On p -elliptic divergence form operators and holomorphic semigroups
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Journal of Evolution Equations
On p-elliptic divergence form operators and holomorphic semigroups Moritz Egert
Abstract. Second-order divergence form operators are studied on an open set with various boundary conditions. It is shown that the p-ellipticity condition of Carbonaro–Dragiˇcevi´c and Dindoš–Pipher implies extrapolation to a holomorphic semigroup on Lebesgue spaces in a p-dependent range of exponents that extends the maximal range for general strictly elliptic coefficients. This has immediate consequences for the harmonic analysis of such operators, including H∞ -calculi and Riesz transforms.
1. Introduction and main results Let A : O → L(Cd ) be a measurable strictly elliptic matrix function on an open set O ⊆ Rd , that is to say, there are constants λ, > 0 such that for almost every x ∈ O, |A(x)ξ | ≤ |ξ |
and
Re(A(x)ξ | ξ ) ≥ λ|ξ |2
(1)
hold for all ξ ∈ Cd . The associated divergence form operator L = − div(A∇ ·) is defined on L2 (O) in the weak sense through a sesquilinear form A∇u · ∇v dx, (2) a : V × V → C, a(u, v) = O
where (mixed) Dirichlet and Neumann boundary conditions are incorporated through the choice of the form domain W01,2 (O) ⊆ V ⊆ W1,2 (O); see Sect. 3 as follows. It is well known that L is maximal accretive [18, VI.6], and therefore −L generates a holomorphic C0 -semigroup of contractions T = (T (t))t≥0 on L2 (O) of angle π/2−ω, where ω := sup arg a(u, u),
(3)
u∈V
see [18, XI.6, Thm. 1.24]. In this paper, we are concerned with extrapolating T by density to a holomorphic C0 -semigroup on Lq (O) for q in an interval around 2. Mathematics Subject Classification: 47D06, 35J15, 47B44 Keywords: Divergence form operators on open sets, p-ellipticity, Holomorphic semigroups, Dissipative operators, Ultracontractivity, Off-diagonal estimates.
M. Egert
J. Evol. Equ.
For real coefficient matrices, it has been long known that T extrapolates to Lq (O) for all q ∈ [1, ∞), see, for example, [1,13,21]. For complex matrices, on the contrary, there is a natural threshold in q that is related to Sobolev embeddings. For |1/2−1/q| < 1/d extrapolation is well known by various different proofs on various classes of open sets [3,7,9,21,23]. In the plane, this covers the full range q ∈ (1, ∞), but in dimensions d ≥ 3 the semigroup may cease from extrapolating if |1/2 − 1/q| > 1/d, even on O = Rd , see [15, Prop. 2.10]. In their groundbreaking paper [5], Maz’ya and Cialdea have considered operators with coefficients A ∈ C1 (O → L(Cd )) and pure Dirichlet boundary conditions on a bounded regular domain. They have found an algebraic condition on the matrix A that is sufficient for T to extrapolate to a contraction semigroup on L p (O) and that is also necessary for the latter to hold if, in addition, Im(A) is symmetric. This was generalized to the setup described above, but still for pure Dirichlet boundary conditions, by Carbonaro and Dragiˇcevi´c [4]. They have elegantly rephrased the condition in [5] as p (A) := essinf
min
x∈O ξ ∈Cd , |ξ |=1
Re(A(x)ξ, J p ξ ) ≥ 0,
where J p : Cd → Cd is t
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