Download Weyl calculus with respect to the Gaussian measure and L^p

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.