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Weyl calculus with respect to the Gaussian measure and Lp -Lq boundedness of the OU semigroup in complex time Jan van Neerven, Pierre Portal Operator semigroups in analysis, Bedlewo, April 24-28, 2017 Setting: • γ – the standard Gaussian measure in Rd , γ(dx) := (2π)−d/2 exp(−|x|2 /2) dx • L – the Ornstein–Uhlenbeck operator L := ∇∗ ∇ = −∆ + x · ∇ with ∇ the realisation of the gradient in L2 (Rd , γ). Facts: • −L generates a C0 -semigroup of contractions on L2 (Rd , γ), given by (in dimension d = 1) e −tL Hn = e −nt Hn , t ≥ 0, n ∈ N, where Hn is the n-th Hermite polynomial (in general dimensions d use multivariate Hermite polynomials). Facts: • −L generates a C0 -semigroup of contractions on L2 (Rd , γ), given by (in dimension d = 1) e −tL Hn = e −nt Hn , t ≥ 0, n ∈ N, where Hn is the n-th Hermite polynomial (in general dimensions d use multivariate Hermite polynomials). • Mehler formula: e −tL Z f (x) = f (e −t x + p 1 − e −2t y ) dγ(y ) Rd Z = Mt (x, y )f (y ) dy Rd with Mt (x, y ) = |e −t x − y |2 1 − exp 2(1 − e −2t ) (2π(1 − e −2t ))d/2 the so-called Mehler kernel. Lp -Theory: • The OU semigroup extends to a C0 -semigroup of contractions on Lp (Rd , γ) for all p ∈ [1, ∞). Lp -Theory: • The OU semigroup extends to a C0 -semigroup of contractions on Lp (Rd , γ) for all p ∈ [1, ∞). • It is an analytic C0 -semigroup of contractions on Lp (Rd , γ) for all p ∈ (1, ∞), of angle φp , where 2 cos φp = − 1, p Lp -Theory: • The OU semigroup extends to a C0 -semigroup of contractions on Lp (Rd , γ) for all p ∈ [1, ∞). • It is an analytic C0 -semigroup of contractions on Lp (Rd , γ) for all p ∈ (1, ∞), of angle φp , where 2 cos φp = − 1, p • The angle φp is optimal. [Epperson 89, nonsymmetric generalisation in [Chill-Fašangová-Metafune-Pallara 05, Maas-van Neerven 07] • The domain of analyticity of e −zL in Lp (Rd , γ) equals the Epperson region Ep = {x + iy ∈ C : | sin(y )| < tan(φp ) sinh(x)} and e −zL is contractive there. [Epperson 89], [Garcı́a Cuerva-Mauceri-Meda-Sjögren-Torrea 01] 4 2 0.2 0.4 0.6 0.8 1 1.2 1.4 -2 -4 The region Ep for p = 4/3 (red) Hypercontractivity • e −tL is contractive from Lp (Rd , γ) to Lq (Rd , γ) if and only if e −2t ≤ [Nelson 66], proof via Mehler kernel. p−1 . q−1 Hypercontractivity • e −tL is contractive from Lp (Rd , γ) to Lq (Rd , γ) if and only if e −2t ≤ p−1 . q−1 [Nelson 66], proof via Mehler kernel. • e −zL is contractive from Lp (Rd , γ) to Lq (Rd , γ) if and only if for all w ∈C (Im(we −z ))2 + (q − 1)(Re(we −z ))2 ≤ (Im w )2 + (p − 1)(Re w )2 . [Epperson 89], proof via two-point hypercontractivity and combinatorial induction arguments involving multivariate Hermite polynomials. Position and momentum On L2 (Rd ), consider the position and momentum operators X = (x1 , . . . , xd ), 1 1 D = ( ∂1 , . . . , ∂d ). i i Position and momentum On L2 (Rd ), consider the position and momentum operators X = (x1 , . . . , xd ), 1 1 D = ( ∂1 , . . . , ∂d ). i i They satisfy the commutation relations [xj , xk ] = [Dj , Dk ] = 0 [xj , Dk ] = iδjk . (1) Position and momentum On L2 (Rd ), consider the position and momentum operators X = (x1 , . . . , xd ), 1 1 D = ( ∂1 , . . . , ∂d ). i i They satisfy the commutation relations [xj , xk ] = [Dj , Dk ] = 0 [xj , Dk ] = iδjk . Note: X (Dj2 + xj2 ) = −∆ + |x|2 j is the quantum harmonic oscillator. (1) Gaussian position and momentum Let m(dx) = (2π)−d/2 dx denote the normalised Lebesgue measure on Rd . • The pointwise multiplier Ef (x) := e(x)f (x) with e(x) := exp(− 41 |x|2 ), is unitary from L2 (Rd , γ) onto L2 (Rd , m). Gaussian position and momentum Let m(dx) = (2π)−d/2 dx denote the normalised Lebesgue measure on Rd . • The pointwise multiplier Ef (x) := e(x)f (x) with e(x) := exp(− 41 |x|2 ), is unitary from L2 (Rd , γ) onto L2 (Rd , m). • The dilation √ √ 2x δf (x) := ( 2)d f is unitary on L2 (Rd , m). Gaussian position and momentum Let m(dx) = (2π)−d/2 dx denote the normalised Lebesgue measure on Rd . • The pointwise multiplier Ef (x) := e(x)f (x) with e(x) := exp(− 41 |x|2 ), is unitary from L2 (Rd , γ) onto L2 (Rd , m). • The dilation √ √ 2x δf (x) := ( 2)d f is unitary on L2 (Rd , m). Consequently, • The operator U := δ ◦ E is unitary from L2 (Rd , γ) onto L2 (Rd , m). On L2 (Rd , γ), consider the Gaussian position and Gaussian momentum operators Q = (q1 , . . . , qd ), P = (p1 , . . . , pd ), where qj := U −1 ◦ xj ◦ U, pj := U −1 ◦ Dj ◦ U. On L2 (Rd , γ), consider the Gaussian position and Gaussian momentum operators Q = (q1 , . . . , qd ), P = (p1 , . . . , pd ), where qj := U −1 ◦ xj ◦ U, pj := U −1 ◦ Dj ◦ U. They satisfy the ‘canonical commutation relations’ [pj , pk ] = [qj , qk ] = 0, [qj , pk ] = iδjk . On L2 (Rd , γ), consider the Gaussian position and Gaussian momentum operators Q = (q1 , . . . , qd ), P = (p1 , . . . , pd ), where qj := U −1 ◦ xj ◦ U, pj := U −1 ◦ Dj ◦ U. They satisfy the ‘canonical commutation relations’ [pj , pk ] = [qj , qk ] = 0, [qj , pk ] = iδjk . We have 1 2 1 (P + Q 2 ) = L + I , 2 2 with L the OU operator (in the physics literature: L is the ‘boson number operator’ and 12 the ‘ground state energy’). NB.: Without the dilation δ, this identity would not come out right. The Weyl calculus For Schwartz functions a : R2d → C define Z a(X , D)f (x) := ab(u, v ) exp(i(uX + vD))f (x) m(du) m(dv ). R2d Here • ab is the Fourier-Plancherel transform of a, • the unitary operators exp(i(uX + vD)) on L2 (Rd , γ) are defined through the action exp(i(uX + vD))f (x) := exp(iux + 21 iuv )f (v + x) (formally apply the Baker-Campbell-Hausdorff formula and use the commutator relations, or note that it give a unitary representation of the Heisenberg group; see [Hall 13]). The Gaussian Weyl calculus For Schwartz functions a : R2d → C define a(Q, P) := U −1 ◦ a(X , D) ◦ U. The Gaussian Weyl calculus For Schwartz functions a : R2d → C define a(Q, P) := U −1 ◦ a(X , D) ◦ U. By explicit computation, Z a(Q, P)f (x) = Ka (x, y )f (y ) dy , Rd where Ka (x, y ) := 1 √ (2 2π)d Z Rd x +y a( √ , ξ) 2 2 x −y × exp(iξ( √ )) exp( 41 |x|2 − 14 |y |2 ) dξ. 2 Recall the identity 12 (P 2 + Q 2 ) = L + 12 I . Thus one would expect 1 e −zL = e − 2 z exp(− 12 z(P 2 + Q 2 )). But this ignores the commutation relations of P and Q. Recall the identity 12 (P 2 + Q 2 ) = L + 12 I . Thus one would expect 1 e −zL = e − 2 z exp(− 12 z(P 2 + Q 2 )). But this ignores the commutation relations of P and Q. Rather, the Weyl calculus gives: Theorem 1. 1 − e −z 1 − e −z d 2 2 e −zL = 1 + exp − (P + Q ) . 1 + e −z 1 + e −z NB.: The RHS can be computed explicitly using the integral representation for the Weyl calculus. Recall the identity 12 (P 2 + Q 2 ) = L + 12 I . Thus one would expect 1 e −zL = e − 2 z exp(− 12 z(P 2 + Q 2 )). But this ignores the commutation relations of P and Q. Rather, the Weyl calculus gives: Theorem 1. 1 − e −z 1 − e −z d 2 2 e −zL = 1 + exp − (P + Q ) . 1 + e −z 1 + e −z NB.: The RHS can be computed explicitly using the integral representation for the Weyl calculus. Theorem 1 suggests the study of the family of operators exp(−s(P 2 + Q 2 )), Res > 0. With as (x) = exp(−s(|x|2 + |y |2 )) we obtain Z exp(−s(P 2 + Q 2 ))f (x) = Kas (x, y )f (y )dy Rd Z 1 = d exp(− 8s1 (1 − s)2 (|x|2 + |y |2 ) + 14 ( 1s − s)xy )f (y )dγ(y ). 2 (2πs)d/2 Rd With as (x) = exp(−s(|x|2 + |y |2 )) we obtain Z exp(−s(P 2 + Q 2 ))f (x) = Kas (x, y )f (y )dy Rd Z 1 = d exp(− 8s1 (1 − s)2 (|x|2 + |y |2 ) + 14 ( 1s − s)xy )f (y )dγ(y ). 2 (2πs)d/2 Rd Note the symmetry in x and y . Consider the Gaussian measure in Rd with variance τ , γτ (dx) := (2πτ )−d/2 exp(−|x|2 /2τ ) dx. Define, for Re s > 0, r± (s) := 1 1 Re ( ± s). 2 s 2 Theorem 2. Let p, q ∈ [1, ∞) and let α, β > 0. If 1 − αp + r+ (s) > 0, 2 βq − 1 + r+ (s) > 0, and 2 2 (∗) (r− (s))2 ≤ 1 − + r+ (s) − 1 + r+ (s) , αp βq then exp(−s(P 2 + Q 2 )) is bounded from Lp (Rd , γα ) to Lq (Rd , γβ ). The case ‘<’ in (∗) follows by a simple application of Hölder’s inequality! To get the result with ‘≤’ in (∗), a Schur estimate is used instead. The case ‘<’ in (∗) follows by a simple application of Hölder’s inequality! To get the result with ‘≤’ in (∗), a Schur estimate is used instead. Corollary. Theorem 2 implies Epperson’s Lp -Lq boundedness criterion. Proof. Substitute z = x + iy and check that Epperson’s criterion implies the positivity conditions of Theorem 2. This involves only elementary (but quite miraculous) high-school algebra. The case ‘<’ in (∗) follows by a simple application of Hölder’s inequality! To get the result with ‘≤’ in (∗), a Schur estimate is used instead. Corollary. Theorem 2 implies Epperson’s Lp -Lq boundedness criterion. Proof. Substitute z = x + iy and check that Epperson’s criterion implies the positivity conditions of Theorem 2. This involves only elementary (but quite miraculous) high-school algebra. The crucial thing is to recognise (we used MAPLE) that 2 2 2 2 2 2 2 2 2 2 2 (q − 1)((1 − (x + y )) + 4y ) + (2 − p − q)(1 − (x + y )) ((1 + x) + y ) 2 | 2 2 2 2 2 4 −(2 − p − q + pq)4y ((1 + x) + y ) + (p − 1)((1 + x) + y ) {z } the positivity condition in Epperson’s criterion factors as 2 2 2 2 2 2 2 4((1 + x) + y ) × (p − q)x(1 + x + y ) + (2p + 2q − 4)x − (pq − 2p − 2q + 4)y . {z } | {z } ≥0 the positivity condition in Theorem 2 | Corollary. For p ∈ (1, ∞), the operator-valued function s 7→ exp(−s(P 2 + Q 2 )) is bounded and holomorphic on the sector Σφp . Corollary. For p ∈ (1, ∞), the operator-valued function s 7→ exp(−s(P 2 + Q 2 )) is bounded and holomorphic on the sector Σφp . Proof. For q = p and z = x + iy , Epperson’s Lp -Lq criterion reduces to p2( x2 x2 − 1) + 4p − 4 > 0, + y2 which is equivalent to saying that s ∈ Σφp . Corollary. For p ∈ (1, ∞), the operator-valued function s 7→ exp(−s(P 2 + Q 2 )) is bounded and holomorphic on the sector Σφp . Proof. For q = p and z = x + iy , Epperson’s Lp -Lq criterion reduces to p2( x2 x2 − 1) + 4p − 4 > 0, + y2 which is equivalent to saying that s ∈ Σφp . NB: s = 1−e −z 1+e −z maps the Epperson region Ep to Σφp ! Thus we recover that Ep is the Lp -domain of holomorphy of e −zL . For p = 1 the following is due to [Bakry, Bolley, Gentil 12] by very different methods (they get contractivity): Corollary. Let p ∈ [1, 2]. For all Re z > 0 the operator exp(−zL) maps Lp (Rd , γ2/p ) into L2 (Rd , γ). For p = 1 the following is due to [Bakry, Bolley, Gentil 12] by very different methods (they get contractivity): Corollary. Let p ∈ [1, 2]. For all Re z > 0 the operator exp(−zL) maps Lp (Rd , γ2/p ) into L2 (Rd , γ). As a consequence, the semigroup generated by −L extends to an analytic C0 -semigroup on Lp (Rd , γ2/p ) of angle 12 π. (Recall that −L extends to an analytic C0 -semigroup on Lp (Rd , γ) of non-trivial angle φp .) For p = 1 the following is due to [Bakry, Bolley, Gentil 12] by very different methods (they get contractivity): Corollary. Let p ∈ [1, 2]. For all Re z > 0 the operator exp(−zL) maps Lp (Rd , γ2/p ) into L2 (Rd , γ). As a consequence, the semigroup generated by −L extends to an analytic C0 -semigroup on Lp (Rd , γ2/p ) of angle 12 π. (Recall that −L extends to an analytic C0 -semigroup on Lp (Rd , γ) of non-trivial angle φp .) THANK YOU FOR YOUR ATTENTION! Some references • D. Bakry, F. Bolley, I. Gentil, Dimension dependent hypercontractivity for Gaussian kernels, Probab. Th. Relat. Fields 154 (2012) 845–874. • R. Chill, E. Fašangová, G. Metafune, D. Pallara, The sector of analyticity of the OU semigroup on Lp spaces with respect to invariant measure, J. London Math. Soc. 71 (2005), 703–722. • J.B. Epperson, The hypercontractive approach to exactly bounding an operator with complex Gaussian kernel, JFA 87 (1989) 1–30. • J. Garcı́a-Cuerva, G. Mauceri, S. Meda, P. Sjögren, J.L. Torrea, Functional calculus for the OU operator, JFA 183 (2001) 413–450. • B. Hall, Quantum mechanics for mathematicians, Springer Verlag, 2013 • J. Maas and J.M.A.M. van Neerven On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions, Archiv Math. (Basel) 89 (2007), 226–236. • J.M.A.M. van Neerven, P. Portal, Weyl calculus with respect to the Gaussian measure and restricted Lp -Lq boundedness of the OU semigroup in complex time, arXiv:1702.03602. • E. Nelson, A quartic interaction in two dimensions, in: “Mathematical theory of elementary particles”, pp. 69–73, MIT Press, 1966.