The heat kernel meets Approximation theory in Dirichlet spaces University of South Carolina with Thierry Coulhon and Gerard Kerkyacharian Paris - June, 2012
Outline 1. Motivation and objectives 2. The setting 3. Realization of the setting in Dirichlet spaces 4. Examples. Heat kernel associated with the Jacobi operator 5. Functional calculus 6. Spectral spaces. Linear approximation theory 7. Construction of frames 8. Definition of Besov and Triebel-Lizorkin spaces 9. Heat kernel characterization of Besov spaces 10. Frame decomposition of Besov spaces 11. Nonlinear n-term approximation from frames 12. Frame decomposition of Triebel-Lizorkin spaces
Motivation and objectives Develop a general theory which includes: 1. Complete Littlewood-Paley theory (Besov and Triebel-Lizorkin spaces) 2. Linear approximation theory 3. Nearly exponentially localized frames and decomposition of spaces 4. Nonlinear n-term approximation from frames 5. Cover most of the existing theory, e.g. ϕ-transform of Frazier-Jawerth, needlets 6. Cover new nontrivial settings
Setting (a) (M, ρ, µ) is a metric measure space with doubling measure: 0 < µ(b(x, 2r)) cµ(b(x, r)) <, x M, r > 0, which implies µ(b(x, λr)) cλ d µ(b(x, r)), r > 0, λ > 1. (b) L is a self-adjoint positive operator on L 2 (M, dµ) s.t. the (heat) kernel p t (x, y) of the associated semigroup P t = e tl obeys p t (x, y) C exp{ cρ2 (x,y) t } µ(b(x, t))µ(b(y, t)) for x, y M, 0 < t 1.
Setting (c) Hölder continuity: There exists a constant α > 0 s.t p t (x, y) p t (x, y ) ( ρ(y, y ) ) α exp{ cρ2 (x,y) C t } t µ(b(x, t))µ(b(y, t)) for x, y, y M and 0 < t 1, whenever ρ(y, y ) t. (d) Markov property: p t (x, y)dµ(y) 1 for t > 0. M
Key implications of the heat kernel properties Davies-Gaffney estimate { P t f 1, f 2 exp c r 2 } f 1 2 f 2 2, t > 0, t for all open sets U j M and f j L 2 (U j ), j = 1, 2, where r := ρ(u 1, U 2 ) and c > 0 is a constant. Finite speed propagation property cos(t L)f1, f 2 = 0, 0 < ct < r, c := 1 2 c, for all open sets U j M, f j L 2 (U j ), j = 1, 2, where r := ρ(u 1, U 2 ).
Key implications of the heat kernel properties (Cont.) Key ingredients Proposition. Let f be even, suppˆf [ A, A] for some A > 0, and ˆf W 2, i.e. ˆf (2) <. Then for δ > 0 and x, y M f (δ L)(x, y) = 0 if cδa < ρ(x, y). Proposition. Let f be a bounded measurable function on R + with supp f [0, τ] for some τ 1. Then f ( L) is an integral operator with kernel f ( L)(x, y) satisfying f ( L)(x, y) c f ( B(x, τ 1 ) B(y, τ 1 ) ), x, y M. 1/2
Realization of the setting in Dirichlet spaces Operator = Quadratic form Let M is a locally compact separable metric space equipped with a positive Radon measure µ such that every open and nonempty set has positive measure. Let L be a positive (non-negative) symmetric operator on (the real) L 2 (M, µ) with domain D(L), dense in L 2 (M, µ). Associate with L the symmetric non-negative form E(f, g) = Lf, g = E(g, f ), E(f, f ) = Lf, f 0, with domain D(E) = D(L). We consider on D(E) the prehilbertian structure induced by f E = f 2 2 + E(f, f ) which in general is not complete (not closed), but closable in L 2.
Realization of the setting (Cont.) Friedrich s extention Let E and D(E) be the closure of E and its domain. E gives rise to a self-adjoint extension L (the Friedrichs extension) of L with domain D( L) consisting of all f D(E) for which there exists u L 2 such that E(f, g) = u, g for all g D(E) and Lf = u. Thus L is positive and self-adjoint, and ( D(E) = D L ), E(f, g) = Lf, Lg. From spectral theory, there is a self-adjoint strongly continuous contraction semigroup P t = e t L on L 2 (M, µ). Then e t L = 0 e λt de λ, where E λ is the spectral resolution associated with L. Moreover, P t has a holomorphic extension P z, Re z > 0.
Realization of the setting (Cont.) Assumption: P t is a submarkovian semigroup If 0 f 1 and f L 2, then 0 P t f 1. Then P t can be extended as a contraction operator on L p, 1 p, preserving positivity, satisfying P t 1 1, and hence yielding a strongly continuous contraction semigroup on L p, 1 p <. Sufficient conditions (Beurling-Deny) For every ε > 0 there exists Φ ε : R [ ε, 1 + ε] such that Φ ε is non-decreasing, Φ ε Lip 1, Φ ε (t) = t for t [0, 1] and Φ ε (f ) D(E) and E(Φ ε (f ), Φ ε (f )) E(f, f ), f D(L). (in fact, this can be done easily only if Φ ε (f ) D(L)). Under the above conditions, (D(E), E) is called a Dirichlet space and D(E) L is an algebra.
Realization of the setting (Cont.) Assumption: E is strictly local and regular E is strictly local E(f, g) = 0 for f, g D(E) if supp f is compact and g = const. on a neighborhood of supp f. E is regular the space C c (M) of continuous functions on M with compact support has the property that the algebra C c (M) D(E) is dense in C c (M) with respect to the sup norm, and dense in D(E) in the norm E(f, f ) + f 2 2. Sufficient conditions (i) D(L) is a subalgebra of C c (M) s.t. E(f, g) = Lf, g = 0 if f, g D(L), supp f is compact, and g is constant on a neighbourhood of the support of f. (ii) For any compact K and open set U s.t. K U there exists u D(L), u 0, supp u U, and u 1 on K.
Realization of the setting (Cont.) Under the above assumptions, there exists a bilinear symmetric form dγ defined on D(E) D(E) with values in the signed Radon measures on M s.t. E(f, g) = dγ(f, g) and dγ(f, f ) 0. M Moreover, if D(L) is a subalgebra of C c (M), then dγ is absolutely continuous with respect to µ, and dγ(f, g)(u) = Γ(f, g)(u)dµ(u), Γ(f, g) = 1 (L(fg) flg glf ) 2 for f, g D(L). Then E(f, g) = M Γ(f, g)(u)dµ(u).
Realization of the setting (Cont.) Definition of intrinsic distance on M: ρ(x, y) = sup{h(x) h(y) : h D(E) C c (M), dγ(h, h) dµ}. where dγ(h, h) dµ iff dγ(h, h) = γ(h)dµ with γ(h) 1. Assumption: We assume that ρ : M M [0, ] is a true metric that generates the original topology on M and (M, ρ) is a complete metric space.
Realization of the setting: Main conditions 1. (M, ρ, µ) obeys the doubling condition: µ(b(x, 2r)) cµ(b(x, r), x M, r > 0. 2. Poincaré inequality: C > 0 s.t. B = B(x, r) f f B 2 Cr 2 dγ(f, f ), B where f B = 1 B B fdµ. These conditions are equivalent to: The semi-group P t is a positive symmetric kernel operator with kernel p t (x, y) s.t. 2B c 1 exp{ c 1ρ 2 (x,y) t } µ(b(x, t))µ(b(y, t)) p t (x, y) c 2 exp{ c 2ρ 2 (x,y) t } µ(b(x, t))µ(b(y, t)) Moreover, p t (x, y) is Hölder continuous.
Examples Classical setting on R d with L = Uniformly elliptic divergence form operators on R d. Uniformly elliptic divergence form operators on subdomains of R d with boundary conditions. Riemannian manifolds and Lie groups. In particular, Compact Riemannian manifolds, Riemannian manifold with non-negative Ricci curvature, Compact Lie groups, Lie groups with polynomial growth and their homogeneous spaces,... Heat kernel on [ 1, 1] associated with the Jacobi operator, Heat kernel on the ball and simplex
Heat kernel theory: References Heat kernel theory people: P. Auscher, M. Barlow, R. Bass, T. Coulhon, B. Davies, A. Grigoryan, E.M Ouhabaz, L. Saloff-Coste, A. Sikora, K. Sturm, N. Varopoulos,... Books: E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989. A. Grigor yan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics vol. 47, 2009. E.M. Ouhabaz, Analysis of heat equations on domains, London Mathematical Society Monographs Series, vol. 31, Princeton University Press, 2005. L. Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, 2002.
Heat kernel associated with the Jacobi operator The case M = [ 1, 1], dµ(x) = w(x)dx, where and w(x) = (1 x) α (1 + x) β, α, β > 1, Lf (x) = [w(x)a(x)f (x)], a(x) = (1 x 2 ), D(L) = C 2 [ 1, 1]. w(x) The theory applies, resulting in a complete strictly local Dirichlet space with an intrinsic metric: ρ(x, y) = arccos x arccos y Doubling measure: B(x, r) r(1 x + r 2 ) α+1/2 (1 + x + r 2 ) β+1/2 Local Poincaré inequality: For any interval I [ 1, 1] I f (x) f I 2 w(x)dx c(diam ρ (I)) 2 where f I = 1 w(i) I f (x) 2 (1 x 2 )w(x)dx I f (x)w(x)dx with w(i) = I w(x)dx
Heat kernel associated with the Jacobi operator Heat kernel: p t (x, y) = k 0 e λ k t P k (x)p k (y), λ k = k(k + α + β + 1), P k is the kth degree Jacobi polynomial normalized in L 2 (w). Theorem. The heat kernel p t (x, y) has Gaussian bounds: c 1 exp{ c 1ρ 2 (x,y) t } B(x, t) B(y, p t (x, y) t) Here c 2 exp{ c2ρ2 (x,y) t } B(x, t) B(y,. t) B(x, t) t(1 x + t) α+1/2 (1 + x + t) β+1/2, ρ(x, y) = arccos x arccos y.
Functional calculus in the general setting Theorem. Let f C 0 (R +) and f (2ν+1) (0) = 0 for ν 0. Then f (δ L), δ > 0, is an integral operator with kernel f (δ L)(x, y) satisfying ) σ f (δ L)(x, y) ( 1 + ρ(x,y) δ c σ σ > 0, B(x, δ) B(y, δ) and if ρ(y, y ) δ ) σ f (δ L)(x, y) f (δ L)(x, y ) ( ρ(y, y ) ) ( α 1 + ρ(x,y) cσ δ δ B(x, δ) B(y, δ) for some α > 0. Here B(x, δ) is the ball with center x and radius δ.
Sup-exponential localization Theorem. For any 0 < ε < 1 there exists κ > 0 and a compactly supported cut-off function ϕ C s.t. for any δ > 0 ϕ(δ L)(x, y) c exp { κ ( ρ(x,y) ) 1 ε } δ, x, y M, B(x, δ) B(y, δ) and if ρ(y, y ) δ ϕ(δ L)(x, y) ϕ(δ L)(x, y ) c( ρ(y,y )) α { ( δ exp κ ρ(x,y) ) 1 ε } δ. B(x, δ) B(y, δ)
Spectral spaces Let E λ, λ 0, be the spectral resolution associated with L. Then L = 0 λde λ Let F λ, λ 0, be the spectral resolution associated with L, i.e. F λ = E λ 2 and hence L = 0 λdf λ. The spectral space Σ λ is defined by Σ λ := {f L 2 : F λ f = f }. This can be extended to define Σ p λ, 1 p : Σ p λ := {f Lp : θ( L)f = f for all θ C 0 (R +), θ 1 on [0, λ]}.
Linear approximation from spectral spaces Let E t (f ) p denote the best approximation of f L p (L := UCB) from Σ p t : E t (f ) p := inf f g p. g Σ p t Bernstein estimate: For any g Σ p t, λ 1, 1 p, L m g p c m t 2m g p, m 1. Jackson estimate: For any f D(L m ) L p, 1 p, and m N E t (f ) p c m t 2m L m f p for t 1.
Construction of frames: Ingredients δ-nets: X M is a δ-net on M if ρ(ξ, η) δ ξ, η X, and X M is a maximal δ-net on M if X is a δ-net on M that cannot be enlarged. Proposition. A maximal δ-net on M always exists and if X is a maximal δ-net on M, then M = ξ X B(ξ, δ) and B(ξ, δ/2) B(η, δ/2) = if ξ η. Furthermore, X is countable or finite and there exists a disjoint partition {A ξ } ξ X of M consisting of measurable sets such that B(ξ, δ/2) A ξ B(ξ, δ), ξ X.
Construction of frames: Ingredients Sampling Theorem. For any 0 < ε < 1 there exists 0 < γ < 1 s.t. if X δ is a maximal δ net on M and {A ξ } ξ Xδ is a companion disjoint partition of M with δ = γ λ, then (1 ε) f 2 2 ξ X δ A ξ f (ξ) 2 (1 + ε) f 2 2 f Σ 2 λ. Cubature Formula. There exists 0 < γ < 1 s.t. for any λ 1 and a maximal δ-net X δ on M with δ = γ λ there exist positive constants (weights) {wξ λ} ξ X δ s.t. M f (x)dµ(x) = ξ X δ w λ ξ f (ξ) for f Σ1 λ, and (2/3) B(ξ, δ/2) w λ ξ 2 B(ξ, δ), ξ X δ.
Frame Properties Frames: {ψ ξ } ξ X, { ψ ξ } ξ X, X = j 0 X j (a) Representation: for any f L p, 1 p, with L := UCB f = f, ψ ξ ψ ξ =, ψ ξ ξ X ξ X f ψ ξ in L p. (b) Frame: The system { ψ ξ } is a frame for L 2 : c 1 f 2 2 ξ X f, ψ ξ 2 c f 2 2, f L2. The same is true for {ψ ξ }.
Frames Properties (c) Space localization: For any ξ X j, j 0, ψ ξ (x) c B(ξ, b j ) 1/2 exp { κ(b j ρ(x, ξ)) β}, and if ρ(x, y) b j ψ ξ (x) ψ ξ (y) c B(ξ, b j ) 1/2 (b j ρ(x, y)) α exp { κ(b j ρ(x, ξ)) β}. Here 0 < κ < 1 and b > 1 are constants. Same holds for ψ ξ. (d) Spectral localization: ψ ξ, ψ ξ Σ p b if ξ X 0 and ψ ξ, ψ ξ Σ p [b j 2,b j+2 ] if ξ X j, j 1, 0 < p. (e) Norms: ψ ξ p ψ ξ p B(ξ, b j ) 1 p 1 2 for 0 < p.
Distributions Distributions in the case µ(m) <. We use as test functions the class D of all functions φ m D(L m ) s.t. P m (φ) := L m φ 2 < for all m 0 Distributions in the case µ(m) =. In this case the class of test functions D is defined as the set of all functions φ m D(L m ) s.t. P m,l (φ) := sup(1 + ρ(x, x 0 )) l L m φ(x) < m, l 0. x M
Definition of Besov spaces Let ϕ 0, ϕ C (R + ), supp ϕ 0 [0, 2], ϕ (ν) 0 (0) = 0 for ν 1, supp ϕ [1/2, 2], and ϕ 0 (λ) + j 1 ϕ(2 j λ) c > 0, λ R +. Set ϕ j (λ) := ϕ(2 j λ) for j 1. Let s R and 0 < p, q. (i) The classical Besov space B s pq = B s pq(l) is defined by ( ( f B s pq := 2 sj ϕ j ( q ) 1/q. L)f ( ) L p) j 0 (ii) The nonclassical Besov space B s pq = B s pq(l) is defined by ( ( f := B(, 2 j ) s/d ϕ Bs pq j ( q ) 1/q. L)f ( ) L p) j 0
Heat kernel characterization of Besov spaces Theorem. Suppose s R, 1 p, 0 < q, and m > s, m Z +. Then and ( f B s pq e L 1 [ ] q dt f p + t s/2 (tl) m/2 e tl f p t 0 ) 1/q ( f B(, 1) s d Bs e L 1 1 f p + B(, t 2 ) s m d (tl) 2 e tl f q pq p 0 dt ) 1/q. t
Frame decomposition of Besov spaces Theorem. Let s R and 0 < p, q. Then for any f B s pq ( f B s pq b jsq[ f, ψ ] ξ ψ ξ p q/p ) 1/q p ξ X j j 0 and for f B s pq ( [ f Bs pq j 0 ξ X j ( B(ξ, b j ) s/d f, ψ ξ ψ ξ p ) p ] q/p ) 1/q. Here b > 1 is from the definition of the frames.
Nonlinear n-term approximation from {ψ ξ } Denote by Ω n is the set of all functions g of the form g = ξ Λ n a ξ ψ ξ, where Λ n X, #Λ n n, and Λ n may vary with g. Let σ n (f ) p := inf g Ω n f g p. The approximation will take place in L p, 1 p <. Suppose s > 0 and let 1/τ := s/d + 1/p. The Besov spaces B s τ := B s ττ play a prominent role. Observe that ( f, ψ ξ ψ ξ τ p f Bs τ For any f L p, 1 p <, ξ X f = ξ X f, ψ ξ ψ ξ in L p. ) 1/τ
Nonlinear n-term approximation (Cont.) Proposition. If f B τ s, then f can be identified as a function f L p and f p f, ψ ξ ψ ξ ( ) c f. p Bs τ ξ X Theorem. If f B s τ, then σ n (f ) p cn s/d f Bs τ, n 1. Open problem: Prove the companion Bernstein estimate: g Bs τ cn s/d g p for g Ω n, 1 < p <. This estimate would allow to characterize the rates of nonlinear n-term approximation from {ψ ξ } ξ X in L p (1 < p < ).
Triebel-Lizorkin space Let ϕ 0, ϕ C (R + ), supp ϕ 0 [0, 2], ϕ (ν) 0 (0) = 0 for ν 1, supp ϕ [1/2, 2], and ϕ 0 (λ) + j 1 ϕ(2 j λ) c > 0, λ R +. Set ϕ j (λ) := ϕ(2 j λ) for j 1. Let s R, 0 < p <, and 0 < q. (i) The classical Triebel-Lizorkin space Fpq s = Fpq(L) s is defined by ( f F s pq := ( 2 js ϕ j ( q ) 1/q L L)f ( ) ). p j 0 (ii) The nonclassical Triebel-Lizorkin space F pq s = F pq(l) s is defined by ( f F s := ( pq B(, 2 j ) s/d ϕ j ( q ) 1/q L L)f ( ) ). p j 0
Frame decomposition of Triebel-Lizorkin spaces Theorem. Let s R, 0 < p < and 0 < q. Then for any f F s pq ( f F s pq b jsq [ f, ψ ξ ψ ξ ( ) ] ) q 1/q L. p ξ X j j 0 and for f F pq s ( f F s pq [ B(ξ, b j ) s/d f, ψ ξ ψ ξ ( ) ] ) q 1/q L. p ξ X Here b > 1 is from the definition of the frames.
Publications 1. T. Coulhon, G. Kerkyacharian, P. Petrushev, Heat kernel generated frames in the setting of Dirichlet spaces, J. of Fourier Anal. Appl. (to appear). 2. G. Kerkyacharian, P. Petrushev, Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces, preprint.
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