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

518 A. Olofsson / Journal of Functional Analysis 236 (2006) 517–545
separable Hilbert space. We denote by An(E) the Hilbert space of all E-valued analytic functions
f(z)=
akz k , z∈D, (0.1)
in the unit disc D with finite norm
k0
f 2
An =
k0
ak 2 μn;k,
where μn;k = 1/ k+n−1 k for k 0. Here the Taylor coefficients ak in (0.1) are elements in E.
The weight sequence {μn;k}k0 is naturally identified as a sequence of moments of a certain
radial measure dμn on the closed unit disc in the sense that
μn;k = |z| 2k
dμn(z) = 1
k + n − 1
, k0. k
For n 2 the measure dμn is given by
¯D
dμn(z) = (n − 1) 1 −|z| 2 n−2 dA(z), z ∈ D,
where dμ2(z) = dA(z) = dxdy/π, z = x + iy, is the usual planar Lebesgue area measure normalized
so that the unit disc D is of unit area. The measure dμ1 is the normalized Lebesgue arc
length measure on the unit circle T = ∂D. The norm of An(E) can also be expressed as
f 2 An
= lim
r→1
¯D
f(rz) 2 dμn(z), f ∈ An(E).
The shift operator Sn on the space An(E) is defined by
(Snf )(z) = zf (z) =
ak−1z k , z∈D, (0.2)
k1
for f ∈ An(E) given by (0.1). It is easy to see that the shift operator Sn is bounded on An(E) of
norm equal to 1 (the weight sequence {μn;k}k0 is decreasing and the ratio μn;k+1/μn;k tends
to 1 as k →∞). The adjoint operator S∗ n of Sn has the form
∗
Snf (z) = μn;k+1
ak+1z k , z∈D, (0.3)
k0
μn;k
where the function f ∈ An(E) is given by (0.1).
Let I be a shift invariant subspace of An(E). By this we mean that I is a closed subspace of
An(E) which is invariant under the shift operator Sn in the sense that Sn(I) ⊂ I. The wandering
subspace EI for I is the subspace
EI = I ⊖ Sn(I)