Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

D. Brydges, A. Talarczyk / Journal of Functional Analysis 236 (2006) 682–711 693
The resulting integrand is jointly measurable and non-negative so integrals and sums can be put
in any order. Therefore both terms give the same contribution and we bound by
dx dy1Uy∩Δ=∅(Fx, 1ΔFx) L2. Δ
Let |Uɛ|= dy1Uy∩Δ=∅. For the future, note that |Uɛ| tends as ɛ → 0tothevolume|U| of U.
Then the preceding expression equals
|Uɛ|
dx(Fx, 1ΔFx) L2 =|Uɛ| dx(Fx,Fx) L2 and the lemma follows by taking ɛ → 0. ✷
Lemma 3.5.
Δ
dx (pUx ϕ,ψ)+
|U|ϕ+ψ+.
Proof. It is sufficient to prove case ψ = ϕ because
dx (pUx ϕ,ψ)+
= dx
(pUx ϕ,pUx ψ)+ dxpUx ϕ+pUx ψ+
pUx ϕ2 + dx
1/2
pUy ψ2 + dy
1/2 .
Now we consider case ϕ = ψ for which there are no absolute values because (pUx ϕ,ϕ)+ 0.
By the Cauchy–Schwartz inequality,
λi(pUx ϕ,ϕ)+ = λipUx ϕ,ϕ λipUx ϕ,
i i i
i
i
+ i
1/2 λj pUx ϕ ϕ+
j
j
+
1/2 = ¯λiλj (pUx ϕ,pUx ϕ)+ ϕ+.
i j
By considering Riemann sums, this implies
dx(pUxϕ,ϕ)+
dx
By Lemma 3.4,
dx
i,j
dy(pUxϕ,pUy ϕ)+
=
dx
=|U|
1/2 dy(pUxϕ,pUy ϕ)+ ϕ+. (3.3)
dy(Fx,Fy) L2 |U|
dx(pUxϕ,pUx ϕ)+
=|U|
Putting this into (3.3), we obtain the lemma for the case ψ = ϕ. ✷
dx(Fx,Fx) L 2
dx(pUx ϕ,ϕ)+.