![Ellipticity and p-ellipticity
Ω ⊆ Rd an open set
A: Ω → Cd×d with L∞ coefficients (Note: non-smooth and complex)
A is elliptic if ∃λ, Λ ∈ (0, ∞) s.t.
Re 〈A(x)ξ, ξ〉Cd > λ|ξ|2
for ξ ∈ Cd
, for a.e. x ∈ Ω
⃒
⃒ 〈A(x)ξ, η〉Cd
⃒
⃒ 6 Λ |ξ| |η| for ξ, η ∈ Cd
, for a.e. x ∈ Ω
Take p ∈ (0, ∞], but think of p ∈ (1, ∞)
A is p-elliptic if additionally ∃c ∈ (0, ∞) s.t.
Re
⟨︀
A(x)ξ, ξ + |1 − 2/p|ξ̄
⟩︀
Cd > c|ξ|2
for ξ ∈ Cd
, for a.e. x ∈ Ω
(Carbonaro–Dragičević, 2016)
2/18](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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- 1. Trilinear embedding for divergence-form operators Vjekoslav Kovač (U. of Zagreb, PMF–MO) Joint work with Andrea Carbonaro (U. of Genova), Oliver Dragičević (U. of Ljubljana), Kristina Ana Škreb (U. of Zagreb, Civil Eng.) Supported by HRZZ UIP-2017-05-4129 (MUNHANAP) Appl Math 20, Brijuni, Sep 15, 2020 1/18
- 2. Ellipticity and p-ellipticity Ω ⊆ Rd an open set A: Ω → Cd×d with L∞ coefficients (Note: non-smooth and complex) A is elliptic if ∃λ, Λ ∈ (0, ∞) s.t. Re 〈A(x)ξ, ξ〉Cd > λ|ξ|2 for ξ ∈ Cd , for a.e. x ∈ Ω ⃒ ⃒ 〈A(x)ξ, η〉Cd ⃒ ⃒ 6 Λ |ξ| |η| for ξ, η ∈ Cd , for a.e. x ∈ Ω Take p ∈ (0, ∞], but think of p ∈ (1, ∞) A is p-elliptic if additionally ∃c ∈ (0, ∞) s.t. Re ⟨︀ A(x)ξ, ξ + |1 − 2/p|ξ̄ ⟩︀ Cd > c|ξ|2 for ξ ∈ Cd , for a.e. x ∈ Ω (Carbonaro–Dragičević, 2016) 2/18
- 3. Properties of p-elliptic matrices Δp(A) := ess inf x∈Ω min ξ∈Cd |ξ|=1 Re ⟨︀ A(x)ξ, ξ + |1 − 2/p|ξ̄ ⟩︀ Cd A is p-elliptic ⇐⇒ Δp(A) > 0 Δp0 (A) = Δp(A) where 1/p0 + 1/p = 1 Ap(Ω) = the class of p-elliptic matrix functions on Ω Ap(Ω) decreases in p ∈ [2, ∞) {elliptic on Ω} = A2(Ω) {real elliptic on Ω} = ⋂︁ p∈[2,∞) Ap(Ω) 3/18
- 4. Divergence-form operators Boundary conditions reflect the choice of U : • Dirichlet: U = H1 0 (Ω) = C∞ c (Ω) in H1(Ω) • Neumann: U = H1(Ω) = W1,2(Ω) • mixed: U = {︀ u|Ω : u ∈ C∞ c (Rd) }︀ in H1(Ω), ⊆ ∂Ω closed Divergence-form operator informally: “LA,U u = −div(A∇u)” Rigorously: 〈LA,U u, v〉L2(Ω) = ∫︁ Ω 〈A∇u, ∇v〉Cd for u ∈ D (LA), v ∈ U , where D (LA,U ) := {︀ u ∈ U : RHS extends boundedly to L2(Ω) }︀ (TA,U t )t>0 is the operator semigroup on L2(Ω) generated by −LA,U 4/18
- 5. Bilinear embeddings A bilinear embedding is any estimate of the form ∫︁ ∞ 0 ∫︁ X |∇Ttf(x)| |∇Ttg(x)| dμ(x) dt 6 C kfkLp(X) kgkLq(X) , where (X, μ) is a measure space, (Tt)t>0 is an operator semigroup, and p, q ∈ (1, ∞) are s.t. 1/p + 1/q = 1 Some history of bilinear embeddings: • Estimates for the Ahlfors–Beurling operator and iterated Riesz transf. (Petermichl–Volberg, 2002; Nazarov–Volberg, 2003): ∫︁ R2 (R2 1 f)(x) g(x) dx = −2 ∫︁ ∞ 0 ∫︁ R2 (︀ ∂x1 Ttf(x) )︀(︀ ∂x1 Ttg(x) )︀ dx dt, where (Ttf)t>0 is the heat extension of f 5/18
- 6. More history of bilinear embeddings • Dimension-free Littlewood–Paley estimates (Dragičević–Volberg, 2006) • Dimension-free estimates for Schrödinger operators (Dragičević–Volberg, 2011, 2012) • Dimension-free estimates for Riesz transforms associated with a Riemannian manifold (Carbonaro–Dragičević, 2013) • Functional calculus for generators of symmetric contraction semigroups (Carbonaro–Dragičević, 2017) • Bilinear embedding for divergence-form operators with complex coefficients (Carbonaro–Dragičević, 2020) – concept of p-ellipticity – connection with Lp -dissipativity of sesquilinear forms 6/18
- 7. Trilinear embedding Take p, q, r ∈ (1, ∞) s.t. 1/p + 1/q + 1/r = 1 Theorem (Carbonaro–Dragičević–K.–Škreb, 2020) Suppose that A, B, C : Ω → Cd×d are mx{p, q, r}-elliptic. Then for f ∈ (Lp ∩ L2)(Ω), g ∈ (Lq ∩ L2)(Ω) and h ∈ (Lr ∩ L2)(Ω) we have ∫︁ ∞ 0 ∫︁ Ω ⃒ ⃒TA,U t f ⃒ ⃒ ⃒ ⃒∇TB,V t g ⃒ ⃒ ⃒ ⃒∇TC,W t h ⃒ ⃒dx dt 6 C kfkLp(Ω) kgkLq(Ω) khkLr(Ω) When Ω = Rd, the same conclusion holds if only: A is p-elliptic, B is q-elliptic and (1 + q/r)-elliptic, C is r-elliptic and (1 + r/q)-elliptic. The embedding constant C only depends on p, q, r and the ∗-ellipticity constants of A, B, C 7/18
- 8. The idea of proof Let us illustrate it in the very special case d = 1, Ω = R, A = B = C = I (one-dimensional heat semigroups) Note that this special case is classical and accessible by tools from harmonic analysis (boundedness of maximal and square functions) The following proof by the so-called Bellman function technique was given by K.–Škreb, 2018 The approach is direct, avoids the need for any classical tools, and extends (after a lot of extra work) to the general case 8/18
- 9. Properties of B Let us find a C1 and “piecewise” C2 function B = B (u, v, w, U, V, W) of 6 real variables s.t. (B 1) Domain: u, v, w, U, V, W > 0, up 6 U, vq 6 V, wr 6 W (B 2) Range: 0 6 B (u, v, w, U, V, W) 6 C (︀ 1 p U + 1 q V + 1 r W )︀ (B 3) Certain concavity: − 1 2 ( d2 B ⏟ ⏞ quadratic form ) (u, v, w, U, V, W) ⏟ ⏞ at a point (4u, 4v, 4w, 4U, 4V, 4W) ⏟ ⏞ on a vector > u|4v||4w| 9/18
- 10. Construction of B WLOG assume q > r and use the ansatz: B (u, v, w, U, V, W) = C (︀ 1 p U + 1 q V + 1 r W )︀ − α(u, v, w) α(u, v, w) = up γ (︁ vq up ⏟ ⏞ t , wr up ⏟ ⏞ s )︁ γ(t, s) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a1 + b1t + c1s; 1 6 s 6 t a2 + b2t + c2s 1 p0 ; s 6 1 6 t a3 + b3t 1 p0 + c3s 1 p0 ; s 6 t 6 1 a4 + b4t 2 q s 1 r − 1 q + c4s 1 p0 ; t 6 s 6 1 a5 + b5t 2 q + c5t 2 q s1− 2 q + d5s; t 6 1 6 s a6 + b6t + c6t 2 q s1− 2 q + d6s; 1 6 t 6 s Adjust the coefficients so that γ is C1 on (0, ∞)2 10/18
- 11. Construction of B γ(t, s) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a + bt + cs; 1 6 s 6 t a(p−1)−c p−1 + bt + cp p−1 s 1 p0 ; s 6 1 6 t a(p−1)−(b+c) p−1 + bp p−1 t 1 p0 + cp p−1 s 1 p0 ; s 6 t 6 1 a(p−1)−(b+c) p−1 + bq 2 t 2 q s 1 r − 1 q + 2cpr−bp(q−r) 2r(p−1) s 1 p0 ; t 6 s 6 1 2ar(p−1)−b(q+r) 2r(p−1) + bq2 2p(q−2) t 2 q + bq(q−r) 2r(q−2) t 2 q s1− 2 q + 2cr−b(q−r) 2r s; t 6 1 6 s a + bq p(q−2) t + bq(q−r) 2r(q−2) t 2 q s1− 2 q + 2cr−b(q−r) 2r s; 1 6 t 6 s Choose a, b, c appropriately Similar to the function constructed by Nazarov–Treil, 1995, used in bilinear embeddings 11/18
- 12. Finalizing the proof Assume f, g, h > 0 and assume for simplicity that B ∈ C2 (mollify it) u(x, t) := (Ttf)(x), . . . , U(x, t) := (Ttfp )(x), . . . b(x, t) := B (︀ u(x, t), v(x, t), w(x, t), U(x, t), V(x, t), W(x, t) )︀ =⇒ (︀ ∂t − 1 2 ∂2 x )︀ b(x, t) = (∇B )(u, v, . . .) · (︀ ∂t − 1 2 ∂2 x )︀ (u, v, . . .) ⏟ ⏞ =0 − 1 2 (d2 B )(u, v, . . .)(∂xu, ∂xv, . . .) Using (B 3) we get ±u(x, t) ∂xv(x, t) ∂xw(x, t) 6 (︀ ∂t − 1 2 ∂2 x )︀ b(x, t) 12/18
- 13. Finalizing the proof Integrating by parts and using B > 0 we get for δ, T > 0: ± ∫︁ R×(δ,T−δ) k(x, T − t) u(x, t) ∂xv(x, t) ∂xw(x, t) dx dt 6 ∫︁ R k(x, δ) b(x, T − δ) dx Letting δ → 0 and using (B 2): ± ∫︁ R×(0,T) k(x, T − t) u(x, t) ∂xv(x, t) ∂xw(x, t) dx dt 6 b(0, T) 6 C p 2πT (︁ 1 p ∫︁ R f(y)p e− y2 2T dy + 1 q ∫︁ R g(y)q e− y2 2T dy + 1 r ∫︁ R h(y)r e− y2 2T dy )︁ Observe lim T→∞ p 2πT k(x, T − t) = 1 uniformly over (x, t) ∈ [−R, R] × (0, T1] and let T → ∞, R → ∞, T1 → ∞ 13/18
- 14. Relation to p-ellipticity For α: C3 → R, A, B, C ∈ Cd×d, (u, v, w) ∈ C3, and (ζ, η, ξ) ∈ (Cd)3 we define the generalized Hessian form of α with respect to (A, B, C), HA,B,C α [(u, v, w); (ζ, η, ξ)], as the standard inner product of ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ReA − ImA ImA ReA ReB − ImB ImB ReB ReC − ImC ImC ReC ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Re ζ Im ζ Re η Im η Re ξ Im ξ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ (Rd )6 14/18
- 15. Relation to p-ellipticity and (︀ Hess(α; (u, v, w)) ⊗ IRd )︀ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Re ζ Im ζ Re η Im η Re ξ Im ξ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ (Rd )6 Here one has to interpret Hess(α; (u, v, w)) as the 6 × 6 real Hessian matrix of the function R6 → R, (ur, ui, vr, vi, wr, wi) 7→ α(ur + iui, vr + ivi, wr + iwi) 15/18
- 16. Relation to p-ellipticity Lemma If α is given by the formula α(u, v, w) := |u|a|v|b|w|c for some a, b, c ∈ [0, ∞〉, then HA,B,C α [(u, v, w); (ζ, η, ξ)] > 1 2 |u|a |v|b |w|c (︁ a2 Δa(A) ⃒ ⃒ ⃒ ζ u ⃒ ⃒ ⃒ 2 + b2 Δb(B) ⃒ ⃒ ⃒ η v ⃒ ⃒ ⃒ 2 + c2 Δc(C) ⃒ ⃒ ⃒ ξ w ⃒ ⃒ ⃒ 2 − 2(Λ(A) + Λ(B)) ⃒ ⃒ ⃒ ζ u ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ η v ⃒ ⃒ ⃒ − 2(Λ(A) + Λ(C)) ⃒ ⃒ ⃒ ζ u ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ξ w ⃒ ⃒ ⃒ − 2(Λ(B) + Λ(C)) ⃒ ⃒ ⃒ η v ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ξ w ⃒ ⃒ ⃒ )︁ 16/18
- 17. A consequence “Vertical” square function, defined as (G A f)(x) := (︂∫︁ ∞ 0 ⃒ ⃒ ⃒∇(TA t f)(x) ⃒ ⃒ ⃒ 2 dt )︂1/2 , is only bounded in a very restrictive range even for real elliptic A (Auscher, 2007) “Conical” square function, defined as (C A f)(x) := (︂∫︁ ∫︁ {|x−y|< p t} ⃒ ⃒ ⃒∇(TA t f)(y) ⃒ ⃒ ⃒ 2 dy dt td/2 )︂1/2 , is bounded in the full range for real elliptic A (Auscher–Hofmann–Martell, 2012) The main theorem here reproves and refines their result 17/18
- 18. Thank you for your attention! 18/18