PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 261–266 https://doi.org/10.2298/PIM1715261C

COMPARISON OF ITERATES OF A CLASS OF DIFFERENTIAL OPERATORS IN ROUMIEU SPACES

Rachid Chaili and Tayeb Mahrouz

Abstract. Considering a class of differential operators with constant coefficients including the hypoelliptic operators, we show that the comparison of the operators implies the inclusion between their spaces of Roumieu vectors.

1. Introduction

The iterate property of linear differential operators has been generalized to the comparison in the sense of inclusion between the spaces of Gevrey vectors of those operators for the first time by Newberger and Zielezny [17], they have given necessary and sufficient conditions to get this comparison considering hypoelliptic differential operators with constant coefficients. This result has been generalized and extended by Bouzar–Chaili [4], to a class of systems of differential operators with constant coefficients including the class of hypoelliptic differential operators. In [11] Juan–Huguet has extended the theorem of Newberger–Zielezny in the spaces of ultradifferentiable functions of type Roumieu.

The aim of this work is to refine these results considering a class of differential operators as in [4] and Roumieu spaces of type M_{p}. For more details on the results concerning the iterate problem see [1, 3, 5–7, 10, 12, 14, 16, 20–22].

Let (M_{p}) be a sequence of positive real numbers satisfying the following conditions logarithmic convexity:

(1.1)  M_{p}^{2} ⩽ M_{p−1} M_{p+1} , ∀p ∈ ℕ^{∗},

non-quasi-analyticity:

(1.2)  ∑_{p=0}^{∞} \frac{M_{p−1}}{M_{p}} < ∞,

stability under derivation and multiplication:

(1.3) ∃A > 0, ∃H > 0, ∃c > 0 : cC_{p}^{j}M_{p−j} M_{j} ⩽M_{p}⩽AH^{p}M_{p−j} M_{j} , ∀p ∈ ℕ, j ⩽ p

2010 Mathematics Subject Classification: 35B65, 35H10.

Key words and phrases: iterates of operators; Roumieu spaces; comparison of operators.

Supported by “Laboratoire d’analyse mathématique et applications”, University of Oran 1.

Communicated by Stevan Pilipović.

Example 1.1. The sequence M_{p} = p!^{s}, s ⩾ 1, called Gevrey sequence of order s, satisfies conditions (1.1)–(1.3).

Let Ω be an open subset of ℝ^{n}, P a linear differential operator with constant coefficients of order m.

Definition 1.1. We call Roumieu vector (or vector of type M_{p}) of the operator P in Ω, any function u ∈ C^{∞} (Ω) such that

∀H compact of Ω, ∃C > 0, ∀l ∈ ℤ_{+} : ∥P^{l}u∥_{L^{2} (H)} ⩽ C^{l+1} M_{lm}

The space of Roumieu vectors of P in Ω is denoted R_{M} (Ω, P).

Definition 1.2. We call Roumieu space in Ω, and we denote R_{M} (Ω), the space of functions u ∈ C^{∞} (Ω) such that

∀H compact Ω, ∃C > 0, ∀α ∈ ℤ_{+}^{n} : ∥D^{α} u∥_{L^{2} (H)} ⩽ C^{|α|+1} M_{|α|}

Example 1.2. If M_{p} = p!^{s}, s ⩾ 1, then R_{M} (Ω) is the Gevrey space of order s in Ω, and it is denoted G^{s}(Ω).

Similarly R_{M} (Ω, P) is denoted G^{s}(Ω, P).

Definition 1.3. We denote H the set of linear differential operators with constant coefficients P satisfying the following condition

(1.4) ∃C > 0, ∃γ ⩾ degP, ∀α ∈ ℤ_{+}^{n} , ∀ξ ∈ ℝ^{n} : |P^{(α)} (ξ)| ⩽ C(1 + |P (ξ)|)^{1− \frac{|α|}{γ}} ,

where P^{(α)} (ξ) = ∂_{ξ}^{α} P (ξ).

Example 1.3. If P is an hypoelliptic operator, it satisfies condition (1.4), and so P ∈ H (see [8]). In particular, if P is elliptic, then P ∈ H and (1.4) is fulfilled for γ = degP.

2. Basic estimates

In this section we recall the notion of comparison of differential operators, see e.g. [9, 19] and we give the basic estimate which is essential for the main result of this work.

Definition 2.1. Let P and Q be two linear differential operators with constant coefficients on ℝ^{n}, we say that P is weaker than Q and we denote P ≺ Q if for any relatively compact open subset Ω of ℝ^{n}, there exists a constant C = C(P, Q, Ω) > 0, such that

∥P υ∥_{L^{2} (Ω)} ⩽ C∥Qυ∥_{L^{2} (Ω)} , ∀υ ∈ ℂ_{0}^{∞} (Ω).

The operators P and Q are said to be equally strong if P ≺ Q ≺ P.

Remark 2.1. If P and Q are equally strong and P ∈ H, then Q ∈ H, further more if P satisfies condition (1.4) for some γ, then Q satisfies also condition (1.4) for the same constant γ, see [8].

For any open subset ω of ℝ^{n} and δ > 0 we set ω_{δ} = {x ∈ ω, d(x, Cω) > δ}.

If f ∈ L_{loc}^{2}(ω), μ > 0 and t > 0, we define

N_{ω,μ,t} (f) = sup_{0<δ⩽t} δ^{μ}∥f∥_{L^{2} (ω_{δ})} .

Without loss of generality, we suppose that ω is a bounded open subset of diameter < 1, and for simplify we denote N_{ω,μ,t} (f) = N_{μ}(f).

We need the following proposition given in Hörmander [8].

Proposition 2.1. Let ω be a bounded open subset of ℝ^{n} and let P ∈ H; then there exists C > 0 such that

(2.1)  ∑_{α} N_{γ−|α|} (P^{(α)} u) ⩽ C(N_{γ}(P u) + ∥u∥_{L^{2} (ω)} ), ∀u ∈ ℂ^{∞} (ω).

The following result is analogous to that of [8, Proposition 4.2].

Proposition 2.2. Let ω be a bounded open subset of ℝ^{n} and let Q , P ∈ H. If Q ≺ P, then there exists C > 0 such that

N_{γ}(Qu) ⩽ C(N_{γ}(P u) + ∥u∥_{L^{2} (ω)} ), ∀u ∈ ℂ^{∞} (ω),

where γ is the constant for which P satisfies condition (1.4).

Proof. Suppose that Q ≺ P, so there exists C > 0 such that

(2.2)  ∥Qυ∥_{L^{2} (ω)} ⩽ C∥P υ∥_{L^{2} (ω)} , ∀υ ∈ ℂ_{0}^{∞} (ω)

Let φ ∈ ℂ_{0}^{∞} (ω_{δ/2}) such that φ(x) = 1 in ω_{δ}, 0 ⩽ φ(x) ⩽ 1 and

|D^{α} φ(x)| ⩽ C_{α} (δ/2)^{−|α|} ,

where C_{α} depends only on n and α. From (2.2), we have for every u ∈ ℂ^{∞} (ω),

∥Qu∥_{L^{2} (ω_{δ})} ⩽ ∥Q(φu)∥_{L^{2} (ω \frac{δ}{2} )} ⩽ C∥P (φu)∥_{L^{2} (ω \frac{δ}{2} )}

By the Leibniz formula P (φu) = ∑ _{α} \frac{1}{α!} D^{α} φP^{(α)} u, we obtain

∥Qu∥_{L^{2} (ω_{δ})} ⩽ C ∑_{α} \frac{C_{α}}{α!} ( \frac{δ}{2})^{−|α|} ∥P^{(α)} u∥_{L^{2} (ω \frac{δ}{2} )}

⩽ ( \frac{δ}{2})^{−γ} ∑_{α} C′_{α}N_{γ−|α|} (P^{(α)} u) ⩽ δ^{−γ} ∑_{α} C′′_{α}N_{γ−|α|} (P^{(α)} u)

Multiplying both sides of this inequality by δ^{γ}, we get

N_{γ}(Qu) ⩽ ∑_{α} C′′_{α}N_{γ−|α|} (P^{(α)} u) ⩽ ( max_{|α|⩽deg P} C′′_{α}) ∑_{α} N_{γ−|α|} (P^{(α)} u),

which gives with (2.1)

N_{γ}Q(u) ⩽ C̃(N_{γ}(P u) + ∥u∥_{L^{2} (ω)} ),

with another constant C̃ > 0. □

Definition 2.2. We denote M_{p} ⊂ N_{p} if
∃L > 0, ∃C > 0 : M_{p} ⩽ CL^{p}N_{p}, ∀p ∈ ℕ

Example 2.1. We have p! ⊂ M_{p}, for all sequences M_{p} satisfying conditions (1.1)–(1.3). In fact we have a more stronger estimate, see [13]

∀L > 0, ∃C > 0 : p! ⩽ CL^{p}M_{p}, ∀p ∈ ℕ

3. Comparison of Roumieu vectors

The main result of this paper is the following theorem.

Theorem 3.1. Let (M_{p}) be a sequence satisfying conditions (1.1)–(1.3), and let Q, P ∈ H such that Q ≺ P. If in addition the sequence (M_{p}) satisfies

(3.1)  (p!)^{d} ⊂ M_{p} where d = \frac{γ(P)}{deg(P)},

then R_{M} (Ω, P) ⊂ R_{M} (Ω, Q).

Proof. Let ω be a bounded open subset. From Proposition 2.2 we have ∃C(Q, P, diamΩ) > 0, ∀t > 0, ∀ρ ⩾ 0, ∀υ ∈ C^{∞} (ω_{ρ}),

supτ^{γ}_{0⩽τ⩽t}∥(Qυ)∥_{L^{2} (ω_{ρ+τ} )} ⩽ C( supτ^{γ}_{0⩽τ⩽t}∥P υ∥_{L^{2} (ω_{ρ+τ} )} + ∥υ∥_{L^{2} (ω_{ρ})} ) ,

hence t^{γ}∥(Qυ)∥_{L^{2} (ω_{ρ+t} )} ⩽ C(t^{γ}∥P υ∥_{L^{2} (ω_{ρ})} + ∥υ∥_{L^{2} (ω_{ρ})} ), and so

(3.2)  ∥Qυ∥_{L^{2} (ω_{ρ+t} )} ⩽ C(∥P υ∥_{L^{2} (ω_{ρ})} + t^{−γ} ∥υ∥_{L^{2} (ω_{ρ})} ), υ ∈ ℂ^{∞} (ω_{ρ}).

Let us show by recurrence that ∃C > 0, ∀k ⩾ 0, ∀δ > 0,

(3.3)  ∥Q^{k}u∥_{L^{2} (ω_{δ})} ⩽ C^{k} ∑_{i=0}^{k} \binom{k}{i}( \frac{k}{δ})^{iγ} ∥P^{(k−i)} u∥_{L^{2} (ω)} , ∀u ∈ ℂ^{∞} (ω)

For k = 1, (3.3) is fulfilled from (3.2), it suffices to take ρ = 0 and t = δ. Suppose that estimate (3.3) is true until the order k and let us prove it at the order k + 1. Replacing in (3.2) t with \frac{δ}{k+1} , ρ with \frac{kδ}{k+1} and υ with Q^{k}(D)u, then we obtain

∥Q^{(k+1)} u∥_{L^{2} (ω_{δ})} ⩽ C(∥P (Q^{k}u)∥_{L^{2} (ω_{ρ})} + t^{−γ} ∥Q^{k}u∥_{L^{2} (ω_{ρ})} )

⩽ C^{k+1} ∑_{i=0}^{k} \binom{k}{i}( \frac{k + 1}{δ} )^{iγ} ∥P^{(k−i)} P u∥_{L^{2} (ω)}

+ C^{k+1} ∑_{i=0}^{k} \binom{k}{i}( \frac{k + 1}{δ} )^{(i+1)γ} ∥P^{(k−i)} u∥_{L^{2} (ω)}

⩽ C^{k+1} ∑_{i=0}^{k} \binom{k}{i}( \frac{k + 1}{δ} )^{iγ} ∥P^{(k+1−i)} u∥_{L^{2} (ω)}

+ C^{k+1} ∑_{i=1}^{k+1} \binom{k}{i−1}( \frac{k + 1}{δ} )^{iγ} ∥P^{(k+1−i)} u∥_{L^{2} (ω)}

But \binom{k}{i} + \binom{k}{i−1} = \binom{k+1}{i}, then

∥Q^{(k+1)} u∥_{L^{2} (ω_{δ})} ⩽ C^{k+1} ∑_{i=1}^{k+1} \binom{k+1}{i}( \frac{k + 1}{δ} )^{iγ} ∥P^{(k+1−i)} u∥_{L^{2} (ω)}

Suppose now that u ∈ R_{M} (Ω, P), so for any compact subset H of Ω there exist a bounded open subset ω and δ > 0 such that H ⊂ ω_{δ} ⊂ ω ⊂ Ω, and therefore there exists B > 0 such that ∥P^{i}u∥_{L^{2} (ω)} ⩽ B^{i+1} M_{im} , i = 0, 1, … Taking into account the relation (km)^{km} ⩽ (km)!e^{km}, we obtain for all i ⩽ k,

k^{iγ} ∥P^{(k−i)} u∥_{L^{2} (ω)} ⩽ B^{k−i+1} (km)^{iγ} M_{(k−i)m}

⩽ B^{k+1} ( (km)^{kmd} \frac{M_{km}}{M_{km}} ) ^{\frac{im}{km}} M_{(k−i)m}

⩽ B^{k+1} ( \frac{(km)!^{d}e^{kmd}}{M_{km}} ) ^{\frac{im}{km}} M_{(k−i)m} (M_{km})^{\frac{im}{km}} ,

which gives from condition (3.1),

(3.4)  k^{iγ} ∥P^{(k−i)} u∥_{L^{2} (ω)} ⩽ B_{1}^{k+1} M_{(k−i)m} (M_{km})^{\frac{im}{km}}

On the other hand we can show from (1.1) that (M_{p})^{1/p} is an increasing sequence. In fact we will prove by induction the equivalent condition

(3.5)  \frac{p + 1}{p} logM_{p} ⩽ logM_{p+1} , ∀p ∈ ℕ^{∗}

It is trivial for p = 1. Suppose that (3.5) is true for p, then from (1.1) we get

2 logM_{p+1} ⩽ logM_{p} + logM_{p+2} ⩽ logM_{p+2} + \frac{p}{p + 1} logM_{p+1} ,

hence

2 logM_{p+1} − \frac{p}{p + 1} logM_{p+1} = \frac{p + 2}{p + 1} logM_{p+1} ⩽ logM_{p+2}

In particular we have, ∀h ⩽ p, (M_{h})^{p} ⩽ (M_{p})^{h}. Applying for h = km − im and p = km, we obtain M_{(k−i)m} ⩽ (M_{km})^{\frac{km−im}{km}} , which implies with (3.4)

k^{iγ} ∥P^{(k−i)} (D)u∥_{L^{2} (ω)} ⩽ B_{1}^{k+1} M_{km} .

Substituting in (3.3) we get

∥Q^{k}u∥_{L^{2} (H)} ⩽ ∥Q^{k}u∥_{L^{2} (ω_{δ})} ⩽ C^{k} ∑_{i=0}^{k} \binom{k}{i}( \frac{1}{δ})^{kmd} B_{1}^{k+1} M_{km} ⩽ B_{2}^{k+1} M_{km}

Thus u ∈ R_{M} (Ω, Q). □

Corollary 3.1. Let P and Q be differential operators belonging to H and equally strong. If (p!)^{d} ⩽ M_{p} with d = \frac{γ(P)}{deg (P)} , then R_{M} (Ω, P) = R_{M} (Ω, Q).

Proof. From Remark 2.1, we have deg(P) = deg(Q) and γ(P) = γ(Q), which implies the result. □

Corollary 3.2. If M_{p} = p!^{s}, Theorem 3.1 coincides with Theorem 1 of Newberger–Zielezny [17] in the class of hypoelliptic operators.

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Department of Mathematics (Received 17 09 2015) University of Sciences and Technology, Oran, Algeria

 rachidchaili@gmail.com; rachid.chaili@univ-usto.dz

Department of Mathematics, University Ibn Khaldoun, Tiaret, Algeria

 mahrouz78@gmail.com