = [0, ). (only analyze behavior near x = 0) ) c Kernels of inverses: K Q (x, x ) = (H(x x ) + const) The example also shows: ( x x The small calculus is not enough. Separation: different kinds of singular behavior of K Q are separated Daniel Grieser (Oldenburg) Lecture 2: Some basic principles of the b-calculus September 20, / 11