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Laure Mamy

Laure Vieublé

Pierre Benoit

Veronique V Chaplain

Sabine Houot

Christian Mougin

Effect of compost amendment and compaction on the fate and ecotoxicological impact of isoproturon in soil

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1 INRA -UR 251 PESSAC, Route de Saint Cyr, F-78026 Versailles 2 INRA-AgroParisTech UMR 1091 EGC, F-78850 Thiverval-Grignon Organic matter decline and compaction are two major related soil degradation processes [1]. Compaction changes the physical properties of soils (porosity, air content…) with consequences on their biological functioning. Compost amendment is a current practice to compensate the loss of organic matter, which could contribute to increase soil aggregate stability and to limit compaction. Furthermore, composts also modify the biological and chemical properties of soils. Therefore, both compost addition and compaction could affect the fate and impact of pesticides in soils, however this is poorly documented.

The objective of this work was to study the effect of compost amendment and soil compaction on the fate and ecotoxicological impact (measured through two enzyme activities) of isoproturon (one of the most used herbicide for cereals in Europe).

 

 Soil cores were prepared at two levels of compaction (Table 1)  The mineralization of isoproturon reached a maximum of 20% (Fig. 1). The main dissipation pathway was the formation of non-extractable residues (Fig. 2)  Compaction had no effect on the fate of isoproturon (Fig. 1 & 2) probably because the reduction in porosity did not affect the habitable pore space accessible to degrading microorganisms [2]


0
 β-glucosidase was significantly affected by compaction except in the soil which was amended with compost. The compost seemed to act as a buffer with regards to compaction  No significant impact of compaction and amendment on urease was observed. This can be due to incubation conditions not limiting for the biological activity or to repeated application of compost in the soil for 10 years  Isoproturon had no effect on β-glucosidase and urease activities probably because of the adaptation of microorganisms after previous applications of the herbicide in the field [4]



The study of the coupled effect of compaction and compost addition on the fate and impact of one herbicide, isoproturon, showed that there were very few modifications compared to the not compacted and not amended soils. However further research should be performed with other composts and biological indicators (microbial biomass, fatty acids…).

  Soil extraction with CaCl 2 then CH 3 OH (0-7-49 d)  Addition of substrates to soil samples  Quantitative determination of the reaction Undisturbed soil cores (5  2 cm) were sampled in the interfurrows of 2 plots (silt loam soil):  They were treated with 14 C-isoproturon and incubated at 28°C in darkness for 60 d  Non-extractable residues (0-7-49 d)



  The unexpected lack of effect of compost can be due to: (i) chemical characteristics of this compost [3], (ii) lack of significant change in the soil characteristics after compost addition, (iii) because the effect of compost on properties such as pesticide degrading activities vary according with the time of sampling and the delay with the last amendments[3]


Pascal Remy

On the summability of the solutions of the inhomogeneous heat equation with a power-law nonlinearity and variable coefficients

Keywords: Summability, Heat equation, Inhomogeneous partial differential equation, Nonlinear partial differential equation, Formal power series, Divergent power series 2010 MSC: 35C10, 35C20, 35K05, 35K55, 40B05

In this article, we investigate the summability of the formal power series solutions in time of the inhomogeneous heat equation with a power-law nonlinearity of degree two, and with variable coefficients. In particular, we give necessary and sufficient conditions for the 1-summability of the solutions in a given direction. These conditions generalize the ones given for the linear heat equation by W. Balser and M. Loday-Richaud in a 2009 article [? ].

Introduction

For several years, various works have been done on the divergent solutions of some classes of linear partial differential equations or integro-differential equations in two variables or more, allowing thus to formulate many results on Gevrey properties, summability or multisummability (e.g. [? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ]).

In the case of the nonlinear partial differential equations, the situation is much more complicated. The existing results concern mainly Gevrey properties, especially the convergence (e.g. [? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ]), and there are very few results about the summation (see [? ? ? ? ? ]).

˚Corresponding author

Email address: pascal.remy@uvsq.fr ; pascal.remy.maths@gmail.com (Pascal Remy)

In this article, we are interested in the summability of the formal solutions of the inhomogeneous semilinear heat equation

# B t u ´apxqB 2 x u ´bpxqu 2 " r f pt, xq up0, xq " φpxq (1.1)
in two variables pt, xq P C 2 , where the coefficients apxq and bpxq, and the initial condition φpxq are analytic on a disc D ρ with center 0 P C and radius ρ ą 0, and where the inhomogeneity r f pt, xq is a formal power series in t with analytic coefficients in D ρ (denoted in the sequel by r f pt, xq P OpD ρ qrrtss) which may be smooth, or not1 . Observe that an important particular case of Eq. (1.1) is the inhomogeneous linear heat equation

# B t u ´apxqB 2 x u " r f pt, xq up0, xq " φpxq (1.2)
obtained for bpxq " 0. Equation (1.1) arises in many physical, chemical, biological, and ecological problems involving diffusion and nonlinear growth such as heat and mass transfer, combustion theory, and spread theory of animal of plant populations. For example, if a chemical reaction generates heat depending on the temperature u, then u satisfies Eq. (1.1). In biological and ecological problems, the nonlinear term u 2 represents the growth of animal or plant population.

Proposition 1.1. Equation (1.1) admits a unique solution r upt, xq P OpD ρ qrrtss.

Proof. Writing the inhomogeneity r f pt, xq in the form

r f pt, xq " ÿ jě0 f j,˚p xq t j j! with f j,˚p xq P OpD ρ q,
and looking for r upt, xq on the same type, one easily checks that the coefficients u j,˚p xq P OpD ρ q are uniquely determined for all j ě 0 by the initial condition u 0,˚p xq " φpxq and by the recurrence relations

u j`1,˚p xq " f j,˚p xq `apxqB 2 x u j,˚p xq `bpxq j ÿ k"0 ˆj k ˙uk,˚p xqu j´k,˚p xq.
In 1999, D. A. Lutz, M. Miyake and R. Schäfke considered the case of Eq. (1.2) with apxq " 1 and r f pt, xq " 0. Using an approach based on the definition of the 1-summability in terms of the Borel transformation, they gave necessary and sufficient conditions on φpxq for r upt, xq to be 1-summable in a fixed direction argptq " θ [? ]. Afterwards, and using the same approach, various authors have extended this result in the case where, either apxq " a P C ˚, or r f pt, xq " 0 [? ? ? ] (see also [? ? ? ] for an extension in higher spatial dimensions). The general case of Eq. (1.2) was treated by W.Balser and M. Loday-Richaud in [? ], but with a different approach based on the definition of the 1-summability in terms of the successive derivatives.

In this article, we propose to extend the results of [? ] to the general equation (1.1). In Section 2, we recall some basic definitions and properties about the 1-summable formal series and we state the main result of our article (Theorem 2.4). This result is proved in Section 3.

1-summability of r upt, xq

All along the article, we consider t as the variable and x as a parameter. Thereby, to define the notion of 1-summability of formal power series in OpD ρ qrrtss, one extends the classical notions of 1-summability of elements in Crrtss to families parametrized by x in requiring similar conditions, the estimates being however uniform with respect to x. Doing that, any formal power series in OpD ρ qrrtss can be seen as a formal power series in t with coefficients in a convenient Banach space defined as the space of functions that are holomorphic on a disc D r (0 ă r ă ρ) and continuous up to its boundary, equipped with the usual supremum norm. For a general study of series with coefficients in a Banach space, we refer for instance to [? ].

1-summable formal series

Among the many equivalent definitions of 1-summability in a given direction argptq " θ at t " 0, we choose in this article a generalization of Ramis' definition which states that a formal series r gpt, xq P Crrtss is 1-summable in direction θ if there exists a holomorphic function g which is 1-Gevrey asymptotic to r

g in an open sector Σ θ,ąπ bisected by θ and with opening larger than π [? , Def. 3.1]. To express the 1-Gevrey asymptotic, there also exist various equivalent ways. We choose here the one which sets conditions on the successive derivatives of g (see [? , p. 171] or [? , Thm. 2.4] for instance).

Definition 2.1 (1-summability). A formal series r upt, xq P OpD ρ qrrtss is said to be 1-summable in the direction argptq " θ if there exist a sector Σ θ,ąπ , a radius 0 ă r ď ρ and a function upt, xq called 1-sum of r upt, xq in direction θ such that 1. u is defined and holomorphic on Σ θ,ąπ ˆDr ; 2. For any x P D r , the map t Þ Ñ upt, xq has r upt, xq "

ÿ jě0 u j,˚p xq t j j!
as Taylor series at 0 on Σ θ,ąπ ;

3. For any proper2 subsector Σ Ť Σ θ,ąπ , there exist constants C ą 0 and K ą 0 such that, for all ℓ ě 0, all t P Σ and all x P D r , ˇˇB ℓ t upt, xq ˇˇď CK ℓ Γp1 `2ℓq. We denote by OpD ρ qttu 1;θ the subset of OpD ρ qrrtss made of all the 1-summable formal series in the direction argptq " θ.

Note that, for any fixed x P D r , the 1-summability of r upt, xq coincides with the classical 1-summability. Consequently, Watson's lemma implies the unicity of its 1-sum, if any exists.

Note also that the 1-sum of a 1-summable formal series r upt, xq P OpD ρ qttu 1;θ may be analytic with respect to x on a disc D r smaller than the common disc D ρ of analyticity of the coefficients u j,˚p xq of r upt, xq. Denote by B ´1 t r u (resp. B ´1 x r u) the anti-derivative of r u with respect to t (resp. x) which vanishes at t " 0 (resp. x " 0). Proposition 2.2 below specifies the algebraic structure of OpD ρ qttu 1;θ .

Proposition 2.2. Let θ P R{2πZ. Then, pOpD ρ qttu 1;θ , B t , B x q is a C-differential algebra stable under the anti-derivatives B ´1 t and B ´1

x .

We refer for instance to [? , Prop. 3.2] for a proof of this result.

With respect to t, the 1-sum upt, xq of a 1-summable series r upt, xq P OpD ρ qttu 1;θ is analytic on an open sector for which there is no control on the angular opening except that it must be larger than π (hence, it contains a closed sector Σ θ,π bisected by θ and with opening π) and no control on the radius except that it must be positive. Thereby, the 1-sum upt, xq is well-defined as a section of the sheaf of analytic functions in pt, xq on a germ of closed sector of opening π (that is, a closed interval I θ,π of length π on the circle S 1 of directions issuing from 0; see [? , 1.1] or [? , I.2]) times t0u (in the plane C of the variable x). We denote by O I θ,π ˆt0u the space of such sections.

Corollary 2.3. The operator of 1-summation

S 1;θ : OpD ρ qttu 1;θ ÝÑ O I θ,π ˆt0u r upt, xq Þ ÝÑ upt, xq
is a homomorphism of differential C-algebras for the derivations B t and B x . Moreover, it commutes with the anti-derivations B ´1 t and B ´1

x .

Let us now turn to the study of the 1-summability of the formal solution r upt, xq P OpD ρ qrrtss of Eq. (1.1).

Main result

Before stating our main result, let us start with a preliminary remark. Write the coefficients apxq and bpxq in the form apxq "

ÿ ně0 a n x n n! , bpxq " ÿ ně0 b n x n n!
and the formal series r upt, xq and r f pt, xq in the form

r upt, xq " ÿ ně0 r u ˚,n ptq x n n! , r f pt, xq " ÿ ně0 r f ˚,n ptq x n n! .
By identifying the terms in x n in Eq. (1.1), we get the identities

$ ' ' & ' ' % a 0 r u ˚,2 ptq " B t r u ˚,0 ptq ´b0 r u ˚,0 ptqr u ˚,0 ptq ´r f ˚,0 ptq a 0 r u ˚,3 ptq `a1 r u ˚,2 ptq " B t r u ˚,1 ptq ´2b 0 r u ˚,0 ptqr u ˚,1 ptq ´b1 r u ˚,0 ptqr u ˚,0 ptq ´r f ˚,1 ptq, (2.1
) for n " 0 and n " 1, and the identities

a 0 r u ˚,n`2 ptq `na 1 r u ˚,n`1 ptq " B t r u ˚,n ptq ´n ÿ k"2 ˆn k ˙ak r u ˚,n`2´k ptq ´ÿ k0`k1`k2"n n! k 0 !k 1 !k 2 ! b k0 r u ˚,k1 ptqr u ˚,k2 ptq ´r f ˚,n ptq
for n ě 2. Consequently, each formal series r u ˚,n ptq is uniquely determined from r u ˚,0 ptq, r u ˚,1 ptq and r f pt, xq. In the case of the linear heat equation (1.2), W. Balser and M. Loday-Richaud proved, under the assumption that pa 0 , a 1 q ‰ p0, 0q, that the terms r u ˚,0 ptq, r u ˚,1 ptq and r f pt, xq allow to fully characterize the 1-summability of the formal solution r upt, xq in a given direction [? ]. In the case of our semilinear heat equation (1.1), Theorem 2.4 below tells us that this characterization remains valid. More precisely, we have: Theorem 2.4. Let argptq " θ P R{2πZ be a direction issuing from 0. Assume that either ap0q ‰ 0, or ap0q " 0 and a 1 p0q ‰ 0. Then, 1. The unique formal series solution r upt, xq P OpD ρ qrrtss of Eq. (1.1) is 1summable in the direction θ if and only if the inhomogeneity r f pt, xq and the coefficients r u ˚,0 ptq and r u ˚,1 ptq are 1-summable in the direction θ. 2. Moreover, the 1-sum upt, xq, if any exists, satisfies Eq. (1.1) in which r f pt, xq is replaced by its 1-sum f pt, xq in the direction θ.

When apxq " Opx 2 q, Theorem 2.4 fails: the formal solution r upt, xq may not be 1-summable in a given direction, while r u ˚,0 ptq, r u ˚,1 ptq and r f pt, xq are 1-summable. Such a situation occurs for example in the case where apxq " x 2 , bpxq " 0, r f pt, xq " 0 and φpxq " 1 1´x . We refer to [? , Counter example 3.5] for the details of the calculations.

Let us now turn to the proof of Theorem 2.4.

3. Proof of Theorem 2.4

3.1. Case ap0q ‰ 0 Ÿ Point 1 (necessary condition). This is straightforward from Proposition 2.2.

We have indeed r u ˚,0 ptq " r upt, 0q, r u ˚,1 ptq " B x r upt, xq |x"0 , and

r f pt, xq " B t upt, xq ´apxqB 2 x upt, xq ´bpxqupt, xq 2 .
Ÿ Point 1 (sufficient condition). To prove that the condition is sufficient, we shall proceed in a similar way as the proof of [? , Thm. 3.4] (see also [? ? ? ]).

By assumption, we have ap0q ‰ 0. Hence, the functions Apxq " 1{apxq and Bpxq " bpxq{apxq are both well-defined and holomorphic on a convenient disc D ρ 1 with 0 ă ρ 1 ď ρ.

Let Let us now assume that r u ˚,0 ptq, r u ˚,1 ptq and r f pt, xq are 1-summable in a given direction θ. Then, r vpt, xq and r gpt, xq are both 1-summable in the direction θ (see Proposition 2.2) and identity (3.1) above tells us it suffices to prove that it is the same for r wpt, xq. To this end, we shall proceed similarly as [? ? ? ? ] through a fixed point method. Of course, as we shall see below, the nonlinear term pB ´2

x r wq 2 induces much more complicated calculations.

Let us set r wpt, xq " recursively determined for all m ě 0 by the system

$ ' & ' % r w 0 " r g r w m`1 " ApxqB t B ´2 x r w m ´2Bpxqr vB ´2 x r w m ´Bpxq m ÿ k"0 pB ´2 x r w k qpB ´2 x r w m´k q
(3.2) Observe that r w m pt, xq P OpD ρ 1 qrrtss for all m ě 0. Observe also that the r w m pt, xq's are of order Opx 2m q in x for all m ě 0, and, consequently, the series r wpt, xq itself makes sense as a formal series in t and x.

Let us now respectively denote by w 0 pt, xq and vpt, xq the 1-sums of r w 0 " r g and r v in direction θ and, for all m ą 0, let w m pt, xq be determined as the solution of System (3.2) in which r v is replaced by v and all the r w m are replaced by w m .

By construction, all the functions w m pt, xq are defined and holomorphic on a common domain Σ θ,ąπ ˆDρ 2 with a convenient radius 0 ă ρ 2 ď ρ 1 .

To end the proof, it remains to prove that the series ÿ mě0 w m pt, xq is convergent, and that its sum wpt, xq is the 1-sum of r wpt, xq in direction θ. According to Definition 2.1, the 1-summability of r w 0 and r v implies that there exists 0 ă r 1 ă minp1, ρ 2 q such that, for any proper subsector Σ Ť Σ θ,ąπ , there exist two positive constants C ą 0 and K ě 1 such that, for all ℓ ě 0 and all pt, xq P Σ ˆDr 1 , the functions w 0 and v satisfy the inequalities Proposition 3.1. The following inequalities

ˇˇB ℓ t w 0 pt,
ˇˇB ℓ t w m pt, xq ˇˇď Cpα `2π 2 Cβq m K ℓ`m Γp1 `2pℓ `mqq |x| 2m p2mq! (3.4)
hold for all ℓ, m ě 0 and all pt, xq P Σ ˆDr 1 .

Proof. The proof proceeds by recursion on m. The case m " 0 is straightforward from the first inequality of (3.3). Let us now suppose that the inequalities (3.4) hold for all 0 ď k ď m for a certain m ě 0. According to the relations (3.2), we deduce from the Leibniz Formula that

B ℓ t w m`1 pt, xq " ApxqB ℓ`1 t B ´2 x w m pt, xq ´2Bpxq ℓ ÿ j"0 ˆℓ j ˙Bj t vpt, xqB ℓ´j t B ´2 x w m pt, xq ´Bpxq m ÿ k"0 ℓ ÿ j"0 ˆℓ j ˙Bj t B ´2 x w k pt, xqB ℓ´j t B ´2 x w m´k pt, xq
for all ℓ ě 0 and pt, xq P Σ ˆDr 1 . Hence, applying the second inequality of (3.3) and the inequalities (3.4) for all the w k 's with k " 0, ..., m, and using the fact that K ě 1 and r 1 ă 1, we get the inequalities

ˇˇB ℓ t w m`1 pt, xq ˇˇď Cpα `2π 2 Cβq m K ℓ`m`1 Γp1 `2pℓ `m `1qq |x| 2m`2 p2m `2q! ˆpα `2CβS m,ℓ `CβS 1 m,ℓ q
for all ℓ ě 0 and pt, xq P Σ ˆDr 1 , where S m,ℓ and S 1 m,ℓ are respectively defined

To prove the second condition of Definition 2.1, we proceed as follows. The removable singularities theorem implies the existence of lim 3.2. Case ap0q " 0 and a 1 p0q ‰ 0

The necessary condition of the first point and the second point result as before from Proposition 2.2 and Corollary 2.3. We sketch here below the proof of the sufficient condition of the first point.

Denote apxq " xa 1 pxq. By assumption, a 1 p0q ‰ 0. Then, the functions A 1 pxq " 1{a 1 pxq and B 1 pxq " bpxq{a 1 pxq are both well-defined and holomorphic on a convenient common disc centered at the origin 0 P C.

Setting By assumption, we have ap0q " a 0 " 0; hence, due to the first equality of (2.1), the constant term in x of B t r vpt, xq ´bpxqr vpt, xq 2 ´r f pt, xq is zero, and, consequently, r gpt, xq is again a formal power series in t and x. Assuming then r vpt, xq and r gpt, xq to be 1-summable in the direction θ, we can prove as previously that r wpt, xq is also 1-summable in the direction θ. Observe that the r w m pt, xq are now recursively determined for all m ě 0 by the system

$ ' & ' % r w 0 " r g r w m`1 " A 1 pxq x B t B ´2 x r w m ´2 B 1 pxq x r vB ´2 x r w m ´B1 pxq x m ÿ k"0 pB ´2 x r w k qpB ´2 x r w m´k q
In particular, the operator

1 x B ´2
x in place of B ´2

x implies that the r w m pt, xq's are now of order Opx m q in x for all m ě 0, instead of Opx 2m q as in the previous case. Still, r wpt, xq is again a formal power series in t and x. The estimates on the derivatives B ℓ t w m given in Proposition 3.1 are modified as follows: for all m, ℓ ě 0 and all pt, xq P Σ ˆDr 1 ,

ˇˇB ℓ t w m pt, xq ˇˇď Cpα 1 `2π 2 Cβ 1 q m K ℓ`m Γp1 `2pℓ `mqq |x| m pm!q 2 ,
where α 1 (resp. β 1 ) stands for the maximum of |A 1 pxq| (resp. |B 1 pxq|) on the closed disc |x| ď r 1 . Consequently, the inequalities (3.5) obtained in the case ap0q ‰ 0 become ˇˇB ℓ t w m pt, xq ˇˇď CK 1ℓ Γp1 `2ℓqpc |x|q m for all ℓ, m ě 0 and all pt, xq P Σ ˆDr 1 , where K 1 and c are the two positive constants defined by K 1 " 4K and c " 16Kpα 1 `2π 2 Cβ 1 q.

The end of the proof is similar to the one of the case ap0q ‰ 0 and is left to the reader. This completes the proof of Theorem 2.4.

Conclusion and directions for further researches

In this article, we presented a method to characterize the 1-summability of the formal power series solutions in time of the inhomogeneous semilinear heat equation (1.1). Although this equation is among the simplest semilinear partial differential equations, it has the merit of easily showing how our approach makes it possible to manage the nonlinear term u 2 .

In the forthcoming article [? ], we will show that this approach can be successfully applied to the more general equation B κ t u ´apt, xqB p x u ´P puq " r f pt, xq where p and κ are two positive integers satisfying p ą κ ě 1, apt, xq is an analytic function at the origin of C 2 , and where P pXq is a polynomial with analytic coefficients at the origin of C 2 . However, even though the general philosophy of studying this equation is similar to the one developed in the present paper for the heat equation (1.1), the calculations are slightly different and more complicated insofar as we are no longer interested in the 1-summability but in the k " κ{pp ´κq-summability. In particular, this leads us, among other things, to use different technical results on the generalized binomial and multinomial coefficients, that is on the binomial and multinomial coefficients with nonnegative real terms. Therefore, it seems that the approach developed in the present paper is interesting to investigate the summability, even the multisummability, of the formal power series solutions of semilinear, even nonlinear, partial differential equations much more general.

Another possible direction of research to investigate the summability of the formal power series solutions of the semilinear partial differential equations is the "Borel-Laplace summation" point of view, as it could already have been done in the linear case; for example [? ? ? ? ? ? ? ] for the linear heat equation.

  , xq and let us choose the solution of Eq. (3.1)

  xq for allx P D r and, thereby, the existence of the Taylor series of w at 0 on Σ 1 for all x P D r (see for instance[? , Cor. 1.1.3.3]; see also[? , Prop. 1.1.11]). On the other hand, considering recurrence relations (3.2) with w m and the 1-sums vpt, xq and gpt, xq instead of r w m , r vpt, xq and r gpt, xq, it is clear that wpt, xq satisfies equation (3.1) with vpt, xq in place of r vpt, xq and right-hand side gpt, xq in place of r gpt, xq and, consequently, so does its Taylor series. Then, since equation (3.1) has a unique formal series solution r wpt, xq, we then conclude that the Taylor expansion of wpt, xq is r wpt, xq. Hence, Condition 2 of Definition 2.1 holds. This achieves the proof of the 1-summability of r wpt, xq and, thereby, the fact that the condition is sufficient. Ÿ Point 2. The fact that the 1-sum upt, xq of r upt, xq in direction θ satisfies Eq. (1.1) with right-hand side the 1-sum f pt, xq of r f pt, xq in direction θ in place of r f pt, xq is a direct consequence of Corollary 2.3. This completes the proof of Theorem 2.4 in the case ap0q ‰ 0.

  xq ˇˇď CK ℓ Γp1 `2ℓq and ˇˇB ℓ t vpt, xq ˇˇď CK ℓ Γp1 `2ℓq. (3.3) Let us now fix a proper subsector Σ Ť Σ θ,ąπ and let us denote by α (resp. β) the maximum of |Apxq| (resp. |Bpxq|) on the closed disc |x| ď r 1 . Proposition (3.1) below provides us some estimates on the derivatives B ℓ t w m .

We denote r f with a tilde to emphasize the possible divergence of the series r f .

A subsector Σ of a sector Σ 1 is said to be a proper subsector and one denotes Σ Ť Σ 1 if its closure in C is contained in Σ 1 Y t0u.

Inequalities (3.4) follow then from Lemmas (3.2) and (3.3) below and from the fact that 2 ď π 2 . This ends the proof.

Lemma 3.2. S m,ℓ ď 1 for all m, ℓ ě 0.

Proof. Lemma (3.2) stems obvious from the identity

ȧnd the combinatorial inequalities ˆ2ℓ `2m 2j ˙ě ˆℓ j ˙ˆℓ `2m j ˙ě ˆℓ j ˙.

Lemma 3.3. S 1 m,ℓ ď π 2 for all m, ℓ ě 0.

Proof. First of all, let us observe that

Applying then the combinatorial inequality

Let us now observe that ℓ ÿ j"0

for all ℓ ě 0: the inequality is clear for ℓ P t0, 1, 2, 3u, and, for ℓ ě 4, we have

Hence,

From Proposition (3.1), we next derive the inequalities

for all ℓ, m ě 0 and all pt, xq P Σ ˆDr 1 , where K 1 and c are the two positive constants defined by K 1 " 4K and c " 4Kpα `2π 2 Cβq. Indeed, applying the inequalities (3.4), we easily have

Let us now choose for Σ a sector containing a proper subsector Σ 1 bisected by the direction θ and opening larger than π (such a choice is already possible by definition of a proper subsector, see Footnote 2). Let us also choose a radius 0 ă r ă minpr 1 , 1{ ? cq and let us set

Thanks to the inequalities (3.5), the series