Properties in infinite dimension

and again, up to subsequences we have T^O(t) f_n²

k
→ T^O(t)(f²) a.e. in O.

T^O(t) g_n²

k
→ T^O(t)(g²) a.e. in O.

Therefore

T^O(t)(|f g|)(x) = lim

k→∞T^O(t) (|f_nkg_nk|) (x)

≤ lim

k→∞

q

T^O(t) f_n²_k
(x) · T^O(t) g²_nk
(x)

=p

T^O(t)(f²)(x) · T^O(t)(g²)(x) for a.e. x ∈ O.

so that

Z

Ω

uψ dγ = lim

n→∞

Z

Ω

uϕ⁺_{n |Ω} dγ ≤ 0.

Now we prove the statement by contradiction: if there exists a measurable set A such that γ(A) > 0 and u_|A > 0, then we take ψ(x) = χ_A(x) ∈ L^∞(Ω) ⊂ L²(Ω, γ). Then

Z

Ω

uψ dγ = Z

A

u dγ > 0 that is a contradiction.

Now we use the cylindrical approximation defined in Section 1.8. Fix n ∈ N, let q ∈ N with q ≥ j(n) = dim Fn and let G = span{h₁, . . . , h_q}. Let πG : X → G be the finite rank projection defined by

πGx =

q

X

i=1

bhi(x)hi.

We denote by γ_G the induced measure γ ◦ π_G⁻¹ in G. If G is identified with R^q through the isomorphism x 7→ (bh₁(x), . . . , bh_q(x)) then γ_G is the standard Gaussian measure in R^q. Setting d = q − dim F_n, let O := O_n× R^d so that π_G(Ω_n) = O; we remark that O is an open convex subset of G. Let L^G be the Ornstein-Uhlenbeck operator defined by (1.21) with Ω = O and γ = γ_G and let T^G(t) be the associated semigroup.

Lemma 8. If w ∈ D(L^G) then the map x 7→ w(πG(x)) belongs to D(L⁽ⁿ⁾) and

L⁽ⁿ⁾(w ◦ π_G) = (L^Gw) ◦ π_G.

Proof. We show that there exists f ∈ L²(Ω_n, γ) such that, given ϕ ∈ W^1,2(Ω_n, γ) we have

Z

Ωn

h∇Hw(πG(x)), ∇Hϕ(x)iHγ(dx) = Z

Ωn

f (x)ϕ(x)γ(dx). (2.4) We remark that

h∇_Hw(π_G(x)), ∇_Hϕ(x)i_H =

q

X

i=1

∂w

∂ξ_i(π_G(x))∂ϕ

∂h_i(x)

Moreover ϕ(x) = ϕ(π_G(x)+(I −π_G)(x)), the space X can be split X = G× eG where eG = (I − π_G)(X) and also γ = γ_G⊗ γ

Ge, see section 1.9. Let ξ ∈ R^q and for every y ∈ eG let

g_y(ξ) := ϕ(ξ₁h₁+ . . . + ξ_qh_q+ y),

then Z

Ωn

h∇_Hw(π_G(x)), ∇_Hϕ(x)i_Hγ(dx)

= Z

Ωn

q

X

i=1

∂w

∂ξ_i(π_G(x))∂ϕ

∂h_i(π_G(x) + (I − π_G)(x))γ(dx)

= Z

Ge

Z

O q

X

i=1

∂w

∂ξi

(ξ)∂g_y

∂ξi

(ξ)γ_G(dξ) γ

Ge(dy)

= Z

Ge

Z

O

L^Gw(ξ)g_y(ξ)γ_G(dξ) γ

Ge(dy) = Z

Ωn

(L^Gw)(π_G(x))ϕ(x)γ(dx).

Then w ◦ π_G ∈ D(L⁽ⁿ⁾) and L⁽ⁿ⁾(w ◦ π_G) = (L^Gw) ◦ π_G. Lemma 9. Let ev ∈ C_b^∞(G). Then the function defined by

g(t)(x) := T^G(t)ev_|O(π_G(x)), x ∈ Ω_n belongs to C((0, ∞); D(L⁽ⁿ⁾)).

Proof. By Lemma 8 it follows that g(t) ∈ D(L⁽ⁿ⁾) for all t > 0. Let us prove that g is continuous at t₀ > 0. For t > 0 we have

Z

Ωn

|g(t)(x) − g(t₀)(x)|²γ(dx)

= Z

Ωn

|T^G(t)ev|O(π_G(x)) − T^G(t₀)ev|O(π_G(x))|²γ(dx)

= Z

O

|T^G(t)ev|O(ξ) − T^G(t₀)ev|O(ξ)|²γ_G(dξ)

that goes to zero as t → t₀ thanks to the strong continuity of T^G(t) in L²(O, γG). Moreover

Z

Ωn

|L⁽ⁿ⁾g(t)(x) − L⁽ⁿ⁾g(t₀)(x)|²γ(dx)

= Z

Ωn

|L⁽ⁿ⁾T^G(t)ve_|O(π_G(x)) − L⁽ⁿ⁾T^G(t₀)ev_|O(π_G(x))|²γ(dx)

= Z

O

|L^GT^G(t)ev|O(ξ) − L^GT^G(t0)ev|O(ξ)|²γG(dξ)

that goes to zero as t → t₀ since t 7→ T^G(t)ev belongs to C((0, ∞), D(L^G)).

Then g ∈ C((0, ∞), D(L⁽ⁿ⁾)).

Theorem 17. For all f ∈ W^1,2(Ω, γ) and all t ≥ 0 we have

|∇_HT^Ω(t)f |_H ≤ e^−tT^Ω(t)|∇_Hf |_H a.e. on Ω. (2.5) Proof. First we prove that (2.5) holds true for the restriction to Ω of any cylindrical regular function. Let f ∈ F C_b^∞(X) and ϕ ∈ F C_b(X) with ϕ ≥ 0.

Then

f (x) = v(l₁(x), . . . , l_k(x)) ϕ(x) = w(lk+1(x), . . . , lk+m(x))

with li ∈ X^∗, v ∈ C_b^∞(R^k) and w ∈ Cb(R^m), w ≥ 0. We want to estimate the integral

Z

Ωn

ϕ(x)|∇HT⁽ⁿ⁾(t)f|Ωn(x)|γ(dx).

Let G := span{F_n, R_γ(l₁), . . . , R_γ(l_k+m)}. Then G is a subspace of H of dimension q ≤ n + k + m; setting d = q − dim F_n let O := O_n × R^d. Let {hi}^q_i=1 ⊂ X^∗ be an orthonormal basis of G such that {h1, . . . , hj} is an orthonormal basis of F_n if dim F_n = j.

Finally we consider:

f (x) =ev(π_G(x)) ϕ(x) =w(πe _G(x)) where ev ∈ C_b^∞(G), w ∈ Ce _b(G).

Now we prove that

(T⁽ⁿ⁾(t)f_|Ωn)(x) = T^G(t)ev_|O(π_G(x)) := g(t)(x) t > 0, x ∈ Ω_n. (2.6) We know that the function t 7→ T^G(t)ev|O belongs to C([ 0, ∞) ; L²(O, γ_G)) ∩ C¹((0, ∞); L²(O, γ_G)) ∩ C((0, ∞); D(L^G)) and it is the unique classical solu-tion of

(u⁰(t) = L^Gu(t), t > 0 u(0) =ev|O

in the space L²(O, γG). Then for fixed t0 ∈ [0, +∞) we have Z

Ωn

|g(t)(x) − g(t₀)(x)|²γ(dx)

= Z

Ωn

|T^G(t)ev|O(π_G(x)) − T^G(t₀)ev|O(π_G(x))|²γ(dx)

= Z

O

|T^G(t)ev|O(ξ) − T^G(t₀)ev|O(ξ)|²γ_G(dξ)

that goes to zero for t → t₀ thanks to the strong continuity of T^G(t) in L²(O, γ_G). Let us prove the differentiability of g at t₀ ∈ (0, +∞). For any t such that t + t₀ > 0 we have

Z

Ωn

   

g(t + t₀)(x) − g(t₀)(x)

t − L^GT^G(t₀)ev|O(π_G(x))    

2

γ(dx)

= Z

Ωn

   

T^G(t + t₀)ev_|O(π_G(x)) − T^G(t₀)ev_|O(π_G(x))

t − L^GT^G(t₀)ev|O(π_G(x))    

2

γ(dx)

= Z

O

   

T^G(t + t₀)ev_|O(ξ) − T^G(t₀)ev_|O(ξ)

t − L^GT^G(t₀)ev_|O(ξ)    

2

γ_G(dξ)

which tends to zero as t → 0. Then g⁰(t₀) = L^GT^G(t₀)ev_|O(π_G(·)). Moreover g⁰ is continuous at any t₀ > 0 with values in L²(Ω_n, γ) since

Z

Ωn

|g⁰(t)(x) − g⁰(t₀)(x)|²γ(dx)

= Z

Ωn

|L^GT^G(t)ve_|O(π_G(x)) − L^GT^G(t₀)ve_|O(π_G(x))|²γ(dx)

= Z

O

|L^GT^G(t)ev|O(ξ) − L^GT^G(t0)ev|O(ξ)|²γG(dξ) that goes to zero as t → t0. By Lemma 9, g ∈ C((0, ∞); D(L⁽ⁿ⁾)) and

g⁰(t) = L^G(T^G(t)ve|O)(π_G(x)) = L⁽ⁿ⁾(T^G(t)ve|O(π_G(x))) = L⁽ⁿ⁾g(t), t > 0.

Moreover

g(0) = T^G(0)ev_|O(π_G(x)) =ev_|O(π_G(x)) = f (x)_|Ωn. Therefore g is a classical solution to

(g⁰(t) = L⁽ⁿ⁾g(t), t > 0 g(0) = f|Ωn,

so that g(t) = T⁽ⁿ⁾(t)f|Ωn and (2.6) is proved.

Now using Lemma 6 we have Z

Ωn

ϕ(x)|∇_HT⁽ⁿ⁾(t)f|Ω_n(x)|_Hγ(dx)

= Z

Ωn

w(πe _G(x))|∇_HT^G(t)(ev_|O)(π_G(x))|_Hγ(dx)

= Z

Ow(ξ)|∇Te ^G(t)(ev_|O)(ξ)|γ_G(dξ)

≤ Z

Ow(ξ)ee ^−tT^G(t)|∇(ev|O)(ξ)|γ_G(dξ)

= Z

Ωn

w(πe _G(x))e^−tT^G(t)|∇_H(ev_|O)(π_G(x))|_Hγ(dx)

= Z

Ωn

ϕ(x)e^−tT⁽ⁿ⁾(t)|∇Hf|Ωn(x)|Hγ(dx)

Therefore for every f ∈ F C_b^∞(X) and ϕ ∈ F C_b(X), ϕ ≥ 0 and every n we have

Z

Ωn

ϕ(x)|∇_HT⁽ⁿ⁾(t)f|Ω_n(x)|γ(dx) ≤ Z

Ωn

ϕ(x)e^−tT⁽ⁿ⁾(t)|∇_Hf|Ω_n(x)|γ(dx).

Now thanks to Proposition 32 we can take the limit as n → ∞ obtaining Z

Ω

ϕ(x)|∇_HT^Ω(t)f|Ω(x)|γ(dx) ≤ Z

Ω

ϕ(x)e^−tT^Ω(t)|∇_Hf|Ω(x)|γ(dx) and by Lemma 7

|∇_HT^Ω(t)f|Ω(x)| ≤ e^−tT^Ω(t)|∇_Hf|Ω(x)| for a.e. x ∈ Ω.

If f ∈ W^1,2(Ω, γ) then there exists {f_n} ∈ F C_b^∞(X) such that f_n|Ω → f in W^1,2(Ω, γ); along a subsequence {fn_k} we have

|∇_HT^Ω(t)f_nk(x)|_H → |∇_HT^Ω(t)f (x)|_H a.e. in Ω and again up to a subsequence we have

T^Ω(t)|∇_Hf_nk(x)|_H → T^Ω(t)|∇_Hf (x)|_H a.e. in Ω.

Therefore

|∇_HT^Ω(t)f (x)|_H = lim

k→∞|∇_HT^Ω(t)f_nk(x)|_H

≤ lim

k→∞e^−tT^Ω(t)|∇Hfn_k(x)|H = e^−tT^Ω(t)|∇Hf (x)|H

for a.e. x ∈ Ω.

Proposition 38. For all f , g ∈ L²(Ω, γ) we have

T^Ω(t)(f g)2

≤ T^Ω(t)(f²)T^Ω(t)(g²) a.e. in Ω

Proof. As before we consider f, g, ϕ ∈ F C_b(X) and using Proposition 37 we prove that

Z

Ωn

ϕ(x)[T⁽ⁿ⁾(t)(f g)(x)]²γ(dx)

≤ Z

Ωn

ϕ(x) q

T⁽ⁿ⁾(t)(f²)(x) · T⁽ⁿ⁾(t)(g²)(x)γ(dx).

The proof is similar to the previous one with the only difference that now there are more functions but there aren’t gradients. Then taking the limit for n → ∞ we get

Z

Ω

ϕ(x)[T^Ω(t)(f g)(x)]²γ(dx) ≤ Z

Ω

ϕ(x)p

T^Ω(t)(f²)(x) · T^Ω(t)(g²)(x)γ(dx).

Since the restrictions to Ω of the functions of F C_b(X) are dense in L²(Ω, γ) we obtain our claim.

Now we turn to the Poincaré inequality. We set m_Ω(ϕ) = 1

γ(Ω) Z

Ω

ϕ dγ, ϕ ∈ L¹(Ω, γ).

We recall that if O ⊂ R^N is an open convex set the then the Poincarè inequality holds (see Proposition 4.2 in [8]), that is

Z

O

|ψ − mO(ψ)|²dγ_N ≤ Z

O

|∇ψ|²dγ_N, ∀ψ ∈ W^1,2(O, γ_N). (2.7) Proposition 39. For each ϕ ∈ W^1,2(Ω, γ) we have

Z

Ω

|ϕ − m_Ω(ϕ)|²dγ ≤ Z

Ω

|∇_Hϕ|²_Hdγ (2.8) Proof. First we prove that (2.8) holds for every f ∈ F C_b¹(X). Therefore

f (x) = v(l₁(x), . . . , l_k(x))

with v ∈ C_b¹(R^k) and l₁, . . . , l_k∈ X^∗. Let G := span{F_n, R_γ(l₁), . . . , R_γ(l_k)}.

Let {h_i}^q_i=1 ⊂ X^∗ be an orthonormal basis of G such that {h₁, . . . , h_j} is an orthonormal basis of Fn if dim Fn = j. Then G is a subspace of H of

dimension q ≤ n + k; setting d = q − dim F_n let O := O_n× R^d. Let {bh_i}^q_i=1 be an orthonormal basis of G such that {bh₁, . . . , bh_j} is an orthonormal basis of F_n if dim F_n= j. Then we have

f (x) = ϕ(πG(x)), ϕ ∈ C_b¹(G), m_Ωn(f ) = 1

γ(Ω_n) Z

Ωn

f (x) γ(dx) = 1 γ_G(O)

Z

O

ϕ(ξ) γ_G(dξ) =: m_O,γG(ϕ),

n→∞lim m_Ωn(f ) = m_Ω(f ).

Applying (2.7) we obtain Z

Ωn

|f (x) − m_Ωn(f )|²γ(dx)

= Z

Ωn

|f (x) − mO,γ_G(ϕ)|²γ(dx) = Z

O

|ϕ(ξ) − mO,γ_G(ϕ)|²γ_G(dξ)

≤ Z

O

|∇ϕ(ξ)|²γ_G(dξ) = Z

Ωn

|∇_Hf (x)|²_Hγ(dx).

Taking the limit as n → ∞ we get Z

Ω

|f (x) − mΩ(f )|²γ(dx) ≤ Z

Ω

|∇Hf (x)|²_Hγ(dx).

Let now f ∈ W^1,2(Ω, γ). There exists a sequence {f_n} ⊂ F C_b¹(X) such that f_n|Ω → f in W^1,2(Ω, γ). It follows that

Z

Ω

|f − m_Ω(f )|²dγ = lim

n→∞

Z

Ω

|f_n− m_Ω(f )|²dγ

≤ lim

n→∞

Z

Ω

|∇Hfn|²_Hdγ = Z

Ω

|∇Hf |²_Hdγ.

The Poincaré inequality (2.8) implies that if ∇_Hϕ vanishes a.e. in Ω, then ϕ is constant a.e. in Ω. By the definition of L, this implies that the kernel of L consists of constant functions. As a consequence of the Poincaré inequality other spectral properties of L follow.

Proposition 40. For all ϕ ∈ L²(Ω, γ) we have

kT^Ω(t)ϕ − m_Ω(ϕ)k_L²_(Ω,γ) ≤ e^−tkϕk_L²_(Ω,γ) (2.9) and consequently

σ(L)\{0} ⊂ {λ ∈ C : Re λ ≤ −1}. (2.10)

Proof. Let f ∈ D(L^Ω) be such that m_Ω(f ) = 0. By using (2.8) we have Z

Ω

L^Ωf · f dγ = − Z

Ω

h∇_Hf, ∇_Hf i_Hdγ = − Z

Ω

|∇_Hf |²_Hdγ ≤ −kf k²_L2(Ω,γ). For every ϕ ∈ L²(Ω, γ), m_Ω(ϕ) = 0, T^Ω(t)ϕ ∈ D(L^Ω) for t > 0. Therefore

d

dtkT^Ω(t)ϕk²_L2(Ω,γ) = 2 Z

Ω

L^ΩT^Ω(t)ϕ T^Ω(t)ϕ dγ ≤ −2kT^Ω(t)ϕk²_L2(Ω,γ), t > 0.

It follows that

kT^Ω(t)ϕk²_L2(Ω,γ) ≤ e^−2tkϕk²_L2(Ω,γ), t > 0. (2.11) Let now ϕ ∈ L²(Ω, γ). Then

Z

Ω

|T^Ω(t)ϕ − m_Ω(ϕ)|²dγ = Z

Ω

|T^Ω(t)(ϕ − m_Ω(ϕ))|²dγ ≤ e^−2t Z

Ω

|ϕ − m_Ω(ϕ)|²

= e^−2t

Z

Ω

|ϕ|²dγ − γ(Ω)[m_Ω(ϕ)]²



≤ e^−2t Z

Ω

|ϕ|²dγ.

Note that ϕ 7→ m_Ω(ϕ) is the orthogonal projection on Ker(L^Ω). Splitting L²(Ω, γ) = Ker(L^Ω) ⊕ (Ker(L^Ω))^⊥, T^Ω(t) maps (Ker(L^Ω))^⊥ = Ker(m_Ω) into itself and the infinitesimal generator of the restriction of T^Ω(t) to (Ker(L^Ω))^⊥ is the part L₀ of L^Ωin (Ker(L^Ω))^⊥. By (2.11), the spectrum of L₀ is contained in {λ ∈ C : Re(λ) ≤ −1}. Since the spectrum of L^Ω consists of the spectrum of L₀ plus the eigenvalue 0, (2.10) follows.

Lemma 10. If ϕ ∈ F C_b¹(X), ϕ ≥ 0, then T^Ω(t)ϕ_|Ω≥ 0 a.e. in Ω.

Proof. Let ϕ ∈ F C_b¹(X), ϕ ≥ 0, then

ϕ(x) = v(l1(x), . . . , lk(x))

with l1, . . . , lk ∈ X^∗ and v ∈ C_b¹(R^k) with v ≥ 0. First we prove that T⁽ⁿ⁾(t)ϕ ≥ 0 a.e. in Ω_n. Let G := span{F_n, R_γ(l₁), . . . , R_γ(l_k)}. Then G is a subspace of H of dimension q ≤ n + k; setting d = q − dim F_n let O := On× R^d. Let {hi}^q_i=1 ⊂ X^∗ be an orthonormal basis of G such that {h₁, . . . , h_j} is an orthonormal basis of F_n if dim F_n = j. Then we have

ϕ(x) =ev(π_G(x)) where ev ∈ C_b¹(R^q) andv ≥ 0 in Re ^q. Therefore, by

(T⁽ⁿ⁾(t)ϕ_|Ωn)(x) = T^G(t)ev_O(π_G(x)), for t > 0, and x ∈ Ωn.

By Proposition 34 we have

(T⁽ⁿ⁾(t)ϕ_|Ωn)(x) ≥ 0 for all x ∈ Ω_n. Thanks to Proposition 32, up to a subsequence,

(T^Ω(t)ϕ|Ω)(x) = lim

n→∞(T⁽ⁿ⁾(t)ϕ|Ωn)(x) ≥ 0 a.e. in Ω.

Now we have all the ingredients to prove the logarithmic Sobolev inequal-ity by the Deuschel-Strook’s method (see [12]).

Proposition 41. For all f ∈ W^1,2(Ω, γ) we have Z

Ω

f²log(f )dγ ≤ Z

Ω

|∇_Hf |²_Hdγ + kf k²_L2(Ω,γ)log(kf k_L²_(Ω,γ)). (2.12) Proof. First we suppose that f ∈ F C_b¹(Ω) and f ≥ c > 0 in Ω. Setting ϕ = f² we have

∇_Hf = 1 2

∇_Hϕ

√ϕ so (2.12) is equivalent to

Z

Ω

ϕ log(ϕ)dγ − Z

Ω

ϕdγ log

Z

Ω

ϕdγ



≤ 1 2

Z

Ω

1

ϕ|∇_Hϕ|²_Hdγ.

We remark that d

dt Z

Ω

T^Ω(t)(ϕ) log(T^Ω(t)(ϕ))dγ = Z

Ω

L^ΩT^Ω(t)(ϕ) log(T^Ω(t)(ϕ))dγ +

Z

Ω

L^ΩT^Ω(t)(ϕ)dγ.

The second term vanishes for the invariance of γ while for the first we recall

that Z

Ω

(L^Ωψ)g(ψ)dγ = − Z

Ω

g⁰(ψ)|∇_Hψ|²dγ with g(ξ) = log ξ and ψ = T^Ω(t)ϕ. Hence

d dt

Z

Ω

T^Ω(t)(ϕ) log(T^Ω(t)(ϕ))dγ = − Z

Ω

1

T^Ω(t)(ϕ)|∇_HT^Ω(t)(ϕ)|²dγ.

Since

(|∇_HT^Ω(t)(ϕ)|_H)² ≤ e^−tT^Ω(t)|∇_H(ϕ)|_H
2

and

T^Ω(t)|∇_H(ϕ)|_H
2

=



T^Ω(t)√

ϕ|∇_Hϕ|_H

√ϕ

2

≤ T^Ω(t)(ϕ)T^Ω(t) |∇_Hϕ|²_H ϕ

 , we have

d dt

Z

Ω

T^Ω(t)(ϕ) log(T^Ω(t)(ϕ))dγ ≥ −e^−2t Z

Ω

T^Ω(t) |∇_Hϕ|²_H ϕ

 dγ

= −e^−2t Z

Ω

|∇_Hϕ|²_H ϕ dγ.

(2.13)

We recall that, by (2.9),

T^Ω(t)ϕ−−−→ m^t→∞ Ω(ϕ), in L²(Ω, γ).

Moreover

t→∞lim Z

Ω

| log(T^Ω(t)ϕ) − log(m_Ωϕ)|²dγ = 0

indeed by assumption, we have ϕ ≥ c² and, by Lemma 10, it follows that T^Ω(t)ϕ ≥ c². Moreover m_Ω(ϕ) ≥ c², and

log(T^Ω(t)ϕ) − log(m_Ω(ϕ)) =

    

Z mΩ(ϕ) T^Ω(t)ϕ

1 t dt

    

≤ 1

min{T^Ω(t)ϕ, m_Ω(ϕ)}|T^Ω(t)ϕ − m_Ω(ϕ)|

≤ 1

c²|T^Ω(t)ϕ − m_Ω(ϕ)|.

Then Z

Ω

T^Ω(t)(ϕ) log(T^Ω(t)(ϕ))dγ −−−→ m^t→∞ _Ω(ϕ) log (m_Ω(ϕ)) Then integrating (2.13) with respect to t between 0 and ∞ we get

m_Ω(ϕ) log (m_Ω(ϕ)) − Z

Ω

ϕ log |ϕ|dγ ≥ −1 2

Z

Ω

|∇_Hϕ|²_H ϕ dγ.

Now let f ∈ W^1,2(Ω, γ) and f ≥ 0 a.e. in Ω. Then there exists a sequence {f_n} ⊂ F C_b¹(Ω) such that f_n ≥ 1/n and f_n → f in W^1,2(Ω, γ) and almost everywhere. Since t²log(t) > −1 for all t > 0, we can apply the Fatou’s

lemma and obtain Z

Ω

f²log(f )dγ ≤ lim

n→∞

Z

Ω

f_n²log(f_n)dγ

≤ lim

n→∞

Z

Ω

|∇_Hf_n|²_Hdγ + kf_nk²_L2(Ω,γ)log(kf_nk_L²_(Ω,γ))



= Z

Ω

|∇_Hf |²_Hdγ + kf k²_L2(Ω,γ)log(kf k_L²_(Ω,γ)).

For a general f ∈ W^1,2(Ω, γ), (2.12) follows from the fact that |f | ∈ W^1,2(Ω, γ) and |∇|f ||H = |∇f |H almost everywhere.

Remark 14. The Log-Sob inequality allows to prove an interesting property of the space W^1,2(Ω, γ), namely if f ∈ W^1,2(Ω, γ) and v is a measurable function with |v(x)| ≤ k(kxkX + 1) for some k > 0 and for a.e. x ∈ Ω, then f v ∈ L²(Ω, γ). To this aim we recall that, by Theorem 6, there exists a constant α > 0 such that

Z

X

e^αkxk2Xdγ < ∞.

Fix c < α/4. Then we have Z

Ω

(f (x)v(x))²dγ

≤ k² Z

Ω

f (x)²(kxk_X + 1)²dγ ≤ 2k² Z

Ω

f (x)²kxk²_Xdγ + 2k² Z

Ω

f (x)²dγ

= k² Z

{x∈Ω: ckxk²_X>log |f (x)|}

f (x)²kxk²_Xdγ + k²

Z

{x∈Ω: ckxk²_X≤log |f (x)|}

f (x)²kxk²_Xdγ + 2k²kf k²_L2(Ω,γ)

≤ k² Z

X

kxk²_X e^2ckxk2Xdγ + k² c

Z

Ω

|f (x)|²log |f (x)|dγ + 2k²kf k²_L2(Ω,γ)

≤ k²

Z

X

kxk⁴_Xdγ

1/2Z

X

e^4ckxk2Xdγ

1/2

+ k² c

Z

Ω

|f (x)|²log |f (x)|dγ + 2k²kf k²_L2(Ω,γ)

≤ C + k² c

Z

Ω

|∇_Hf |²_Hdγ + kf k²_L2(Ω,γ)log(kf k_L²_(Ω,γ))



+ 2k²kf k²_L2(Ω,γ), that is f v ∈ L²(Ω, γ). The previous estimates shows that the linear function Λv : f 7→ f v, maps bounded subsets of W^1,2(Ω, γ) into bounded subsets of L²(Ω, γ); this implies that Λ_v is continuous.

Chapter 3

Maximal Sobolev Regularity for Ornstein-Uhlenbeck equation

In this chapter we study the elliptic equation λu−L^Ωu = f in an open convex subset Ω of an infinite dimensional separable Banach space X endowed with a centered non-degenerate Gaussian measure γ, where L^Ω is the Ornstein-Uhlenbeck operator. We prove that for λ > 0 and f ∈ L²(Ω, γ) the weak solution u belongs to the Sobolev space W^2,2(Ω, γ). Moreover, under some regularity assumption on the boundary of Ω, we prove that u satisfies a gen-eralized Neumann boundary condition in the sense of traces at the boundary of Ω.

3.1 Finite-dimensional estimates

Let O be an open smooth convex subset of Rⁿ, with fixed n. We assume that O = {x ∈ Rⁿ : g(x) < 0}

where g is a smooth convex function whose gradient does not vanish at the boundary ∂O. We denote by ν(x) the exterior normal vector to ∂O at x, ν(x) = _|∇g(x)|^∇g(x). Let µ be the standard Gaussian measure in Rⁿ and let L^O be the associated Ornstein-Uhlenbeck operator, that is

L^Oψ(x) =

n

X

i=1

D_iiψ(x) −

n

X

i=1

x_iD_iψ(x) for ψ ∈ D(L^O).

In this section we consider the following problem







λψ − L^Oψ = f in O ⊂ Rⁿ,

∂u

∂ν = 0 on ∂O (3.1)

where f ∈ L²(O, µ) and λ > 0.

Let us introduce a weighted surface measure on ∂O:

dσ = N (x)dHⁿ⁻¹

where N (x) = (2π)^−n/2exp(−|x|²/2) is the Gaussian weight and dHⁿ⁻¹ is the usual Hausdorff (n − 1) dimensional measure. Using the surface measure dσ the integration by parts formula reads as:

Z

O

D_kϕ ψ dµ = − Z

O

ϕD_kψ dµ + Z

O

x_kϕψ dµ + Z

∂O

D_kg

|∇g|ϕψ dσ, (3.2) for each ϕ, ψ ∈ W^1,2(O, µ) one of which with bounded support, so the bound-ary integral is meaningful. Indeed W^1,2(O, µ) ⊂ W_loc^1,2(O, dx) and the trace at the boundary of any function in W^1,2(O, µ) belongs to L²_loc(∂O, dHⁿ⁻¹) = L²_loc(∂O, dσ).

Applying (3.2) with ϕ replaced by D_kϕ and summing up, we find Z

O

L^Oϕψ dµ = − Z

O

h∇ϕ, ∇ψidµ + Z

∂O

h∇ϕ, ∇gi

|∇g| ψ dσ (3.3) for every ϕ ∈ W^2,2(O, µ) , ψ ∈ W^1,2(O, µ) one of which with bounded support.

Now we give dimension free estimates for the weak solution u ∈ W^1,2(O, µ) to (3.1) with λ > 0 and f ∈ L²(O, µ). We can suppose f ∈ C_c^∞(O) because C_c^∞(O) is dense in L²(O, µ). In this case, thanks to classical results about elliptic equations with smooth coefficients we know that the weak solution u of (3.1) belongs to C^∞(O) ⊂ W_loc^2,2(O, µ). Since O can be unbounded, we introduce a smooth cutoff function θ : Rⁿ → R such that

0 ≤ θ(x) ≤ 1, |∇θ(x)| ≤ 2 ∀x ∈ Rⁿ, θ ≡ 1 in B(0, 1), θ ≡ 0 outside B(0, 2) and we set, for R > 0

θ_R(x) = θ(x/R), x ∈ Rⁿ.

For the W^1,2 estimates we take u as a test function in the definition of weak solution and we get

λ Z

O

u²dµ + Z

O

|∇u|²dµ = Z

O

f u dµ, (3.4)

then Z

O

u²dµ ≤ 1

λ²kf k²_L2(O,µ), Z

O

|∇u|²dµ ≤ 1

λkf k²_L2(O,µ). (3.5) The following lemma takes into the account the convexity of O.

Lemma 11. If u ∈ C²(O) satisfies h∇u, νi = 0 on ∂O then hD²u · ∇u, νi ≤ 0 on ∂O.

Proof. We recall that ∂O = g⁻¹(0) where g : Rⁿ → R is a smooth convex function. Let τ ∈ Rⁿ such that hτ, ν(x)i = 0 for x ∈ ∂O, then we have

h∂ν

∂τ(x), τ i ≥ 0 ∀x ∈ ∂O. (3.6)

Indeed

∂ν

∂τ = ∂

∂τ

 ∇g

|∇g|



= 1

|∇g|

∂

∂τ(∇g) + ∂

∂τ

 1

|∇g|



∇g

= 1

|∇g|D²g · τ + h∇

 1

|∇g|



, τ i∇g,

then for x ∈ ∂O we have h∂ν

∂τ(x), τ i = 1

|∇g(x)|hD²g(x) · τ, τ i + h∇

 1

|∇g(x)|



, τ ih∇g(x), τ i

= 1

|∇g(x)|hD²g(x) · τ, τ i ≥ 0

since D²g is a positive semi-definite symmetric matrix. Now we recall that h∇u, νi = 0 on ∂O therefore

∂

∂τ(h∇u(x), ν(x)i) = hD²u(x) τ, ν(x)i + h∇u(x), ∂ν

∂τ(x)i = 0, x ∈ ∂O for each τ ∈ Rⁿ such that hτ, νi = 0 on ∂O. If we take τ = ∇u(x) then we get

hD²u(x) · ∇u(x), ν(x)i = −hτ,∂ν

∂τ(x)i ≤ 0, x ∈ ∂O.

Now we can give an estimate of the second order derivatives of u.

Proposition 42. For every f ∈ C_c^∞(O) and ε > 0 there exists R₀ > 0 such that for R > R₀ the solution u to (3.1) satisfies

(1 − ε) Z

O

θ_R²Tr[(D²u)²]dµ ≤  2 + ε

λ

kf k²_L2(O,µ). (3.7)

Proof. Recall that u ∈ C^∞(O); differentiating (3.1) with respect to x_h yields λD_hu − ∆D_hu − hx, ∇(D_hu)i + D_hu = D_hf.

Multiplying by D_huθ_R² we obtain

(λ + 1) (D_hu)²θ²_R− ∆D_hu · D_huθ_R² − hx, ∇(D_hu)iD_huθ_R² = D_hf D_huθ_R². Integrating over O and using (3.3) yields

Z

O

(λ + 1)(D_hu)²θ_R²dµ + Z

O

|∇(D_hu)|²θ²_Rdµ + 2 Z

O

θ_Rh∇(D_hu), ∇θ_RiD_hu dµ

= Z

∂O

h∇(Dhu), ∇giDhu

|∇g| θ²_Rdσ + Z

O

D_hf D_huθ²_Rdµ.

Summing over h we obtain Z

O

(λ + 1)|∇u|²θ²_Rdµ + Z

O

Tr[(D²u)²]θ_R²dµ + 2 Z

O

hD²u · ∇u, ∇θ_Riθ_R dµ

= Z

∂O

hD²u · ∇u, ∇gi

|∇g| θ²_Rdσ + Z

O

h∇f, ∇uiθ_R²dµ.

Since f has compact support, for R large enough θR ≡ 1 on the support of f . For such R we obtain

    Z

O

h∇f, ∇uiθ²_Rdµ    

=    

− Z

O

L^Ouf dµ    

=     Z

O

(λu − f )f dµ    

≤ 2kf k²_L2(O,µ). Moreover

    Z

O

hD²u∇u, ∇θ_Riθ_R dµ    

≤ Z

O n

X

i,j=1

|D_iju D_ju D_iθ_R|θ_R dµ

≤ 1 2

Z

O n

X

i,j=1

|D_iju|²|D_iθR|θ_R² dµ + 1 2

Z

O n

X

i,j=1

|D_ju|²|D_iθR| dµ

≤ 1 2

k|∇θ|k∞

R Z

O

θ_R²Tr[(D²u)²]dµ + 1 2

k|∇θ|k∞

R Z

O

|∇u|²dµ

≤ 1 2

Z

O

Tr[(D²u)²]θ²_Rdµ + 1

2λkf k²_L2(O,µ)

 k|∇θ|k∞

R .

Taking R large enough, such that k|∇θ|k∞/R ≤ ε, we get (1 − ε)

Z

O

θ²_RTr[(D²u)²]dµ ≤  2 + ε

λ

kf k²_L2(O,µ)+ Z

∂O

θ_R²hD²u · ∇u, ∇gi

|∇g| dσ.

By using Lemma 11, the statement follows.

Theorem 18. For each λ > 0 there exists C = C(λ) > 0, independent of n and O, such that for each f ∈ L²(O, µ) the weak solution u of problem (3.1) belongs to W^2,2(O, µ), and satisfies

kuk_W^2,2_(O,µ) ≤ Ckf k_L²_(O,µ). (3.8) Proof. Let f ∈ C_c^∞(O). Taking the limit as R → ∞ in (3.7) and using the monotone convergence theorem, we get

(1 − ε) Z

O

Tr[(D²u)²]dµ ≤ 2 + ε

λ

kf k²_L2(O,µ).

Now taking the limit as ε → 0 we get Z

O

Tr[(D²u)²]dµ ≤ 2kf k²_L2(O,µ). (3.9) Taking into account (3.4), (3.5), and (3.9) we obtain

kuk²_W2,2(O,µ) = kuk_L²_(O,µ)+ k|∇u|k_L²_(O,µ) + kTr[(D²u)²]k_L²_(O,µ)

≤ 1 λ² + 1

λ + 2



kf k²_L2(O,µ)

which is (3.8) with C(λ) = _λ¹2+_λ¹+2. For f ∈ L²(O, µ) the statement follows approaching it by a sequence of functions belonging to C_c^∞(O).

We recall that (1.26) holds, and we prove that the condition h·, ∇f (·)i ∈ L²(O, µ) can be omitted.

Proposition 43. If f ∈ W^2,2(O, µ) then the map x 7→ hx, ∇f (x)i belongs to L²(O, µ), moreover the map

f 7→ h·, ∇f (·)i is continuous from W^2,2(O, µ) to L²(O, µ).

Proof. Let f ∈ W^2,2(O, µ), then Z

O

|h∇f, xi|²dµ = Z

O n

X

i=1

(D_if x_i)²dµ

by assumption D_if ∈ W^1,2(O, µ) and if c < 1/4, by using (1.28), we have Z

O

(D_if (x))²x²_ie^−|x|2/2dx

= Z

{x∈O: cx²_i>log |Dif (x)|}

(Dif (x))²x²_ie^−|x|2/2dx +

Z

{x∈O: cx²_i≤log |Dif (x)|}

(D_if (x))²x²_ie^−|x|2/2dx

≤ Z

O

e^2cx2ix²_ie^−|x|2/2dx + Z

O

1

c|D_if |²log |D_if |e^−|x|2/2dx

≤ C₁+ 1 c

Z

O

|∇D_if |dµ + 1 2

Z

O

(D_if )²dµ log

Z

O

(D_if )²dµ



. Summing over i from 1 to n we have h∇f, xi ∈ L²(O, µ).

3.2 Maximal Sobolev regularity for infinite

Nel documento Ornstein-UhlenbeckoperatorinconvexdomainsofBanachspaces UniversitàdegliStudidiParma (pagine 61-78)