S Konate

M.-A Orsoni

SAMPLING CONSTANTS IN GENERALIZED FOCK SPACES

Keywords: 2010 Mathematics Subject Classification. 30H20 Dominating sets, reverse Carleson measure, Fock space, sampling

We prove several results related to a Logvinenko-Sereda type theorem on dominating sets for generalized doubling Fock spaces. In particular, we give a precise polynomial dependence of the sampling constant on the relative density parameter γ of the dominating set. Our method is an adaptation of that used in [HKKO21] for the Bergman spaces and is based on a Remez-type inequality and a covering lemma related to doubling measures.

Introduction

Sampling problems are central in signal theory and cover, for instance, sampling sequences and so-called dominating sets which allow to recover the norm of a signal -defined by an integration over a given domain -from the integration on a subdomain (precise definitions will be given later). They have been considered in a large variety of situations (see e.g. the survey [START_REF] Fricain | Harmonic analysis, function theory, operator theory, and their applications[END_REF]), including the Fock space and its generalized versions (see [Sei04, MMOC03, OC98, Lin00, JPR87, LZ19]). In this paper we will focus on the second class of problems, i.e. dominating sets. Conditions guaranteeing that a set is dominating were established rather long ago (in the 70's for the Paley-Wiener space and in the 80's for the Bergman and Fock spaces, see e.g. the survey [START_REF] Fricain | Harmonic analysis, function theory, operator theory, and their applications[END_REF] and references therein). More recently, people got interested in estimates of the sampling constants which give quantitative information on the tradeoff between the cost of the sampling and the precision of the estimates. The major paper on this connection is by Kovrijkine [Kov01] who gave a method to consider this problem in the Paley-Wiener space. Subsequently, his method was adapted to several other spaces (see [HJK20, JS21, HKKO21, GW21, BJPS21]). In this paper, inspired by the methods in [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF], we will discuss the case of generalized doubling Fock spaces. For these, the paper [START_REF] Marco | Interpolating and sampling sequences for entire functions[END_REF] provides a wealth of results that allow to translate the main steps of [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF] to this new framework.

Notations. As usual, A B (respectively A B) means that there exists a constant c > 0 independent of the relevant variables such that A ≤ cB (resp. A ≥ cB). Similarly, A B stands for A B and A B.

1.1. Doubling subharmonic functions and generalized Fock spaces. A function φ : C -→ R of class C 2 is said to be subharmonic if ∆φ ≥ 0, and doubling subharmonic if it is subharmonic and the (non-negative) measure µ := ∆φ is doubling, i.e. there exists a constant C such that for every z ∈ C and r > 0, µ(D(z, 2r)) ≤ Cµ(D(z, r)).

(1.1)

Here D(z, r) denotes the standard open euclidean disk of center z ∈ C and radius r > 0.

The minimal constant satisfying (1.1) is denoted by C µ and called doubling constant for µ. A basic example of doubling subharmonic function is φ(z) = |z| 2 for which µ = ∆φ is equal to 4 times the Lebesgue measure and C µ = 4. The reader may consider this simpler case for the time of the introduction. Doubling subharmonic functions induce a new metric on the complex plane C which is more adapted to the complex analysis we shall work with. We postpone the advanced results on doubling measures to Section 2 but let us introduce some basic objects related to it. First, we can associate with φ as above a function ρ :

C -→ R + , such that µ(D(z, ρ(z))) = 1.
To get an idea, assuming φ is suitably regularized, we have ∆φ ρ -2 (see [MMOC03, Theorem 14 and Remark 5]). Next, denote D r (z) := D(z, rρ(z)) the disk adapted to the new metric and write D(z) := D 1 (z) the disk with unit mass for the measure µ. Finally, with these definitions in mind we can introduce the following natural density. A measurable set E is (γ, r)-dense (with respect to the above metric), if for every z ∈ C

|E ∩ D r (z)| |D r (z)| ≥ γ.
Here |F | denotes planar Lebesgue measure of a measurable set F . We will just say that the set is relatively dense if there is some γ > 0 and some r > 0 such that the set is (γ, r)-dense. We shall see in the next subsection that relative density is the right notion to characterize dominating sets.

Let dA be the planar Lebesgue measure on C. For a doubling subharmonic function φ : C -→ R and 1 ≤ p < +∞, we will denote by L p φ (F ) the Lebesgue space on a measurable set F ⊂ C with respect to the measure e -pφ dA. We will also use the notation

f p L p φ (F ) := F |f | p e -pφ dA.
Finally, we define the doubling Fock space, by

F p φ = {f ∈ Hol(C) : f p L p φ (C) := C |f | p e -pφ dA < +∞}.
These spaces appear naturally in the study of the Cauchy-Riemann equation (see [START_REF] Christ | On the ∂ equation in weighted L 2 norms in C 1[END_REF][START_REF] Constantin | Some spectral properties of the canonical solution operator to ∂ on weighted Fock spaces[END_REF]) and are well studied objects (see e.g. [START_REF] Oliver | Toeplitz operators on doubling Fock spaces[END_REF] for Toeplitz operators on these spaces). When φ(z) = |z| 2 , F p φ is the classical Fock space (see the textbook [START_REF] Zhu | Analysis on Fock spaces[END_REF] for more details). 1.2. Dominating sets and sampling constants. A measurable set E ⊂ C will be called dominating for F p φ if there exists C > 0 such that

E |f | p e -pφ dA ≥ C p C |f | p e -pφ dA, ∀f ∈ F p φ . (1.2)
The biggest constant C appearing in (1.2) is called the sampling constant.

The notion of dominating sets can be defined for several spaces of analytic functions and their characterization has been at the center of intensive research starting with a famous result of Panejah [START_REF] Panejah | On some problems in harmonic analysis[END_REF][START_REF] Panejah | Certain inequalities for functions of exponential type and a priori estimates for general differential operators[END_REF], and Logvinenko and Sereda [START_REF] Logvinenko | Equivalent norms in spaces of entire functions of exponential type[END_REF] for the Paley-Wiener space, which consists of all functions in L 2 whose Fourier transform is supported on a fixed compact interval. For corresponding results in the Bergman space we refer to [START_REF] Luecking | Inequalities on Bergman spaces[END_REF][START_REF] Luecking | Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives[END_REF][START_REF] Luecking | Dominating measures for spaces of analytic functions[END_REF][START_REF] Green | Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains[END_REF]. As for Fock spaces, this has been done by Jansen, Peetre and Rochberg in [START_REF] Janson | Hankel forms and the Fock space[END_REF] for the classical case ϕ(z) = |z| 2 and by Lou and Zhuo in [LZ19, Theorem A] for generalized doubling Fock spaces (see also [START_REF] Ortega-Cerdà | Sampling measures[END_REF] for a characterization of general sampling measures). It turns out that a set E is dominating for the doubling Fock spaces if and only if it is relatively dense. This characterization also holds for the other spaces of analytic functions mentioned above with an adapted notion of relative density.

Once the dominating sets have been characterized in terms of relative density, a second question of interest is to know whether the sampling constants can be estimated in terms of the density parameters (γ, r). Kovrijkine answered this question in [START_REF] Kovrijkine | Some results related to the Logvinenko-Sereda theorem[END_REF] giving a sharp estimate on the sampling constants for the Paley-Wiener spaces, with a polynomial dependence on γ (see also [START_REF] Reznikov | Sharp constants in the Paneyah-Logvinenko-Sereda theorem[END_REF] for sharp constants in some particular geometric settings). Such a dependence is used for example in control theory (see [START_REF] Egidi | Sharp geometric condition for null-controllability of the heat equation on R d and consistent estimates on the control cost[END_REF][START_REF] Egidi | An abstract Logvinenko-Sereda type theorem for spectral subspaces[END_REF][START_REF] Beauchard | Spectral estimates for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations[END_REF] and the references therein). Kovrijkine's method involves Remez-type and Bernstein inequalities. It has been used in several spaces in which the Bernstein inequality holds (see for instance [START_REF] Hartmann | Quantitative estimates of sampling constants in model spaces[END_REF] for the model space, [START_REF] Beauchard | Spectral estimates for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations[END_REF] for spaces spanned by Hermite functions, or [START_REF] Egidi | An abstract Logvinenko-Sereda type theorem for spectral subspaces[END_REF] for general spectral subspaces), and also to settings where a Bernstein inequality is not at hand (e.g. Fock and polyanalytic Fock spaces [START_REF] Jaming | Planar sampling sets for the short-time Fourier transform[END_REF] or Bergman spaces [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF][START_REF] Green | Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains[END_REF]). In [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF], the authors developed a machinery based on a covering argument to circumvent Bernstein's inequality and the aim of this paper is to adapt this method to doubling Fock spaces. 1.3. Main results. It can be deduced from Lou and Zhuo's result [LZ19, Theorem A] that if E is (γ, r)-dense then there exist some constants 0 < ε 0 < 1 and c > 0 depending only on r such that inequality (1.2) holds for every

C p ≤ cγε 2(p+2) γ 0 .
This last estimate gives an exponential dependence on γ. In the spirit of the work [START_REF] Kovrijkine | Some results related to the Logvinenko-Sereda theorem[END_REF], we improve it providing a polynomial estimate in γ with a power depending suitably on r.

Theorem 1.1. Let φ be a doubling subharmonic function and 1 ≤ p < +∞. Given r > 1, there exists L such that for every measurable set E ⊂ C which is (γ, r)-dense, we have

f L p φ (E) ≥ γ c L f L p φ (C) (1.3) for every f ∈ F p φ .
Here, the constants c and L depend on r, and for L we can choose

L r log 2 (Cµ) + 1 p (1 + log(r))
where C µ is the doubling constant (see (1.1)) and the implicit constant depends only on the space (i.e. only on φ).

Remark 1.2. When φ(z) = |z| 2 , we have log 2 (C µ ) = 2 and we get the same result as that given in [JS21, Theorem 4.6] for the classical Fock space with an explicit dependence on p in addition.

Observe that we are mainly interested in the case when r is big, for instance r ≥ 1 (in case E is (γ, r)-dense for some r < 1 we can also show that E is ( γ, 1)-dense for some γ γ where underlying constants are universal).

The proof of Theorem 1.1 follows the scheme presented in [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF]. We will recall the necessary results from that paper. The main new ingredients come from [START_REF] Marco | Interpolating and sampling sequences for entire functions[END_REF] and concern a finite overlapping property and a lemma allowing us to express the subharmonic weight locally as (the real part of) a holomorphic function.

We should mention that in [LZ19, Theorem 7], in the course of proving that the relative density is a necessary condition for domination, it is shown that γ C p . Hence, we cannot expect better than a polynomial dependence in γ of the sampling constant C. In this sense, our result is optimal.

As noticed in a remark in [START_REF] Luecking | Inequalities on Bergman spaces[END_REF]p.11] and after Theorem 2 in [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF] for the Bergman space, there is no reason a priori why a holomorphic function for which the integral E |f | p e -pφ dA is bounded for a relative dense set E should be in F p φ . Outside the class F p φ relative density is in general not necessary for domination. For this reason, we always assume that we test on functions f ∈ F p φ .

As a direct consequence of Theorem 1.1, we obtain a bound for the norm of the inverse of a Toeplitz operator T ϕ . We remind that for any bounded measurable function ϕ, the Toeplitz operator T ϕ is defined on F 2 φ by T ϕ f = P(ϕf ) where P denotes the orthogonal projection from L 2 φ (C) onto F 2 φ . As remarked in [LZ19, Theorem B], for a non-negative function ϕ, T ϕ is invertible if and only if E s = {z ∈ C : ϕ(z) > s} is a dominating set for some s > 0. Tracking the constants we obtain Corollary 1.3. Let ϕ be a non-negative bounded measurable function. The operator T ϕ is invertible if and only if there exists s > 0 such that E s = {z ∈ C : ϕ(z) > s} is (γ, r)-dense for some γ > 0 and r > 0. In this case, we have

||T -1 ϕ || ≤ ϕ -1 ∞ 1 -1 - s ϕ ∞ 2 γ c 2L .
Notice that the right-hand side behaves as γ -2L as γ → 0. For the sake of completeness, we will give a proof of the reverse implication at the end of Section 4.

The paper is organized as follows. In Section 2 we recall several results from [START_REF] Marco | Interpolating and sampling sequences for entire functions[END_REF] concerning doubling measures and subharmonic functions. In Section 3, we introduce planar Remez-type inequalities which will be a key ingredient in the proof of Theorem 1.1. Finally, we prove the covering lemma and Theorem 1.1 in Section 4 and deduce Corollary 1.3.

Reminders on doubling measures

A non-negative doubling measure µ (see (1.1) for the definition) determines a metric on C to which the usual notions have to be adapted. In this section, we recall several geometric measure theoretical tools in connection with doubling measures that we will need later and which come essentially from the paper [MMOC03, Chapter 2]. Actually, the results of the present section can be translated in terms of the distance induced by the metric ρ(z) -2 dz ⊗ dz but we will not exploit this point of view here.

We start with a standard geometric estimate whose main part is due to [Chr91, Lemma 2.1].

Lemma 2.1. [MMOC03, Lemma 1] Let µ be a doubling measure on C. For any disks D(z, r) and D(z , r ) such that r > r and z ∈ D(z, r), we have

1 2C µ r r κ ≤ µ(D(z, r)) µ(D(z , r )) ≤ C 2 µ r r log 2 (Cµ)
.

where κ = 1 C 2 µ +2 . Here x denotes the smallest integer bigger or equal to x and correspondingly, x is the biggest integer less or equal to x. The paper [START_REF] Marco | Interpolating and sampling sequences for entire functions[END_REF] refers to [START_REF] Christ | On the ∂ equation in weighted L 2 norms in C 1[END_REF] for the proof of this result. We mention that this latter paper only contains the left hand side estimate. For the convenience of the reader and in order to better understand the constants, we provide a detailed proof completing Christ's argument.

Proof. Let us start with the right hand side inequality. By the triangular inequality, we have D(z, r) ⊂ D(z , 2r). Now, note that

2r = 2 log 2 ( r r )+1 r ≤ 2 log 2 ( r r ) +1 r .
Hence iterating the doubling inequality (1.1), we get

µ(D(z, r)) ≤ µ(D(z , 2r)) ≤ C log 2 ( r r ) +1 µ µ(D(z , r )) ≤ C 2 µ r r log 2 (Cµ)
µ(D(z , r )).

As mentioned above, the left hand side inequality is given in [Chr91, Lemma 2.1]. We reproduce the proof here making the constant κ more precise. Since z ∈ D(z, r), we have D(z , r) ⊂ D(z, 2r) and so

µ(D(z, r)) µ(D(z , r )) ≥ C -1 µ µ(D(z, 2r)) µ(D(z , r )) ≥ C -1 µ µ(D(z , r)) µ(D(z , r )) . (2.1)
Hence it suffices to prove that for any z ∈ C, µ(D(z , r)) µ(D(z , r )) ≥ 1 2 r r κ whenever r > r . Let k ≥ 3 be an integer to be fixed later. First assume that r r = 2 k . Then we can construct pairwise disjoint disks D 1 , D 2 , . . . D k-2 such that D j has radius 2 j r , D(z , r ) ⊂ 3D j and D j ⊂ D(z , r). A way to do that is to choose the disks D j centered along a radius of D(z , r) and ordered in such a way that the radius increases away from z (see Figure 1).

Now, since µ(D(z , r )) ≤ µ(3D j ) ≤ C 2 µ µ(D j ), we get µ(D(z , r)) ≥ k-2 j=1 µ(D j ) + µ(D(z , r )) ≥ [(k -2)C -2 µ + 1]µ(D(z , r )).
Hence, fixing k to be the smallest integer such that (k -2)C -2 µ + 1 ≥ 2 i.e k = C 2 µ + 2 , we obtain µ(D(z , r)) ≥ 2µ(D(z , r )) whenever r r = 2 k . Let us treat the general case r > r now. If r r ≥ 2 k , we reduce the problem to the previous setting decomposing D(z , r) into a chain of disks

D(z , r ) = D(z , r 0 ) ⊂ D(z , r 1 ) ⊂ • • • ⊂ D(z , r m ) ⊂ D(z , r)
where r i = 2 k r i-1 for 1 ≤ i ≤ m, and m = log 2 (r/r ) k . Hence we can iterate the process and get for every z ∈ C,

µ(D(z , r)) ≥ µ(D(z , r m )) ≥ 2µ(D(z , r m-1 )) ≥ . . . ≥ 2 m µ(D(z , r )) = 2 log 2 ( r r )/k µ(D(z , r )) ≥ 1 2 r r 1/k µ(D(z , r )).
Combined with (2.1), this leads to the left hand side inequality of Lemma 2.1 with

κ = 1 k = 1 C 2 µ + 2 . If r r < 2 k , then it is clear that 1 2 r r κ < 1 ≤ µ(D(z ,r))
µ(D(z ,r )) since r > r . Again, combined with (2.1), this yields the expected constant in the left hand side inequality.

Remark 2.2. If instead of z ∈ D(z, r), we have D(z, r) ∩ D(z , r ) = ∅ as in [MMOC03,
Lemma 1], we get the same inequalities with C µ replaced by C 2 µ in the left hand inequality and C 2 µ replaced by C 3 µ in the right hand inequality. Also, when z = z , we can remove C µ from the left hand side and replace C 2 µ by C µ in the right hand side. Remark 2.3. In the classical Fock space, µ is the Lebesgue measure (up to a multiplicative constant) and so C µ = 4. Note that in this case, we obtain an equality with the right hand side and the factor C 2 µ disappears. This is due to the invariance by translation and the homogeneity of Lebesgue measure.

As a direct consequence of Lemma 2.1 and Remark 2.2, picking z = z, and replacing r by rρ(z) and r by ρ(z), we get the following useful estimate: for all z ∈ C and r > 1,

1 2 r κ ≤ µ(D r (z)) ≤ C µ r log 2 (Cµ) .
(2.2)

We are now looking for an upper bound and a lower bound for ρ(z) ρ(w) whenever w ∈ D r (z). Lemma 2.4. For every z ∈ C, we have

∀r > 0, ∀w ∈ D r (z), ρ(z) ρ(w) ≥ 1 1 + r . (2.3) and ∀r > 1, ∀w ∈ D r (z), ρ(z) ρ(w) max r log 2 (Cµ) κ -1 , 1 = r max log 2 (Cµ) κ -1, 0 . (2.4) Proof. A classical fact is that ρ is a 1-Lipschitz function (see [OP16, equation 2.4] for a simple proof): |ρ(z) -ρ(w)| ≤ |z -w|, ∀z, w ∈ C.
Hence, for every z ∈ C, we get inequality (2.3) by the triangular inequality. To obtain the upper bound, we use both inequalities of Lemma 2.1 distinguishing the cases rρ(z) ≥ ρ(w) and rρ(z) < ρ(w). We get for every w ∈ D r (z),

rρ(z) ρ(w) max µ(D r (z)) 1/κ , µ(D r (z)) 1/ log 2 (Cµ) .
Therefore inequality (2.2) implies for r > 1

rρ(z) ρ(w) max r log 2 (Cµ) κ , r ,
and hence (2.4) follows.

Notice that for C µ large enough, we have log 2 (Cµ) κ -1 ≥ 0, which means that the maximum in the last inequality (2.4) is equal to r log 2 (Cµ) κ

-1 . This holds exactly when

C µ ≥ 4 √ 2.
In the proof of our main theorem, we will need a particular covering of the complex plane. Let us explain how we can construct it. We say that a sequence (a n ) n∈N is ρ-separated if there exists δ > 0 such that

|a i -a j | ≥ δ max (ρ(a i ), ρ(a j )), ∀i = j.
This means that the disks D δ (a n ) are pairwise disjoint. We will cover C by disks satisfying a finite overlapping property and such that the sequence formed by their centers is ρseparated. In [START_REF] Marco | Interpolating and sampling sequences for entire functions[END_REF], the authors construct a decomposition of C into rectangles R k : C = k R k , and two such rectangles can intersect at most along sides. For these rectangles, so-called quasi-squares, there exists a constant e > 1 depending only on C µ such that the ratio of sides of every R k lies in the interval [1/e, e]. Denoting by a k the center of R k , Theorem 8(c) of [START_REF] Marco | Interpolating and sampling sequences for entire functions[END_REF] claims in particular that there is r 0 ≥ 1, such that for every k, ρ(a k )

r 0 ≤ diam R k ≤ r 0 ρ(a k ).
In other words

D 1/(2Cr 0 ) (a k ) ⊂ R k ⊂ D r 0 /2 (a k ) (2.5) with C = √ 1 + e 2 .
As a consequence we get the required covering C = D r 0 /2 (a k ) and the sequence (a k ) is ρ-separated (take δ = 1/(2Cr 0 )).

We end this section by focusing on the particular case of a doubling measure µ given by the Laplacian ∆φ of a subharmonic function φ which is the case of interest in this paper. We will need to control the value of φ in a disk by the value at its center. For this, we can use [MMOC03, Lemma 13] that we state as follows: for every σ > 0, there exists A = A(σ) > 0 such that for all k ∈ N,

sup z∈D σ (a k ) |φ(z) -φ(a k ) -h a k (z)| ≤ A(σ) (2.6)
where h a k is a harmonic function in D σ (a k ) with h a k (a k ) = 0. Moreover, in view of the proof of [MMOC03, Lemma 13] we have

A(σ) sup k∈N µ (D σ (a k )) σ log 2 (Cµ) (2.7)
where the inequalities come from [MMOC03, Lemma 5(a)] and the estimate (2.2). This last result will allow us to translate the subharmonic weight locally into a holomorphic function.

Remez-type inequalities

Let us introduce a central result of the paper [START_REF] Andrievskii | Remez-type inequalities in the complex plane[END_REF]. Let G be a (bounded) domain in C and let 0 < s < |G| (Lebesgue measure of G). Denoting Pol n the space of complex polynomials of degree at most n ∈ N, we introduce the set

P n (G, s) = {p ∈ Pol n : |{z ∈ G : |p(z)| ≤ 1}| ≥ s}. Next, let R n (z, s) = sup p∈Pn(G,s) |p(z)|.
This expression gives the biggest possible value at z of a polynomial p of degree at most n and being at most 1 on a set of measure at least s. In particular Theorem 1 from [START_REF] Andrievskii | Remez-type inequalities in the complex plane[END_REF] claims that for z ∈ ∂G, we have

R n (z, s) ≤ c s n (3.1)
where the constant c depends only on the (square of the) diameter of G. This result corresponds to a generalization to the two-dimensional case of the Remez inequality which is usually given in dimension 1. In what follows we will essentially consider G to be a disk or a rectangle. By the maximum modulus principle, the above constant gives an upper estimate on G for an arbitrary polynomial of degree at most n which is bounded by one on a set of measure at least s. Obviously, if this set is small (s close to 0), i.e. p is controlled by 1 on a small set, then the estimate has to get worse.

Remark 3.1. Let us make another observation. If c is the constant in (3.1) associated with the unit disk G = D = D(0, 1), then a simple argument based on homothety shows that the corresponding constant for an arbitrary disk D(0, r) is cr 2 (considering D(0, r) as underlying domain, the constant c appearing in [AR07, Theorem 1] satisfies c > 2m 2 (D(0, r))). So, in the sequel we will use the estimate

R n (z, s) ≤ cr 2 s n , (3.2)
where c does not depend on r.

Up to a translation, the following counterpart of Kovrijkine's result for the planar case has been given in [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF]: Lemma 3.2. Let 0 < r < R be fixed. There exists a constant η > 0 such that the following holds. Let w ∈ C, let E ⊂ D r (w) be a planar measurable set of positive measure and let

z 0 ∈ D r (w). For every φ analytic in D R (w), if |φ(z 0 )| ≥ 1 and M = sup z∈D R (w) |φ(z)| then sup z∈D r (w) |φ(z)| ≤ cr 2 ρ(w) 2 |E| η log M sup z∈E |φ(z)|,
where c does not depend on r, and

η ≤ c R 4 (R -r) 4 log R R -r
for an absolute constant c .

The corresponding case for p-norms is deduced exactly as in Kovrijkine's work.

Corollary 3.3. Let 0 < r < R be fixed. There exists a constant η > 0 such that the following holds. Let w ∈ C, let E ⊂ D r (w) be a planar measurable set of positive measure and let z 0 ∈ D r (w). For every

φ analytic in D R (w), if |φ(z 0 )| ≥ 1 and M = sup z∈D R (w) |φ(z)| then for p ∈ [1, +∞) we have φ L p (D r (w)) ≤ cr 2 ρ(w) 2 |E| η log M + 1 p φ L p (E) .
The estimates on η are the same as in the lemma. The constant c does not depend on r.

Proof of Theorem 1.1

The key ingredient in the proof of Theorem 1.1 and a main contribution of this paper is the existence and the estimate of the covering constant. Indeed, we have shown in Section 2 that the disks (D s (a k )) k∈N cover the complex plane C for s > r 0 /2. Now, we shall prove that there exists a covering constant N depending on the covering radius s ≥ r 0 /2. This means that the number of overlapping disks cannot exceed N . We denote by χ F the characteristic function of a measurable set F in D. We are now in a position to prove the covering lemma.

Lemma 4.1. For s > 1, there exists a constant N := N (s) such that

k χ D s (a k ) ≤ N.
Moreover, there exists some universal constant c ov := c ov (φ) > 0 such that

N ≤ c ov s α where α = min 2 max 1 + log 2 (C µ ) κ , 2 ; max log 2 (C µ ) κ , 1 log 2 (C µ ) ,
C µ is the doubling constant and κ is given in Lemma 2.1.

Obviously, the constant N is at least equal to 1 and since N is non-decreasing in s, we are only interested in the behavior when s goes to ∞. We also remind the reader that in order to cover the complex plane C, we need to require that s ≥ r 0 /2. Proof of Lemma 4.1. The proof is inspired by that of [AS11, Lemma 2.5]. Let s > 1 and z ∈ C. Denote Γ(z) = {k ∈ N, z ∈ D s (a k )}. Our goal is to estimate #Γ(z). Write θ = log 2 (Cµ) κ . Recall that for z ∈ D s (a k ), we have by (2.3) and (2.4)

ρ(z) 1 + s ≤ ρ(a k ) ρ(z) max(s θ-1 , 1). (4.1)
Now, pick δ > 0 such that the disks D δ (a n ) are disjoint (recall that (a n ) n∈N is ρ-separated).

By the triangular inequality, there exists a constant C > 0 such that D δ (a k ) ⊂ D C max(s θ ,s) (z) whenever z ∈ D s (a k ) (see Figure 2). Indeed, if w ∈ D δ (a k ) then

|w -z| ≤ |w -a k | + |z -a k | ≤ δρ(a k ) + sρ(a k ) (δ + s)ρ(z) max(s θ-1 , 1) ρ(z) max(s θ , s),
where we have used that δ > 0 is fixed and s > 1. Hence, since D δ (a k ) are disjoint and with (4.1) in mind, it follows

#Γ(z) ≤ # k ∈ N, D δ (a k ) ⊂ D C max(s θ ,s) (z) ≤ D C max(s θ ,s) (z) inf k∈Γ(z) |D δ (a k )| = sup k∈Γ(z) C max(s θ , s)ρ(z) 2 (δρ(a k )) 2 max(s θ , s)(1 + s) 2 s 2 max(1+θ, 2)
where we have used that max(s θ , s) = s max(θ,1) for s > 1. Moreover, notice that µ D δ (a k ) 1 since 0 < δ < 1 is fixed. Hence an analogous computation, replacing the Lebesgue measure by the measure µ and using the right hand side of inequality (2.2), leads to

#Γ(z) ≤ # k ∈ N, D δ (a k ) ⊂ D C max(s θ ,s) (z) ≤ µ D C max(s θ ,s) (z) inf k∈Γ(z) µ [D δ (a k )] max(s θ , s) log 2 (Cµ)
s max(θ, 1) log 2 (Cµ) .

Finally, taking the minimum between the two estimates, we get for s > 1 #Γ(z) min s 2 max(1+θ, 2) , s max(θ, 1) log 2 (Cµ) = s min[2 max(1+θ, 2), max(θ, 1) log 2 (Cµ)] .

Since the estimates are uniform in z, the lemma follows.

As in [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF] for the Bergman space, we now introduce good disks. Fix r 0 /2 ≤ s < t where r 0 /2 is the radius such that the disks D r 0 /2 (a k ) cover the complex plane. For K > 1 the set

I K-good f = {k : f L p φ (D t (a k )) ≤ K f L p φ (D s (a k
)) } will be called the set of K-good disks for (t, s) (in order to keep notation light we will not include s and t as indices). This set depends on f .

The following proposition has been shown in [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF] for the Bergman space, but its proof, implying essentially the finite overlapping property, is exactly the same. Proposition 4.2. Let r 0 /2 ≤ s < t. For every constant c ∈ (0, 1), there exists K such that for every f ∈ F p φ we have

k∈I K-good f f p L p φ (D s (a k )) ≥ c f p L p φ (C) .
One can pick K p = N (t)/(1 -c) where N (t) corresponds to the overlapping constant from Lemma 4.1 for the radius t.

We are now in a position to prove the theorem.

Proof of Theorem 1.1. Take s = max(r, r 0 /2) and t = 4s. As noted in [START_REF] Hartmann | Dominating sets in Bergman spaces and sampling constants[END_REF], if E is (γ, r)-dense then E is ( γ, s)-dense, where γ = cγ and c is a multiplicative constant. Hence we can assume that E is (γ, s)-dense instead of (γ, r)-dense.

Let h a k be the harmonic function introduced in (2.6) for σ = t. Since h a k is harmonic on D t (a k ), there exists a function H a k holomorphic on D t (a k ) with real part h a k . Now, given f with f L p φ (C) = 1, let g = f e -(Ha k +φ(a k )) for k ∈ I K-good f , and set

h = c 0 g, c 0 = πs 2 ρ(a k ) 2 D s (a k ) |g| p dA 1/p .
Clearly there is z 0 ∈ D s (a k ) with |h(z 0 )| ≥ 1. Set R = 2s. We have to estimate the maximum modulus of h on D R (a k ) in terms of a local integral of h. To that purpose, we use that h ∈ A p (D t (a k )). Indeed, applying (2.6) we have

D t (a k ) |h| p dA = πs 2 ρ(a k ) 2 D s (a k ) |g| p dA D t (a k ) |g| p dA = πs 2 ρ(a k ) 2 D s (a k ) |f | p e -p(ha k +φ(a k )) dA D t (a k ) |f | p e -p(ha k +φ(a k )) dA ≤ πs 2 ρ(a k ) 2 e 2pA(t) D s (a k ) |f | p e -pφ(z) dA(z) D t (a k ) |f | p e -pφ(z) dA(z). ≤ πs 2 ρ(a k ) 2 e 2pA(t) K p
where the last inequality comes from the fact that k is K-good for (s, t). Therefore, taking into account this last estimate, the subharmonicity of |h| p yields where C , C depend only on the space. In view of (2.7), we get log M s log 2 (Cµ) + 1 p (1 + log(s)).

M p := max z∈D R (a k ) |h(z)| p ≤ 1 πs 2 ρ(a k ) 2 D t (a k ) |h| p dA ≤ c s K p
Finally, noticing that r s = max(r, r 0 /2) for r > 1, this implies log M r log 2 (Cµ) + 1 p (1 + log(r)).

Finally, we give a short proof of the reverse implication in Corollary 1.3 which is a straightforward adaptation to the doubling Fock space of Luecking's proof of [Lue81, Corollary 3].

Proof. First, assume that ϕ ≤ 1. Therefore s ≤ 1. Since E is (γ, r)-dense, it is a dominating set. So

C ϕ 2 |f | 2 e -2φ dA ≥ s 2 E |f | 2 e -2φ dA ≥ s 2 C 2 f 2 L 2 φ (C) ,
where C is the sampling constant. Then

(I -T ϕ )f 2 L 2 φ (C) = T 1-ϕ f 2 L 2 φ (C) = P[(1 -ϕ)f ] 2 L 2 φ (C) ≤ (1 -ϕ)f 2 L 2 φ (C) ≤ C (1 -ϕ 2 )|f | 2 e -2φ dA ≤ (1 -s 2 C 2 ) f 2 L 2 φ (C)
Hence I -T ϕ < 1. So T ϕ is invertible and

T -1 ϕ ≤ 1 1 -I -T ϕ ≤ 1 1 - √ 1 -s 2 C 2 .
Using Theorem 1.1, we obtain

||T -1 ϕ || ≤ 1 1 -1 -s 2 γ c 2L .
Finally, for a general ϕ, let ψ = ϕ ϕ ∞ . Then ψ ≤ 1, E = {ψ ≥ s ϕ ∞ = s } and T ϕ = ϕ ∞ T ψ . So, applying the previous discussion to ψ, it follows

T -1 ϕ = T -1 ψ ϕ -1 ∞ ≤ ϕ -1 ∞ 1 -1 -(s ) 2 γ c 2L = ϕ -1 ∞ 1 -1 - s ϕ ∞ 2 γ c 2L
.

Figure 1 .

 1 Figure 1. The disks D j , D(z , r ) and D(z , r).

Figure 2 .

 2 Figure 2. The disks D s (a k ), D C max(s θ ,s) (z) and D δ (a k ).

  where c s = e 2pA(4s) since t = 4s. Now, setting E = E ∩ D s (a k ) we get using Corollary 3.3 applied to h:D s (a k ) |h(z)| p dA(z) ≤ cs 2 ρ(a k ) 2 | E| pη log M +1 E |h(z)| p dA(z)Again, by homogeneity we can replace in the above inequality h by g. Note also that πs 2 ρ(a k ) 2 /| E| is controlled by 1/γ. This yieldsD s (a k ) |f | p e -pφ(z) dA(z) ≤ e pA(t) D s (a k ) |g(z)| p dA(z) ≤ e pA(t) cs 2 ρ(a k ) 2 | E| pη log M +1 E |g(z)| p dA(z) ≤ e pA(t) c 1 γ pη log M +1 E |g(z)| p dA(z) ≤ e 2pA(t) c 1 γ pη log M +1 E |f (z)| p e -pφ(z) dA(z),where c 1 is an absolute constant. Summing over all K-good k, and using Lemma 4.1 and Proposition 4.2 we obtain the required resultc f L p φ (C) c 1 γ η log M +1/p f L p φ (E)where in view of Lemma 3.2 η ≤ c × 2 4 log 2 and log(M ) ≤ log(c 1/p s K)Here c comes from Proposition 4.2, and c ov and α from Lemma 4.1. In particular, fixing c ∈ (0, 1) we obtain log M ≤ 2A(4s) + 1 p (C + C log(s))

Acknowledgment. This work is part of the first named author's thesis supervised by Andreas Hartmann. The authors would like to thank him for many helpful discussions.

The research of the first author is partially supported by Banque Mondiale via the Projet d'Appui au Développement de l'Enseignement Supérieur du Mali. The research of the second author is partially supported by the project ANR-18-CE40-0035 and by the Joint French-Russian Research Project PRC CNRS/RFBR 2017-2019.

University of Segou, Mali E-mail address: gnatiosia@gmail.com Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence, France E-mail address: Marcu-Antone.Orsoni@math.u-bordeaux.fr


Guillaume Jacques

Elevational deixis in the Kiranti verb

Keywords: Kiranti, elevational deixis, motion verbs, Thulung, Khaling, Belhare, Bantawa, case marking

This article deals with elevational deixis in Kiranti languages, a feature which is pervasive in these languages in both the verbal and nominal domains. The system of elevation is described in most grammars of these languages as tripartite, following the typologically common system comprising up(wards), same level/across, down(wards) elevations.

This work reviews the available data on elevational deixis in the verbal system, and has two main contributions. First, it shows that motion verbs unspecified for elevation are an essential part of the elevation marking paradigm, and are obligatory in some specific elevational configurations. Second, it argues that on the one hand elevationally marked motion verbs are cognate as whole sets across Kiranti, and probably reconstructible to proto-Kiranti and beyond, and on the second hand that a subgroup of Kiranti including Chintang, Athpare, Belhare and Yamphu have innovated a second set of elevationally-marked motion verbs.

Introduction

Elevational (or topographical) deixis has been described as a pervasive typological feature in Trans-Himalayan/Sino-Tibetan [START_REF] Post | Topographical deixis in Trans-Himalayan (Sino-Tibetan) languages[END_REF][START_REF] Post | The distribution, reconstruction and varied fates of topographical deixis in Trans-Himalayan (Sino-Tibetan): Implications for the reconstruction of an early Trans-Himalayan environment[END_REF], including Lolo-Burmese (Bradley 1979: 354-356;[START_REF] Bradley | Deictic patterns in Lisu and southeastern Tibeto-Burman[END_REF], Rgyalrongic [START_REF] Zhang | Le dialecte de Brag-bar du rgyalrong situ et sa contribution à la typologie de l'expression des relations spatiales: le mouvement associé et l'orientation[END_REF], Bodish (Hyslop 2017: 161) and Tani [START_REF] Post | The distribution, reconstruction and varied fates of topographical deixis in Trans-Himalayan (Sino-Tibetan): Implications for the reconstruction of an early Trans-Himalayan environment[END_REF].

It is particularly conspicuous in Kiranti languages, where a tripartite (upwards, same level, downwards) elevational contrast [START_REF] Bickel | Deictic transposition and referential practice in Belhare[END_REF][START_REF] Jacques | The auditory demonstrative in Khaling[END_REF][START_REF] Michailovsky | Kiranti languages[END_REF]) is coded by locational/directional adverbs, demonstratives and case markers as well by motion verbs. 1 The

Elevation distinctions

Kiranti languages distinguish between three marked elevations: high/upward, low/downward, and same level/across.

Examples (1) and (2), from Khaling and Thulung respectively, illustrate the coding of upward elevation, using a combination of locational adverbs, case markers and motion verbs. 'Because of this, I brought my children up here.' (Soamaya's life) cord to which 'All languages with level or across elevationals also have down and up elevationals."

2 Some of the texts cited in this article are included in the Pangloss collection [START_REF] Michailovsky | Documenting and researching endangered languages: The Pangloss collection[END_REF].

The Bantawa example (3) illustrates the coding of downward elevation, through both a case marker and a downward motion verb.

(3) Bantawa hyu-cok-yu down-floor-loc.up dʰa-Ø-kʰa-Ø go.down-n.pst-see-n.pst 'Please, come down to the lower floor! ' Doornenbal 2009: 85) An example from Chintang (Paudyal 2015: 279) illustrates the expression of same level elevation through a demonstrative and a motion verb. As the name implies, same level elevation refers to the elevation at a location or motion at the same level as the speaker or deictic center; in Kiranti languages, it has been referred to with terms such as 'across', 'horizontal' or 'same level'. Same level elevation is, significantly, distinct from neutral or unspecified elevation, which is coded using distinct verbs, adverbs and case markers.

(4) Chintang

Type of elevation coding

There are two main interpretations for the elevation coded in Kiranti languages.

First, and by far most frequently, it can refer to a geophysical (uphilldownhill) contrast, a common interpretation in mountainous areas such as the Himalayas.

Examples frequently include geographical or landscape features, in relation to which and along which elevation-coded motion occurs. This can be seen in ( 5) and ( 6).

(5) Thulung (Rai 2016: 303) In the following Khaling example, elevation is coded through upward and downward motion verbs, as well as exploiting local knowledge about the elevations of the ʣhēːs (highland area) and the wʌjʌ (lowland area) relative to the Khaling homeland, which serves as the deictic center. 'The highland insect comes down from the highlands saying "rachachacha". The lowland insect goes up from the lowlands saying "nyeyeyeyeye".' (Solme Lamalit,(128)(129)(130) Second, the up vs. down elevational contrast can refer to a vertical axis. This interpretation is found in a number of different situations, such as descriptions of birds flying (8), meteorological events, such as rain falling (9) or the position of stars (10), and events or descriptions involving body posture (11). Both geophysical and vertical elevation are coded using the same morphology in most Kiranti languages. The only documented exception is Mewakhola Limbu [START_REF] Michailovsky | On Limbu directionals and locative expressions[END_REF] which has contrasting elevationcoding adverbs opposing vertical vs. geophysical (up vs. uphill/upstream3 ) for upper and lower elevations; note however that for elevation-coding postpositions, the contrast in types of elevation is neutralized for lower elevation, for which there is a single marker covering both geophysical and vertical interpretations.

Domains of elevation coding

One interesting feature of elevation coding in Kiranti languages is the possibility of marking elevational location/direction in multiple grammatical domains simultaneously, something that can be thought of as elevational concord. All languages have demonstratives, locational/directional adverbs, and motion verbs, and some additionally have locational/directional case markers. Among the languages that do not have locational/directional case markers are Hayu, Athpare, Yakkha, Wambule and Bahing, for which the locational/directional adverbs and/or demonstratives are often cognate with the locational/case markers in the languages which have them. Post (2020: 381) proposes reconstructed forms for both the adverbs and the elevationcoding case markers: an upward form in *tV, a downward form in *mV or *yV, and a same-level form in *yV or *nV (all of these are reconstructed with a back-rounded vowel), with far more variability to be found among same-level formatives than among upward and downward .

The examples presented thus far show the phenomenon of elevation concord, as do additional examples (12,13,14). 'That [child] did not come down to the house.' One of the logical consequences of being able to encode elevation in multiple domains is the possibility of discordant marking, namely situations in which elevation is not consistently marked across the same utterance. This will be discussed in section 4.

Motion verbs

The focus of this article is the coding of elevational deixis through sets of lexical verbs, and this section presents the motion verbs used to code elevation. As with other grammatical domains, motion verbs code three distinct elevations-high/upward, same level, down/downward are found. In addition to these are motion verbs unspecified for elevation. These are considered to make up the fourth element of the paradigm, for reasons that will be explained below.

The contrast in elevation coding can be seen in ( 15) for Belhare, where distinct motion verbs are used to express upward, downward and same level motion.

(15) Belhare a. come.across-ipfv:npst 'From there, the ox is coming over to the tree that has been planted.' (Bickel 1996: 126)

Venitive and itive verbs and elevation

We distinguish between venitive motion verbs, describing motion towards the speaker or towards the deictic center (which we gloss 'come' and which are seen in the Belhare examples above), and itive verbs, describing motion away from the speaker or from the deictic center (which we gloss 'go'). 4One characteristic of Kiranti languages is that while they all have elevationcoding venitive verbs, they rarely have elevation-coding itive verbs. Belhare is one of two languages (to the best of our knowledge) with complete sets of both, as can be seen in the elevation-coding itive verbs in the examples below ( 16).

(16) Belhare a. 3.n.sg-go.across-telic-pst 'Hold on! These two guys are going only now, their friends went much earlier!' (Bickel 1996: 137) Another language with elevation-coding itive verbs in addition to venitives is Bantawa. The itive verbs of both Bantawa and Belhare are presented together in Table 1. In addition to Bantawa and Belhare, a few other languages show signs of having lexicalized elevation-coding itive verbs. Athpare has the same contrast but only for 'upwards' elevation: Ebert (1997b) has both kat-'come from below' and thaŋ-'climb up, go up', like Belhare. A cognate of the latter verb is also found in Yakkha (thaŋ-'climb', Schackow 2015: 534). In the available data on Athpare, there is, however, no evidence of 'same level' and 'downwards' coding on itive verbs, and the Athpare data is best seen as evidence of a possible source for one member of the Belhare set coding elevation and motion away-in the form of a verb coding elevational change but not motional deixis.

Situations like the ones in Athpare and Yakkha, with single members of sets for motion away verbs, raise the issue of the interpretative difficulties with using grammatical descriptions: how should one interpret glosses such as 'ascend', 'climb'? In one sense these are indeed verbs that combine features of motion and elevation, and are as such comparable to other material considered here; on the other hand, the gloss may be an approximation for an action that includes motion and elevational features but is not limited to these, also encoding other aspects of the action such as manner.

Wambule does not have a system of distinct verbal roots to express itive motion, but has compound itive verbs ga-lwa 'go up' and pi-lwa-'go (same level)' built by compounding the corresponding venitive verbs (Table 2) with the elevationally-unspecified lwa-'go' (Opgenort 2004: 794). The elevationally-marked motion verb roots gaand pimay thus be interpreted as venitive by default, but are neutral as to motion deixis in this language. The meaning 'go down' in Wambule is expressed using the roots doand dwak-, which are however also translated as 'fall' and 'come down' and may not be dedicated itive motion verbs.

The most frequent elevation-coding motion verbs in Kiranti are therefore venitive verbs. Table 2 shows the verb roots corresponding to these meanings in most of the Kiranti languages.5

? Thulung get- jok- bik- rok- Wambule ga- ywa- pi- blak- Khaling kʰoŋ- je- pi- ɦo- Dumi kʰoŋ- yi- pi- ho- Koyi kʰo- gʰuʔ- bhiʔ- hu- Kulung tʰoŋ- yu- ban- ta- Camling saŋ- i- ban- ? Puma tʰoŋ- i- ben- ta- Bantawa tʰaŋ- yɨ- ban- ta- Chintang kat- kuŋs- tʰap- ti- Athpare kat- uŋs- ap- ta- Belhare kat- uŋs- ap- ta- Yakkha keʔ- uks- ap- ta- Yamphu kat- uks- ap- ? Limbu tʰaŋ- yu- phɛn- ta-
It is important to highlight that it is the verb roots themselves that express the combination of elevational and motional deixis. 6 The verbs in each language form a suppletive paradigm, in that the verbs have different lexical roots. Three factors lead us to consider that the sets of verbs in Table 2 constitute paradigms. First, their use in obligatory when conditions are met, the required conditions being motion towards the speaker/deictic center (as well as away from it, for Bantawa and Belhare) combined with the specification of elevation which is either upwards, horizontal, or downwards. The elevationunspecified member of the sets is not, in fact, a basic motion verb (it is unspecified, not underspecified, for elevation), as it cannot be used as a substitute for one of the elevation-coding motion verbs: it can be considered to be part of the paradigm, being selected only in specific conditions (when Driem 1993; Michailovsky 2012), Koyi [START_REF] Lahaussois | Koyi Rai: An initial grammatical sketch[END_REF], Kulung [START_REF] Tolsma | A grammar of Kulung[END_REF], Camling (Ebert 1997a), Chintang [START_REF] Paudyal | Aspects of Chintang syntax[END_REF], Athpare (Ebert 1997b), Belhare [START_REF] Bickel | Aspect, mood, and time in Belhare: studies in the semantics-pragmatics interface of a Himalayan language[END_REF][START_REF] Bickel | Spatial operations in deixis, cognition, and culture: where to orient oneself in Belhare[END_REF], Bantawa [START_REF] Doornenbal | A grammar of Bantawa[END_REF], Yakkha [START_REF] Schackow | A grammar of Yakkha[END_REF], Yamphu [START_REF] Rutgers | Yamphu, grammar, texts and lexicon[END_REF], Limbu [START_REF] Michailovsky | Limbu-English dictionary[END_REF].

6 Note, however, that associated motion, which also encodes elevation in some Kiranti languages, is expressed through a series of derivational suffixes [START_REF] Jacques | Associated motion in Sino-Tibetan/Trans-Himalayan[END_REF].

elevation change is unknown, or in the conditions described in section 4.2).

In other words, while the elevation-coding motion verbs each code specific elevation and motion deixis, the elevation-unspecified motion verb covers the rest of the semantic space for motion without elevation change (upwards, downwards) or specification (across) (except for situations with neutralization, as described in section 4.2), without overlapping on the elevation coding motion verbs. This property of elevation-unspecified verbs is not reported in typological surveys on elevational deixis7 (Forker 2020; Post 2020), and may in fact not be relevant beyond the Kiranti languages.

Second, these verbs form a closed class: the languages in our sample only have a single verbal root that meets each of the conditions described (motion + change in elevation). There is however some diachronic evidence about sources for some of these verbs ( §6), some of the sources being verbs which code elevation but not motion (verbs such as 'ascend', 'climb'.)

Third, the verbs in these sets have a tendency to grammaticalize into verbal derivational affixes (often called V2's in South Asia).

The specific contexts where the elevation-unspecified motion verb is obligatory are those where the elevation is unknown. This will typically occur in questions. This can be seen in ( 17) from Thulung, with the elevationunspecified motion verb romu used to ask where the children are coming from, on account of the elevational path being unknown to the speaker. This contrasts with the verb which follows, namely the same level motion verb bimu, which is used, in an imperative form, because the elevational aspect of the path between their current position and the speaker is clear.

? Thulung kʰet- sɵt- pʰit- ret- Khaling kʰoŋt- jet- pit- ɦot- Dumi kʰoŋt- yit- pit- hot- Koyi kʰo- gʰuʔ- bhiʔ- hu- Kulung ? yut- ban- tat- Camling said- it- baid- ? Bantawa tʰakt- yɨt- batt- tat- Chintang katt- kukt- tʰapt- tat- Athpare katt- ukt- apt- tat- Belhare katt- ukt- apt- ta- Yakkha ket- ukt- apt- taʔ- Limbu tʰakt- yuːt- phɛtt- taːt-

Caused accompanied motion verbs

In addition to motion verbs, which are intransitive, Kiranti languages also have cognate verbs which express caused accompanied motion: these are transitive verbs (glossed 'bring' and 'take' respectively for motion towards and away from the speaker/deictic center), and they convey the fact that both subject and object are affected by the motion event. The existence of intransitive and transitive motion verb sets is due to the fact that the caused accompanied motion verbs are derived from the corresponding motion verbs by suffixation of the applicative/causative -t suffix [START_REF] Michailovsky | Tibeto-Burman dental suffixes: Evidence from Limbu (Nepal)[END_REF][START_REF] Jacques | Derivational morphology in Khaling[END_REF]. In rarer cases, there is an aspiration alternation between the initial consonant of the motion verb and that of the caused association motion verbs (for instance Thulung get-'come up' vs. kʰet-'bring up'), or suppletion (Thulung jok-'come down' vs. sɵt-'bring down').

Both the motion verbs and caused accompanied motion verbs can, depending on the language, be the locus for the expression of different elevational contrasts. Table 3 lists venitive caused accompanied motion verbs, as well as the elevation-unspecified counterparts, across the various languages.

For languages without an elevation-coding set of itive verbs, there are also no itive caused accompanied motion verbs. Bantawa, however, has a nearly complete set of venitive and itive verbs, shown in Table 4.

A few Kiranti languages (in particular Khaling and Belhare) have gram- maticalized associated motion markers from motion and caused accompanied motion verbs [START_REF] Jacques | Associated motion in Sino-Tibetan/Trans-Himalayan[END_REF], and the elevation contrast is thus also reflected in complex predicates taking these markers.

Since the grammatical encoding of elevation is completely parallel between motion verbs, caused accompanied motion verbs and associated motion markers, the remainder of the paper will focus on the former.

Mismatches in the encoding of elevational deixis

All the examples provided above show elevational concord across the various grammatical domains where elevation is expressed in a given utterance.

It happens, however, that one comes across examples in narratives and conversations that appear to contain a mismatch in elevational deixis in the materials: these examples feature discordances in the elevation coded across the locational/directional case markers, demonstratives, adverbs, and motion verbs, combining an elevation-coding formative with an elevationunspecified formative.

Three distinct situations involving mismatches are detailed in the following sections.

Elevationally neutral locative with elevationally marked motion verbs

One type of apparent mismatch involves the neutral locative marker paired with an elevationally marked verb, as in examples (18,19,20). 'We came up here.' (Rutgers 1998: 416, S18) However, unlike elevationally unspecified motion verbs, which are an essential part of the elevational paradigm in Kiranti languages (see section 4.2), the elevationally neutral locational/ directional case marker is not part of the same paradigm as its elevation-coding counterparts. We make this claim on account of the neutral locative marker often combining with elevation-coding case-markers and demonstratives to build compound elevation-coding formatives. This is seen in examples ( 21) and ( 22). fall-res-pst-evd '[The snake] was a bit further up' (Rutgers 1998: 82) For this reason, we consider that the neutral locative case marker is actually not unspecified but underspecified for elevation, hence the use of the term 'neutral'.

Therefore, the combination of a neutral case marker with elevationcoding verbs do not represent genuine mismatches in elevation-coding.

Neutralization of deixis

When a scenario involves displacement between elevations, and the deictic center is distinct from the endpoint of the motion, two possible situations can result:

• When the deictic center is not at the same elevation as the endpoint of motion, elevational deixis is neutralized, and only motional deixis is expressed: a motion verb unspecified for elevation is used.

• When the deictic center is at the same elevation but distant from both source and goal, the motional deixis is neutralized, and only elevational deixis is taken into account: an elevation coding motion verb can be used -even when the motion is deictically the opposite of that expressed by the verb. This most likely results from most of the languages not having itive verbs that encode elevation.

In both cases, mismatches in elevational deixis can result between verbs and other domains, as exemplified below.

Neutralization of elevational deixis

Examples ( 23), ( 24) and ( 25) illustrate uses of elevationally-unspecified motion verbs in contexts where the starting point of a path is lower or higher than the deictic center, and thus where an elevationally marked motion verb could have been expected instead. In ( 23), the deictic center (the protagonist referred to as "mother") is in a tree, while the children move on the ground. The reason for using the elevationally unspecified verb is not simply due to the fact that the mother can only perceive them by hearing and is not even sure of their identity (or their number), but rather because the children are moving towards to deictic center but remain on low elevation, thus not being closer to the elevational center. It is impossible here to use either kʰoŋ-'come up' (since they are not coming up the tree) or pi-'come (same level)' since they are not moving in the air at the same height as the tree. In ( 24), the deictic center is a house located at a lower elevation than the fields. The use of the unspecified venitive here is due to two factors. First, the direction from which the boar will come is not known in advance. Second, as in the case of ( 23), the boar indeed gets closer to the deictic center from the point of view of absolute distance, but its endpoint is not on the same elevational level. Using the 'upwards' venitive |kʰoŋ| 'come up' here would imply that the boar was coming from a lower elevation to a place located at the same elevation as the house.

Example (25) from Thulung illustrates a similar situation but with three levels of elevation. The use of different elevation coding verbs paints a picture allowing the audience to situate the actors in their topography: A Tibetan man brings his daughter down to marry the main character, who is staying in an encampment situated at a higher location than his village (but lower than the Tibetan's). The main character then goes back to his village without his bride. The 'downwards' venitive verb 'bring down' is predictably used for the second motion, but for the first one, the unspecified venitive verb reʈis used instead. The reason for this is that the deictic center is in the main character's village, and that the first motion event is between two locations that are both elevationally higher than -and never reaching the elevation of -the deictic center.

Neutralization of motion deixis

Neutralization of motion deixis is considerably rarer than the previous case, and the only examples we have found are in elicited material from Bantawa taken from Doornenbal (2009: 109).

In the scenarios represented in Figures 1 and2, there are three characters, Prem, Syam and Ram. In Figure 1, 8 Syam adresses Prem (who is located lower) to tell him to move towards his own position (the red arrow), and uses the upwards venitive verb tʰaŋ-(26a). To tell him to go to Ram (the green arrow), he has the choice between the elevationally unspecified itive (26c), along with a distally marked elevation coding adverb, or, more surprisingly, the upwards venitive verb (26b), accompanied by a nominal phrase indicating that the direction is upwards toward Ram. What is surprising about (26b) is that even though the motion involves an upward direction, as expected for the verb |thaŋ|, this is a venitive verb, and yet the motion takes Prem away from, rather than towards, the deictic center (which is the speaker, namely Syam). Prem only comes closer to the deictic center from the point of view of elevational deixis, since he ends up at the same elevation as Syam.

The converse situation applies, of course, to venitive verbs with downward elevation, as seen in ( 27a) and ( 27b). Because Bantawa has elevation-coding itive verbs as well, the same phenomenon is found with those verbs. In a second scenario (Figure 2), Syam tells Ram to go to Prem, who is located higher that either of them. Here, the upwards itive verb lontmust be used (28). This is because the itive verb is used in situations where both elevation and motion are excentric, i.e. away from the deictic center.

(28) Bantawa mu-dʰutni that-upwards lont-a! go.up-pst 'Go up over there!' (go up, towards that direction, upwards)

The above discussion can be summarized in Table 5. The elevationally marked venitive verbs can be used in two cases: (i) when the subject moves closer to the deictic center of the motion (examples 26a and 27a)) or (ii) when it moves to the same elevation as the deictic center, without however moving closer to it (examples 26b and 27b). In case (ii), the elevationally unspecified itive verb kʰatcan be used instead (examples 26c and 27c), but not the elevationally marked ones. With this distinction, the data in Table 5 can be accounted for with the following rules:

1. Elevationally marked venitive verbs can be used both for motion towards the motional deictic center or towards the elevational deictic center.

2. Motion towards the elevational deictic center, but away from (or not closer to) the motional deitic center can also be expressed with the unspecified itive verb.

3. The elevationally marked itive verbs are only used for motion away from both deictic centers.

In languages where itive verbs lack the elevational contrast, the distinction between elevational and motional deictic centers is still relevant.

In particular, the elevationally unspecified venitive verbs are the only option when the moving entity gets closer to the motional deictic center in terms of absolute distance, but remains at a different elevation (higher or lower), and thus remains equally distant from the elevational deictic center.

This neutralization of either the motional or elevational component in situations such as those described above is one of the factors that can explain what may otherwise appear to be discordant uses of deixis across different word classes within a sentence. The choices in fact come about in scenarios involving more than two elevations or motion not involving the deictic center.

Omission of allative and ablative marking

Another cause for discordant elevational deixis is the fact that allative (for languages which have such a case marker) and ablative case markers are frequently omitted and replaced by elevation coding case markers. Because there is no way to identify the source and goal, in the absence of allative and ablative case markers, there can appear to be discordant, or non-agreeing, elevation marking across a sentence.

In (29) and in (30), Thulung uses an elevation-coding locational/directional case marker ('down(wards)' here), rather than an ablative, to indicate the source; when combined with the upward venitive verb, the use of the case marker appears discordant, until it is clearly identified as the source rather than the goal. 'They brought water from down in the river and drank it. ' [Thulung origins,215] In these two examples, it is the elevation-coding markers which make it possible, together with the motion verbs, to parse the sentence correctly and identify key elements of the path such as source and goal.

Fluidity of the deictic center

In section 4.2.2 above, we saw that neutralization of elevational and motion deixis in specific contexts can cause confusion between itive and venitive motion. A similar confusion can arise through another distinct phenomenon, namely shifts in motion deixis within a narrative.

In (31) for instance, the main character leaves his father (who has been transformed into a stone) and goes back to his village; he predictably uses the venitive verb kʰɵŋ-ʌ-t-ʌ 'I came up' when talking to his co-villagers, and invites them to return to the place where he left his father using the itive kʰoɔç-ki 'let us go'. However, when describing their trip back to this place (of lower altitude), the venitive verb jāː-t-nu (literally 'they came down') occurs instead of the itive, apparently to highlight the fact that the main character is among the persons going to that place.

(31) Khaling In the following Thulung passage (32a) and (32b), made up of material which is separated by two sentences in a narrative, the deictic center shifts from the Thulung homeland of Mukli -as evidenced by the adverbs ojʉ, ola (down(wards) and up(wards) respectively), the itive caused accompanied motion verb lʌ:ɖʉ and the up(wards) locative case marker -la in (32a) -to the main protagonist, as seen by the use of locational/directional case markers -ra (neutral) and -jʉ (downwards) combined with -go ('inside'), showing that they are from the perspective of the cave; and then back again, with the use of the downward venitive verb jomu. This fluidity in the deictic center, and its ability to shift even within a single sentence, leaves speakers with a great amount of flexibility in the description of the physical environment in which narratives occur.

"mɛbenʌ then ʔuŋ-ʌ 1sg-erg ni top gʰrok-tû-ŋ-t-ʌ-nʌ leave-put-1sg→3-pst-1sg-lnk kʰɵŋ-ʌ-t-ʌ-m come.up-1sg-pst-1sg-nmlz

Diachronic evolution

All Kiranti languages have elevational motion verbs, and some of these verb forms are cognate across the subgroup.

In this section, we show that nearly all elevational motion verbs in Kiranti fit into two distinct tripartite cognate sets, and propose several historical hypotheses to account for the observed data.

Two sets of elevation deixis verbs

Among Kiranti languages, only Bantawa and Belhare are described as having a tripartite elevation constrast for both venitive and itive verbs (section 3.1).

The system of Belhare (Table 6) is particularly interesting, because both its itive and venitive series have cognates occurring as a set in other languages (at least in East Kiranti): the Belhare itive set is cognate with the Limbu venitive set, while the Belhare venitive set is cognate with its Chintang, Yakkha and Yamphu counterparts. By contrast, the itive system of Bantawa is completely distinct from those found in its closest relatives, Camling and Puma.9

Since there is some discrepancy between languages as to the motion deixis of these verb roots, the unspecified terms 'set A' and 'set B' are used in the following.

Set A

Set A occurs as a complete set in Belhare and Limbu, and in Bantawa it is split between the venitive and the itive verbs. Languages west of Bantawa have set A verbs, but lack the 'upwards' verbs.

All three verb roots have cognates elsewhere in Trans-Himalayan/Sino-Tibetan, in particular in Rgyalrongic.

The 'upwards' motion verb, reconstructed *tʰaŋ in Jacques (2017),10 is clearly related to the locative adverb thaŋ 'up, overhead' in Limbu (Michailovsky 2015: 115). Outside of Kiranti, we find cognates of this verb in the whole Trans-Himalayan/Sino-Tibetan family, but with a final stop: Chinese has the cognate verb 陟 ʈik 'ascent' (Old Chinese *trək, Baxter and Sagart 2014; on the vowel correspondence see [START_REF] Gong | The system of finals in proto-Sino-Tibetan[END_REF], and in Gyalrongic there are cognate nouns or adverbs, such as Japhug taʁ 'up' (from *taq).

The 'downwards' motion verb had the form *ju proto-Kiranti but its rhyme presents irregular correspondences in some languages such as Khaling and Dumi due to the fact that it underwent fusion with person indexation suffix, as is the case in Limbu, where its conjugation is irregular (Jacques 2018,2020). These verbs are mainly used with itive meaning, but also occur in contexts incompatible with the itive, and are better analyzed as having neutral deixis. There are several similar-looking 'same level' venitive verb roots in set A. The first one is reflected by Hayu pʰi-, Thulung bik-(with an irregular -k, possibly analogical, see Lahaussois 2011), Khaling and Dumi pi-, and can be reconstructed as *piin proto-Kiranti. The second one corresponds to Belhare pʰen-'go across' and Limbu pʰɛn-'come across' (possible proto-Kiranti reconstructions for this etymon would be *pʰɛn or *pɛn).

The West and East Kiranti forms, although they do not derive from exactly the same proto-form, are very probably related. Although alternations between *ɛ and *i and the addition of -n in Limbu are not regular morphological processes, there are other examples of this correspondence between Limbu and Khaling. In particular note the Khaling verb pʰi-'be spoiled (of rice)' and Limbu pʰɛn-'be spoiled' (Jacques 2017: 203). While the expla-nation for these alternations still eludes us, it is not an obstacle to positing cognacy between these roots. 11

Furthermore, Jacques (2017: 203) analyzes the Bantawa itive verb bitt-'go (same level)' as the cognate of *pi-(Bantawa bregularly goes back to proto-Kiranti *p-). The origin of the coda -ttis completely clear, but it is not an obstacle against positing an etymological relationship between these forms. 12 Outside of Kiranti, cognates of this verb root are found in Rgyalrongic (Cogtse Situ pi, stem II of the verb 'come', Lin 2003).

Set B

The set B verbs are restricted to a subset of Eastern Kiranti languages (Table 8), excluding Limbu. The 'upwards' verb can be reconstructed as *gaton the basis of Eastern languages. 13 It is perhaps tempting to compare Thulung get-'come up' and Wambule ga-'come up' to this set, but as far as we know there are no good examples of Thulung -et corresponding to East Kiranti -at, and Wambule normally 11 Another root form reflected by banin Kulung, Camling and Bantawa 'come across' goes back to *panfollowing [START_REF] Michailovsky | Manner vs place of articulation in the Kiranti initial stops[END_REF] laws. It is unrelated to the previous forms, as although the initial consonant is the same, there are no other examples of such a correspondence. These verbs are instead to be compared with the non-independent Khaling verb root -pɛ-(from *pa) which appears in the bipartite verb ɦɵ-n-pɛ-nɛ 'reach'.

12 An interesting gap in Bantawa is the absence of a corresponding itive caused accompanied motion verb (Doornenbal 2009: 125), which would be the equivalent of Khaling pit-'bring across', and whose expected form in Bantawa would be †bit-. In view of this gap, it is possible that Bantawa bittactually reflects a double derivation from the same root: first derived into a caused accompanied motion verb by the -t applicative suffix, and then back to an intransitive form by middle derivation.

13 Yakkha -eʔregularly corresponds to proto-Kiranti *-at, as shown by kʰeʔ-'go' from *kʰat-.

preserves a trace of the final *-t. It is more straightforward to compare Thulung get-'come up' to Limbu kɛt-'to arrive, to reach (a destination), to occur' (33). The 'same level' verb has a tʰinitial in Chintang corresponding to a zero initial in the other languages. As pointed out by [START_REF] Michailovsky | Preliminaries to the comparative study of the Kiranti subgroup of Tibeto-Burman[END_REF], proto-Kiranti *t debuccalizes in Yamphu and Belhare, and this is true of Yakkha too (for instance *tuŋ-'drink' yields Belhare and Yakkha uŋand Yamphu uks-). Thus, a proto-form *tapcould account for the forms of all languages except Athpare, where *tdoes not debuccalize regularly (cf Athpare tʰuŋs-'drink'). This suggests that Athpare ap-'come across' is not a native word, but a borrowing, perhaps from Belhare.

The 'come down' verb has kin Chintang corresponding to zero initial in other languages, but there is no evidence that velars debuccalize in Yamphu and Belhare, so that this set is unexplained.

Given the absence of clear cognates of these verbs outside of Eastern Kiranti, it is likely that set B constitutes an innovation for this subgroup.

Historical shifts

The set A paradigm is most likely reconstructible to proto-Kiranti and is possibly older than Kiranti, judging from cognates with the same meaning in Thangmi and Gyalrongic.

Two main hypotheses can be proposed for the function of the set A verbs in proto-Kiranti.

First, one can suppose that the verbs in set A were venitive, as they are in most Kiranti languages. In this hypothesis, the Chintang-Yamphu group has undergone a shift whereby former venitive motion verbs became itive, through elevational contexts where motional deixis is neutralized (Table 5), or as a result of the fluidity that can characterize motion deixis (section 5). Set B verbs were innovated and became a new set of venitive verbs.

Second, it is also possible that set A verbs had an elevational contrast but were neutral with respect to motion deixis, as seems to be the case in Wambule (section 3.1). Limbu on the one hand and Western languages on the other could have independently innovated by interpreting set A verbs as venitive, whereas they became itive in the Chintang-Yamphu group.

In any case, in both hypotheses, set A is reconstructible to proto-Kiranti, while set B is an innovation. Moreover, while motional deixis may vary across languages, elevational deixis is remarkably stable.

'

  Come over here! (Your mother told me to ask you.)'

  who stayed above the footpath'

  kite) came down (from heaven) to earth and gave his grandmother (the comb).' (Origin, 66) (9) Khaling wɵ rain je come.down:n.pst:3sg 'It is raining.' An upward location along a vertical axis is seen in (10), referring to the position of a star in the sky. (10) Khaling pɛri-bʉ-tʉ heaven-loc-loc.up tsʌī top sukra Polar.Star mōː-tɛ be-pst:3sg 'Up in the sky was the Polar Star' (Ur-Mother 44) In (11) the up(ward) case marker is used to express a vertical posture with respect to the ground. -leg-loc.up lamdi-pa walk-act.pcp dʉs-ta-m-ka become-pst.3sg-nmlz-erg 'Because she had become able to walk up on her legs, ....' [about a child learning to walk].

  basket.shelf-loc.up ʔʌm-si make.sleep-refl.n.pst.3sg ʔei intj lʉː-tɛ-su tell-pst:3→3-du 'In that case you will sleep up there on the shelf, they told him.

  (17) Thulung ba-tsi stay-imp.2du a-tsʉsʉ-tsip 1sg.poss-grandchild-du ba:-laŋka where-abl re foc rok-tsi come-pst.2du bik-tsi come.across-imp.2du bik-tsi come.across-imp.2du rwak-ta say-pst.3sg retsʌ it.seems ʔe hs 'She said "Stay, stay, Grandchildren, where did you come from? Come, come."' (Vulture, 79)

'

  When they come down to the husker and the handmill, nobody must leave any flour or rice!'(Ebert 2000come.up.s2-1sg.so-pa-also sa:ŋwa buffalo kɔtt-aŋ keep.s2-1sg.so.pa pit cattle kɔtt-aŋ-aŋ keep.s2-1sg.so.pa-and way-aŋ be.s2-1sg.so.pa 'I became an orphan, so I came up to my maternal uncle's house and stayed looking after the buffalo and the cattle.' (https://doi.org/ 10.24397/pangloss-0004194,

  above-loc-loc.up sip-pet:-tt-ae:

  noise [made by] children, their du mother knew that [her] children had come here.'(Solme Lamalit, 71)

'

  The boar will come and eat a lot of lentils in the field high up.'(Ogress 156)

  reʈ-ɖʉ-m-ka bring.unspec-pst.(3→3)sg-nmlz-erg me-sɵɖ-ʉ-ja neg-bring.down-(3→3)sg-irr retsʌ seem.to ʔe hs 'Because he [Tibetan] brought him [main protagonist] a Tibetan girl, he did not bring her down to his home [lower down, in the ancestral village].'

8

  We thank Lai Yunfan for the artwork.

Figure 1 :

 1 Figure 1: Motion along a slope in Bantawa (first scenario)

  'Go downwards to where Syam is!'

Figure 2 :

 2 Figure 2: Motion along a slope in Bantawa (second scenario)

  bring.water-ant.cvb pe-m-thal-miri eat-3pl-hab-pst.3pl→3

  lʉː-tɛ-nu-nʌ say-pst.3→3-pl-lnk kʰɵleŋʌ all ʔu-ŋoɔpsu-ɦɛm-kolo 3sg.poss-clan.members-pl-comit jāː-t-nu-nʌ come.down-pst.3-pl-lnk sên-tɛ-nu see-pst.3→3-pl"Then I left him there and came back. So, clan members, let us go look for him" he said to them, and they all came down and looked.'Khamnime, 36)

  me-go-la-ŋa dist.dem-inside-loc.up-int ʣʉl-thal-lʉ place-hab-pst.(3→3)sg ʔe hs me-la dist.dem-loc.up burkhum-go-la cave-inside-loc.up thʌk-ʣɵl-thal-lʉ hide-antic-hab-pst.(3→3)sg ʔe hs pʉ:-thal-lʉ eat-hab-pst.(3→3)sg ʔe hs bai-thal-la stay-hab-pst.(3→3)sg ʔe hs 'He used to take food from down here, from home, and used to place it up there, he used to hide it up inside a cave up there and eat it and stay there.' [Foundation 16] b. burkhum-go-ra cave-inside-loc kwa: mud dha-i-thal-lʉ-ma dig-3sg-hab-pst.(3→3)sg-conj dhwa:-go-jʉ earth-inside-loc.down thʌk-ʣɵl-lʉ-ma hide-antic-pst.(3→3)sg-conj jok-thal-la come.down-hab-pst.3sg ʔe hs 'He used to dig down into the mud in the cave and hide it [the food], and then come down.' [Foundation 19]

Table 1

 1

		: Itive motion verbs
	Upwards Downwards Across Unspecified
	Belhare Bantawa lont-tʰaŋ-	yu-dʰa-	pʰen-bitt-	kʰat-kʰat-

Table 2 :

 2 Venitive verbs Upwards Downwards Across Unspecified

	Hayu	dzɔk-	yu-	phi-

Table 3 :

 3 Caused accompanied motion verbs (venitive)

		Upwards Downwards Across Unspecified
	Hayu	?	yut-	pit-

Table 4

 4

		: Bantawa elevational motion verbs
		Up	Down Across Unspecified
	come go	tʰaŋ-yɨ-lont-dʰa-	ban-bitt-	ta-kʰat-
	bring (transitive) tʰakt yɨt take (transitive) lont-dʰant--batt	tat-

kʰatt-

Table 5

 5

		: Elevation deixis vs. Motion deixis
	Example Verb	Elevation	Motion Adverb/Noun
	(26a) (26b)	tʰaŋ-ven+up tʰaŋ-ven+up	ven+up ven+up	ven iti	ven+up iti+up
	(26c)	kʰat-iti+unspec ven+up	iti	loc:up
	(28)	lont-up+iti	iti+up	iti	iti+up
	(27a)	yɨ-ven+down	ven+down ven	ven+down
	(27b) (27c)	yɨ-ven+down kʰat-iti+unspec ven+down iti ven+down iti	iti+down loc:down

Table 6 :

 6 Elevational deixis in Belhare intransitive motion verbs (based on

	Bickel 1999: 94)
	Itive 'go' (set A) Venitive 'come' (set B)
	Upwards Downwards yu-tʰaŋ-Across pʰen-Unspecified kʰat-	kat-uŋ-ap-ta-

Table 7

 7

	: Set A motion verbs
	Upwards	Downwards Across
	Hayu Thulung Khaling Dumi Koyi Kulung Camling saŋ-tʰoŋ-Bantawa tʰaŋ-Puma tʰoŋ-Athpare tʰaŋ-(itive) Belhare tʰaŋ-(itive) yu-(itive) yu-jok-je-yi-gʰuʔ-? yu-i-yɨ-i-Yakkha tʰaŋ-(itive) Yamphu saŋ-(itive) yu-(itive) Limbu tʰaŋ-yu-	phi-bik-pi-pi-bhiʔ-bitt-(itive) pʰen-(itive) ? pʰɛn-

2017: 203). This verb is related to the adverbial root reflected by Limbu yo 'down, downhill, downstream'. Outside of Kiranti, cognates of both the motion verbs and their -t applicatives are found in Thangmi (yu-'come down', Turin 2012), but also further away in Situ Gyalrong: Cogtse Situ jə' go downwards', jut 'take downwards'

(Lin 2017: 61)

and Bragbar Situ ɟə, ɟə, ɟú 'se déplacer vers le bas', ɟət, ɟət, ɟút 'transporter vers le bas'

(Zhang

Table 8 :

 8 Venitive motion verbs Upwards Downwards Across

	Chintang kat-Athpare kat-Belhare kat-Yakkha keʔ-Yamphu kat-	kuŋs-uŋs-uŋs-uks-uks-	tʰap-ap-ap-ap-ap-

The riverine (upstream-downstream) dimension, which is linguistically encoded in other Trans-Himalayan/Sino-Tibetan languages, in particular in Gyalrongic[START_REF] Sun | Parallelisms in the verb morphology of Sidaba rGyalrong and Lavrung in rGyalrongic[END_REF][START_REF] Lin | A dimension missed: East and west in Situ rGyalrong orientation marking[END_REF][START_REF] Lai | Grammaire du khroskyabs de Wobzi[END_REF][START_REF] Zhang | Le dialecte de Brag-bar du rgyalrong situ et sa contribution à la typologie de l'expression des relations spatiales: le mouvement associé et l'orientation[END_REF] and Tani (Post 2019, 2020), is rarely mentioned in descriptions of Kiranti languages.

It is worth pointing out that the connection found in some Sino-Tibetan languages between 'home' and 'uphill'(Post 2019: 243) does not appear to be strongly developed in the Kiranti materials we have access to. AlthoughSchackow (2015: 182) mentions that any location outside the Himalayas is referred to as 'downhill' in Yakkha, this statement may simply be an expression of a topographic fact. Examples such as 25, where the homeland is at with a lower elevation, as reflected in the choice of motion verb, do not suggest that there is any particular connection between 'home' and 'uphill'.

Languages in the table are organized from West to East, with Hayu being the westernmost language. The sources are the following: Hayu[START_REF] Michailovsky | La langue hayu[END_REF], Thulung[START_REF] Lahaussois | Thulung Rai[END_REF], Wambule[START_REF] Opgenort | A grammar of Wambule[END_REF], Khaling(Jacques et al. 2015), Dumi (van

It is possible that what Schapper (2014) refers to as 'unelevated' terms have a similar use.

The 'upwards' itive verb of Bantawa lont-'go up' also means 'go out'(Doornenbal 2009: 350), and is cognate to Puma lon-'come out', which does not belong to the elevational deixis paradigm.

Concerning the correspondence of Camling sto tʰin other languages,[START_REF] Michailovsky | Preliminaries to the comparative study of the Kiranti subgroup of Tibeto-Burman[END_REF], who reconstructs *Xt for the set reconstructed here as *tʰ, argues for a consonant cluster, but also indicates that *tʰis a possible reconstruction, which however implies a specific sound change in Camling (perhaps *tʰ → *θ → s).

Abbreviations

In adding to the Leipzig glossign rules, this work uses the following abbreviations: act active, antic anticipatory, comit comitative, conf confirmative, conv converb, int intensifier, lnk linker, pcp participle, pft perfect, rep reported speech, res resultative, subj subject,

Conclusion

This paper contributes to the study of elevational deixis in both typological and historical perspectives.

Kiranti languages have typologically fairly common elevational systems (Forker 2020) involving 'upwards', 'level/across' and 'downwards' elevations in the verbal system (section 3). However, the Kiranti data contribute to the typology of elevational deixis by showing that the elevation-unspecified motion verbs form, at least for Kiranti, an integral part of the elevation system, in addition to up(wards), down(wards) and level/across categories: they fill the gaps in the semantic spaces not expressible by elevation-specified verbs.

Elevational motion verbs are largely cognate across Kiranti as entire sets (section 6.1), and it is likely that this category is reconstructible to proto-Kiranti. Cognate verbs always have the same elevational deixis, but may be itive in some languages and venitive in others. The synchronic shifts of deixis documented in sections 4.2 and 5 are not only of interest for synchronic description and typology, but also offer plausible models to interpret the data in section 6 from a historical perspective and reconstruct proto-Kiranti.