Salem Ben Said

Jean-Louis Clerc

Khalid

Salem Ben

S A Ïd

Khalid Koufany
email: khalid.koufany@univ-lorraine.fr

CONFORMALLY COVARIANT BI-DIFFERENTIAL OPERATORS FOR DIFFERENTIAL FORMS

Keywords: 2000 Mathematics Subject Classification. Primary 43A85. Secondary 58J70, 22E46, 58A10 Differential forms, conformal group, covariant bi-differential operators, principal series, Riesz distribution

Conformally covariant bi-differential operators for differential forms

Introduction

The Rankin-Cohen brackets are the most famous examples of conformally covariant bi-differential operators. For a presentation of these operators from our point of view based on harmonic analysis of the group SLp2, Rq, we refer the reader to the introduction of [START_REF] Ben Saïd | Conformally covariant bi-differential operators on a simple real Jordan algebra[END_REF]. In [START_REF] Ovsienko | Generalized transvectants Rankin-Cohen brackets[END_REF] Ovsienko and Redou introduced their analogs for conformal analysis on R n .

A new construction of these covariant bi-differential operators was proposed by Beckmann and the second present author in [START_REF] Beckmann | Singular invariant trilinear forms and covraint (bi)-differential operators under the conformal group[END_REF], where (although implicitly) the source operator method was introduced. In our situation, the source operator is a differential operator on R n ˆRn , covariant for the diagonal action of the conformal group SO 0 p1, n `1q. The covariant bi-differential operators are obtained by composing the source operator with the restriction map from R n ˆRn to the diagonal. This technique has shown to be very efficient to produce new examples of covariant differential operators in many different contexts. In [START_REF] Ben Saïd | Conformally covariant bi-differential operators on a simple real Jordan algebra[END_REF], we constructed covariant bi-differential operators in the context of simple real Jordan algebras. The article [START_REF] Clerc | Another approach to Juhl's conformally covariant differential operators from S n to S n´1[END_REF] contains an alternative construction of the covariant differential operators introduced by Juhl [START_REF] Juhl | Families of conformally covariant differential operators, Q-curvature and holography[END_REF] in the context of the restriction of R n to R n´1 . In the same geometric context, Fischmann, Ørsted and Somberg [START_REF] Fischmann | Bernstein-Sato identities and conformal symmetry breaking operators[END_REF] recently obtained a new construction of the covariant differential operators for differential forms, previously obtained by Kobayashi, Kubo and Pevzner in [START_REF] Kobayashi | Conformal symmetry breaking operators for differential forms on spheres[END_REF] and by Fischmann, Juhl and Somberg in [START_REF] Fischmann | Conformal symmetry breaking differential operators on differential forms[END_REF]. Finally, it is worthwhile mentioning that a more general notion of symmetry breaking operators (not necessarily differential) has been studied by Kobayashi and his collaborators (see, e.g., [START_REF] Kobayashi | Symmetry breaking for representations of rank one orthogonal groups[END_REF], [START_REF] Kobayashi | Symmetry breaking for representations of rank one orthogonal groups[END_REF], [START_REF] Kobayashi | Differential symmetry breaking operators I. General theory and F-method[END_REF]).

In the present paper, we construct bi-differential operators acting on spaces of differential forms which are covariant for the conformal group of R n ; more precisely for the group G " SO 0 p1, n `1q. To build these bi-differential operators, we use again the source operator method. The source operators are constructed as a composition of the multiplication operator by the function }x ´y} 2 (using its transformation rule under the action of the conformal group) and classical Knapp-Stein intertwining operators. These intertwining operators on differential forms were studied recently in [START_REF] Fischmann | A family of Riesz distributions for differential forms on Euclidean space[END_REF][START_REF] Fischmann | Bernstein-Sato identities and conformal symmetry breaking operators[END_REF] and some of their results are used (and sometimes reproved) in the present article.

The construction relies ultimately on two main identities stated in Theorem 3.2 and Theorem 3.3 (see also Theorem 3.4), the second one being the Euclidean Fourier transform of the first one. As they involve purely Euclidean harmonic analysis they are presented in Section 3, independently of the conformal context. In Section 4 we give some background on the conformal group of R n needed to describe the noncompact model for the principal series representations of SO 0 p1, n `1q in Section 5. The conformal properties of the source operator are given in Section 6, where harmonic analysis of the group SO 0 p1, n `1q plays a crucial role.

The corresponding covariant bi-differential operators are constructed in Section 7. The lack of a manageable decomposition of the tensor product σ k b σ , 0 ď k, ď n, where σ k denotes the representation of SOpnq on the spacecomplex-valued Λ k of complexvalued alternating k-forms on R n , prevents us from giving explicit formulas for the corresponding bi-differential operators, but this can be done at least for the Cartan factor Λ k` with 0 ď k ` ď n, appearing in Λ k b Λ , (see (7.1)).

Background on differential forms

Let x¨, ¨y be the standard Euclidean scalar product in R n and } ¨} the corresponding norm. Let pe 1 , e 2 , . . . , e n q be the standard orthonormal basis of R n and let pe 1 , e 2 , . . . , e nq be its dual basis.

For 0 ď k ď n, we denote by Λ k " Λ k pR n ˚q b C the vector space of complex-valued alternating multilinear k-forms on R n . A basis of the space Λ k is given by te I :" e i1 ^¨¨¨^e ik : 1 ď i 1 ă ¨¨¨ă i k ď nu.

If ω P Λ k and η P Λ , then ω ^η P Λ k` . Furthermore, ω ^η " p´1q k η ^ω.

(2.1)

The exterior algebra Λ :"

À 8 k"0 Λ k "
À n k"0 Λ k is an associative algebra, graded with respect to the degree k.

The interior product of a k-form ω with a vector x P R n is the pk ´1q-form defined by ι x ωpx 1 , . . . , x k´1 q " ωpx, x 1 , . . . , x k´1 q.

Moreover,

ι e j pe i1 ^¨¨¨^e ik q " # 0 if j " any i r p´1q r´1 e i1 ^¨¨¨^x e ir ^¨¨¨^e ik if j " i r , (2.2)
where the "cap" over e ir means that it is deleted from the exterior product. One may check that ι x ι y `ιy ι x " 0.

(2.3) Given x P R n , the exterior product of a k-form ω with the linear form x ˚is the pk `1q-form defined by

ε x ω " x ˚^ω.
From the associativity of the wedge product and (2.1), it follows

ε x ε y `εy ε x " 0. (2.4)
There is the following useful anticommutation relation

ε x ι y `ιy ε x " xx, yy Id Λ , x, y P R n . (2.5)
This is a fairly straightforward consequence of (2.2). Further, by (2.5), (2.4) and (2.3) we have ε x ι y ε x " xx, yyε x , ι y ε x ι y " xx, yyι y .

We denote by ι j the interior product with the basis vector e j and by ε j the exterior products with e j . The following lemma is needed for later use.

Lemma 2.1. On Λ k we have n ÿ j"1 ε j ι j " k Id Λ , n ÿ j"1 ι j ε j " pn ´kq Id Λ .
Proof. Let I " ti 1 , . . . , i k u Ă t1, 2, . . . , nu. In view of (2.2) and the fact that e j ^ej " 0 for every j, clearly we have

´ÿ jPI ε j ι j ¯eI " ke I , ´ÿ jPI ι j ε j ¯eI " 0.
Similarly, ´ÿ jRI ε j ι j ¯eI " 0, ´ÿ jRI ι j ε j ¯eI " pn ´kqe I .

The lemma is now a matter of putting pieces together.

From now on, we will identify the dual of R n with R n . For 0 ď k ď n, let

E k pR n q " C 8 `Rn , Λ k ˘» C 8 pR n q b Λ k pR n q
be the space of smooth complex-valued differential forms of degree k on R n . An element of E k pR n q can be uniquely represented as

ωpxq " ÿ 1ďi 1 㨨¨ăi k ďn ω i 1 ,¨¨¨,i k pxqe i 1 ,¨¨¨,i k , (2.6)
where the coefficients ω I are complex-valued smooth functions on R n . In particular, E 0 pR n q " C 8 pR n q. The direct sum

EpR n q :" n à k"0 E k pR n q
is the linear space of all smooth differential forms.

From the properties of the interior product ι x and the exterior product ε x on Λ k which we discussed above, one gets analogue properties on E k pR n q.

We now define the exterior differential d :

E k pR n q ÝÑ E k`1 pR n q by d " n ÿ m"1 ε m B m ,
and the co-differential δ :

E k`1 pR n q ÝÑ E k pR n q by δ " ´n ÿ m"1 ι m B m ,
where B m is the directional derivative in the direction of the basis vector e m . We set δ " 0 on E 0 pR n q " C 8 pR n q. In the light of (2.3), (2.4) and (2.5), direct computations show d ˝d " δ ˝δ " 0 and dι j `ιj d " B j , δε j `εj δ " ´Bj , 0 ď j ď n.

(2.7)

We close this section by introducing the Hodge Laplacian on differential forms, defined by ∆ :" ´pdδ `δdq "

n ÿ m"1 B 2 m .
(2.8)

Henceforth we will denote the Hodge Laplacian by Q `B Bx ˘where

Qpxq :" x 2 1 `¨¨¨`x 2 n .
(2.9)

The main identities for Riesz distributions on differential forms

For s a complex parameter, consider for ϕ P SpR n q the formula

@ R s , ϕ D " π ´n 2 2 ´n´s Γ `´s 2 Γ `n`s 2 ˘żR n ϕpxq}x} s dx. (3.1)
For ´n ă Re s ă 0, this formula defines a tempered distribution depending holomorphically on s. The normalization factor is chosen for convenience, as we shall see below (see (3.2)). By standard argument (see [START_REF] Gelfand | Spaces of fundamental and generalized functions[END_REF]) it can be analytically continued to C, yielding a meromorphic family of tempered distributions, called the Riesz distributions.

We follow the following convention for the Fourier transform on Schwartz functions ϕ P SpR n q:

Fpϕqpξq " ż R n ϕpxqe ixx,ξy dx, which extends to the space of tempered distributions S 1 pR n q. It is known (see [START_REF] Gelfand | Spaces of fundamental and generalized functions[END_REF] or [16, p. 38]) that the image of R s by the Fourier transform is FpR s qpξq " }ξ} ´n´s .

(3.

2)

The classical Riesz distributions offer a good motivation for defining Riesz distributions for differential forms on R n with coefficients in the Schwartz space SpR n q.

For 0 ď k ď n, let SE k pR n q (resp. S 1 E k pR n q) be the space of differential k-forms represented as in (2.6) with coefficients ω i 1 ,...,i k in SpR n q (resp. S 1 pR n q ). We pin down that we may extend the Fourier transform F on the space SE k pR n q by acting on the coefficients ω i 1 ,...,i k of the form (2.6).

For 0 ď k ď n and a complex parameter s, let R k s be the Riesz distribution on differential forms defined by

xR k s , ωy " π ´n 2 2 ´s´n`1 Γ `´s 2 `1Γ `s`n 2 ˘żR n }x} s´2 pι x ε x ´εx ι x qωpxqdx, (3.3)
with ω P SE k pR n q. For ´n ă Re s ă 0, this formula defines a tempered distribution depending holomorphically on s. By [4, §3.2], R k s can be analytically continued to C, giving a meromorphic family of tempered distributions. When k " 0, the identity ι x ε x `εx ι x " }x} 2 Id E implies immediately that R 0 s is nothing but the classical Riesz distribution (3.1).

In [START_REF] Fischmann | A family of Riesz distributions for differential forms on Euclidean space[END_REF]Theorem 3.2] the authors proved that the Fourier transform of R k s is given by FpR k s qpξq " }ξ} ´s´n´2 ´´ps `2kqι ξ ε ξ `ps `2n ´2kqε ξ ι ξ ¯(3.4)

for s not a pole. Due to the facts ι ξ " 0 and ι ξ ε ξ " }ξ} 2 Id E on SE 0 pR n q " SpR n q, it follows that (3.4) for k " 0 coincides with (3.2).

In order to simplify notation, it is convenient to let

Z k s :" FpR k ´s´n q (3.5) i.e. Z k s pxq " }x} s´2 ´ps `n ´2kqι x ε x ´ps ´n `2kqε x ι x ¯.
We should point that in all arguments below we first prove the desired result when the complex parameter s is so that everything makes sense, and then we extend it meromorphically to the complex plane C.

We shall need the following crucial result (which might be of some interest in its own right):

Theorem 3.1. The distribution Z k s satisfies the following properties:

Z k s pxq " Z k s´2 pxq ´ak,s ι x ε x `bk,s ε x ι x ¯, (3.6)
B Bx j Z k s pxq " Z k s´2 pxq ´sx j Id `ck,s ι x ε j `dk,s ε x ι j ¯, (3.7)
Q ´B Bx ¯Zk s pxq " sps `nqZ k s´2 pxq, (3.8)
with

a k,s " s `n ´2k s `n ´2k ´2, b k,s " s ´n `2k s ´n `2k ´2 , (3.9)
and

c k,s " 2s s `n ´2k ´2 , d k,s " 2s s ´n `2k ´2. (3.10)
Above Q `B Bx ˘denotes the Hodge Laplacian (see (2.8)).

Proof.

(1) On one hand, we may rewrite Z k s pxq as

Z k s pxq " }x} 2 Z k s´2 pxq ´2sps `nq}x} s´2 pι x ε x ´εx ι x q.
On the other hand, using the fact ι x ε x `εx ι x " }x} 2 Id E , we may rewrite the term }x} 2 pι x ε x ´εx ι x q as follows:

}x} 2 pι x ε x ´εx ι x q " ´ps `n ´2k ´2qι x ε x ´ps ´n `2k ´2qε x ι x ¯´a 1 k,s ι x ε x `b1 k,s ε x ι x ¯, (3.11)
where

a 1 k,s " 1 s `n ´2k ´2, b 1 k,s " 1 s ´n `2k ´2.
Thus,

Z k s pxq " }x} 2 Z k s´2 pxq `2Z k s´2 pxq pa 1 k,s ι x ε x `b1 k,s ε x ι x q.
Using again the identity ι x ε x `εx ι x " }x} 2 Id E to deduce the first statement.

(2) First we have

B Bx j pι x ε x q " ι j ε x `ιx ε j " x j Id E ´εx ι j `ιx ε j , B Bx j pε x ι x q " ε j ι x `εx ι j " x j Id E `εx ι j ´ιx ε j ,
where, for abbreviation, ι j (resp. ε j ) denotes ι e j (resp. ε e j ). Above we have used the identity (2.5). Then B Bx j Z k s pxq " ps ´2qx j }x} s´4 ´ps `n ´2kqι x ε x ´ps ´n `2kqε x ι x }x} s´2 ´ps `n ´2kqpx j Id E ´εx ι j `ιx ε j q ´ps ´n `2kqpx j Id E `εx ι j ´ιx ε j q

" ps ´2qx j }x} s´4 ´`ps ´2 `n ´2kqι x ε x ´ps ´2 ´n `2kqε x ι x ˘`2pι x ε x ´εx ι x q }x} s´2 ´2pn ´2kqx j Id E `2spι x ε j ´εx ι j q " ps ´2qx j Z k s´2 pxq `2x j }x} s´4 ´pn ´2kq}x} 2 Id E `ps ´2qpι x ε j ´εx ι j q 2s}x} s´2 pι x ε j ´εx ι j q
" ps ´2qx j Z k s´2 pxq `2x j Z k s´2 pxq `2s}x} s´2 pι x ε j ´εx ι j q. Now using again the trick (3.11) we obtain

B Bx j Z k s pxq " sx j Z k s´2 pxq `2sZ k s´2 pxqpa 1 k,s ι x ε j `b1 k,s ε x ι j q, and (3.7) follows.
(3) The definition of Z k s pxq and the intertwining property

Q `B Bx ˘˝F " ´F ˝} ¨}2 imply Q ´B Bx ¯Zk s pxq " ´Fp} ¨}2 R k ´s´n qpxq " sps `nqFpR k ´s´n`2 qpxq " sps `nqZ k s´2 pxq.
For 0 ď k, ď n, we define the space E k, pR n ˆRn q as the space of smooth functions on R n ˆRn with values in Λ k b Λ . More generally, denote by DE k, pR n ˆRn q (resp. SE k, pR n ˆRn q, S 1 E k, pR n ˆRn q) the space of differential forms of bidegree pk, q on R n ˆRn with coefficients in DpR n ˆRn q (resp. SpR n ˆRn q, S 1 pR n ˆRn q) Theorem 3.2. For every ω P SE k, pR n ˆRn q, the following formula hols true

Q ˆB Bx ´B By ˙`Z k s pxq b Z t pyq ωpx, yq ˘" Z k s´2 pxq b Z t´2 pyq D k, s,t ωpx, yq,
where D k, s,t is the differential operator on pk, q-differential forms given by

D k, s,t " ´ak,s ι x ε x `bk,s ε x ι x ¯b ´a ,t ι y ε y `b ,t ε y ι y ¯˝Q ˆB Bx ´B By 2 n ÿ j"1 ´sx j Id E k `ck,s ι x ε j `dk,s ε x ι j ¯b ´a ,t ι y ε y `b ,t ε y ι y ¯˝´B Bx j ´B By j 2 n ÿ j"1 ´ak,s ι x ε x `bk,s ε x ι x ¯b ´ty j Id E `c ,t ι y ε j `d ,t ε y ι j ¯˝´B Bx j ´B By j 2 n ÿ j"1 ´sx j Id E k `ck,s ι x ε j `dk,s ε x ι j ¯b ´ty j Id E `c ,t ι y ε j `d ,t ε y ι j sps `nq Id E k b ´a ,t ι y ε y `b ,t ε y ι y ¯`tpt `nq ´ak,s ι x ε x `bk,s ε x ι x ¯b Id E .
The coefficients a k,s , b k,s , c k,s and d k,s are given by (3.9) and (3.10) (and similarly when the subscripts k, s are replaced by , t).

Proof. A routine calculation gives

Q ˆB Bx ´B By ˙´Z k s pxq b Z t pyqωpx
"Q ˆB Bx ˙Zk s pxq b Z t pyqωpx, yq `Zk s pxq b Q ˆB By ˙Z t pyqωpx, yq ´2 n ÿ j"1 B Bx j Z k s pxq b B By j Z t pyqωpx, yq "sps `nqZ k s´2 pxq b Z t pyqωpx, yq `tpt `nqZ k s pxq b Z t´2 pyqωpx, yq ´2Z k s´2 pxq b Z t´2 pyq ! n ÿ j"1 ´sx j Id E k `ck,s ι x ε j `dk,s ε x ι j ¯b ´ty j Id E `c ,t ι y ε j `d ,t ε y ι j ¯)ωpx, yq "Z k s´2 pxq b Z t´2 pyq ! sps `nq Id E k b ´a ,t ι y ε y `b ,t ε y ι y ¯`tpt `nq ´ak,s ι x ε x `bk,s ε x ι x ¯b Id E ´2 n ÿ j"1 ´sx j Id E k `ck,s ι x ε j `dk,s ε x ι j ¯b ´ty j Id E `c ,t ι y ε j `d ,t ε y ι j ¯)ωpx, yq.
Secondly, the identity (3.6) gives

Z k s pxq b Z t pyqQ ˆB Bx ´B By ˙ωpx, yq " Z k s´2 pxq b Z t´2 pyq !´a k,s ι x ε x `bk,s ε x ι x ¯b ´a ,t ι y ε y `b ,t ε y ι y ¯Q ˆB Bx ´B By ˙)ωpx, yq.
Finally, using again (3.6) and (3.7) we obtain

r Q ˆB Bx , B By ˙`Z k s pxq b Z t pyqωpx, yq ˘" 2Z k s´2 pxq b Z t´2 pyq ! n ÿ j"1 ´sx j Id E k `ck,s ι x ε j `dk,s ε x ι j ¯b ´a ,t ι y ε y `b ,t ε y ι y ¯Bω Bx j px, yq `n ÿ j"1 ´ak,s ι x ε x `bk,s ε x ι x ¯b ´ty j Id E `c ,t ι y ε j `d ,t ε y ι j ¯Bω By j px, yq ´n ÿ j"1 ´ak,s ι x ε x `bk,s ε x ι x ¯b ´ty j Id E `c ,t ι y ε j `d ,t ε y ι j ¯Bω Bx j px, yq ´n ÿ j"1 ´sx j Id E k `ck,s ι x ε j `dk,s ε x ι j ¯b ´a ,t ι y ε y `b ,t ε y ι y ¯Bω By j px, yq ) "2Z k s´2 pxq b Z t´2 pyq ! n ÿ j"1 ´sx j Id E k `ck,s ι x ε j `dk,s ε x ι j ¯b ´a ,t ι y ε y `b ,t ε y ι y ¯´B Bx j ´B By j ¯ωpx, yq ´n ÿ j"1 ´ak,s ι x ε x `bk,s ε x ι x ¯b ´ty j Id E `c ,t ι y ε j `d ,t ε y ι j ¯´B Bx j ´B By j ¯ωpx, yq ) .
This finishes the proof of the theorem.

For s P C, let

J k s ωpxq :" ż R n
R k s px ´yqωpyqdy, ω P SE k pR n q.

(3.12)

We may see J k s as a convolution operator with the distribution R k s , J k s ω " R k s ˚ω. So (3.12) defines a meromorphic family of operators from SE k pR n q to S 1 E k pR n q.

Recall the following formulas for the Fourier transform : This is merely the Fourier transform version of Theorem 3.2, using (3.5) and formulas (3.13), (3.14) and (3.15). We omit details.

F
Next, we need to rewrite E k, s,t in its normal form (i.e. multiplications after differentiations). Before stating the result, let us introduce for 0 ď k ď n, 1 ď j ď n the following differential operators:

l k,s :" α k,s δd `βk,s dδ, ∇ k,s,j :" ´2α k,s B Bx j ´4pn ´2kqε j δ `4pn ´2kqdι j ¯´´s α k,s B Bx j `γk,s ε j δ `δk,s dι j " p2 ´sqα k,s B Bx j ´p4pn
´2kq `γk,s qε j δ `p4pn ´2kq ´δk,s qdι j " p2 ´sq " ps `n ´2kqps ´n `2k ´2q B Bx j `2ps ´n `2kqε j δ `2ps `n ´2kqdι j ı ,

where the coefficients α k,s , β k,s , γ k,s and δ k,s are given by (3.9) and (3.10). Similarly, we introduce the operators l ,t and ∇ ,t,j with respect to the y-variable.

Theorem 3.4. The operator E k, s,t in Theorem 3.3 can be rewritten in the following normal form:

E k, s,t " ´}x ´y} 2 l k,s b l ,t `2 n ÿ j"1 px j ´yj q t∇ k,s,j b l ,t ´lk,s b ∇ ,t,j u `2 n ÿ j"1 ∇ k,s,j b ∇ ,t,j
`pt ´2qpt ´n ´2qpt ´n `2 qpt `n ´2 q l k,s b Id E `ps ´2qps ´n ´2qps ´n `2kqps `n ´2kq Id E k b l ,t .

(3.20)

The proof is straightforward, but long and tedious. We first need some elementary formulas .

Lemma 3.5. Let ω be a k-form. Then, for fixed y P R n ,

δ dpx j ´yj qω " px j ´yj q δ d ω ´2 B Bx j ω ´εj δ ω `d ι j ω d δpx j ´yj qω " px j ´yj q d δ ω `εj δ ω ´d ι j ω δ d }x ´y} 2 ω " }x ´y} 2 δ d ω `2 n ÿ j"1 px j ´yj qp´2 B Bx j ´εj δ `d ι j qω ´2pn ´kqω d δ }x ´y} 2 ω " }x ´y} 2 d δ ω `2 n ÿ j"1
px j ´yj qpε j δ ´d ι j qω ´2kω.

Derivations are taken with respect to the x-variable.

Proof. This is a direct consequence of the identities # dpx j ´yj qω " ε j ω `px j ´yj q d ω δpx j ´yj qω " ´ιj ω `px j ´yj q δ ω, and # d }x ´y} 2 ω " }x ´y} 2 d ω `2 ř n j"1 px j ´yj q ε j ω δ }x ´y} 2 ω " }x ´y} 2 δ ω ´2 ř n j"1 px j ´yj q ι j ω.

We may now start the proof of Theorem 3.4. In the light of Lemma 3.5 (and its version with respect to y) together with the anti-commutator laws in (2.7) we arrive at the expressions below for the first three terms of the operator E k, s,t in Theorem 3. px j ´yj q ´´2pn ´2kqpε j δ ´dι j q `αk,s B Bx j ¯b ´α ,t δd `β ,t dδ

4 n ÿ j"1
px j ´yj q ´αk,s δd `βk,s dδ ¯b ´´2pn ´2 qpε j δ ´dι j q `α ,t B By j

8 n ÿ j"1
´´2pn ´2kqpε j δ ´dι j q `αk,s B Bx j ¯b ´´2pn ´2 qpε j δ ´dι j q `α ,t B By j 2ppn

´ qα ,t ` β ,t q ´αk,s δd `βk,s dδ ¯b

Id E `2ppn ´kqα k,s `kβ k,s q Id E k b ´α ,t δd `β ,t dδ ¯.
The second term of the operator E k, s,t in (3.19) can be rewritten as:

´2 n ÿ j"1 px j ´yj q ´sα k,s B Bx j `γk,s ε j δ `δk

¯.

It remains to sum up all terms to finish the poof of Theorem 3.4

We close this section by writing the operator E k, s,t in the particular case k " " 0. Here the operator E 0,0 s,t will act on the space SE 0,0 pR n ˆRn q " SpR n ˆRn q. Since δ " 0 and ι j " 0 on scalar functions, the operators l 0,s and ∇ 0,s,j reduce to l 0,s " α 0,s δd " ´α0,s Q `B Bx ˘, (see (2.8)), and ∇ 0,s,j " p2 ´sqα 0,s B Bx j .

Similar identities hold with respect to y. Hence Up to the normalization constant ps `nqps ´n ´2qpt `nqpt ´n ´2q and the change of variables s by 2s and t by 2t, the operator E 0,0 s,t coincides with the differential operator obtained in [START_REF] Ben Saïd | Conformally covariant bi-differential operators on a simple real Jordan algebra[END_REF]Proposition 10.3] to build covariant bi-differential operators under the (diagonal) action of the Lie group Opn `1, 1q.

E 0,0 s,t " α 0,s α 0,t ! ´}x ´y} 2 Q `B Bx ˘b Q `B By ˘´2 n ÿ j"1 pt ´2qpx j ´yj qQ `B Bx ˘b B By j `2 n ÿ j"1 ps ´2qpx j ´yj q B Bx j b Q `B

Background on the conformal group of R n

Let R 1,n`1 be the n `2-dimensional real vector space equipped with the Lorentzian quadratic form rx, xs " x 2 0 ´px 2 1 `¨¨¨`x 2 n`1 q, x " px 0 , x 1 , . . . , x n`1 q.

Let Ξ be the isotropic cone defined by Ξ " tx P R 1,n`1 zt0u : rx, xs " 0u.

For x P R 1,n`1 zt0u, denote by rxs " R ˚x the ray through x and consider the space of isotropic rays, i.e. the quotient space Ξ{R ˚.

The subspace tx P R 1,n`1 : x 0 " 0u will be identified with R n`1 under the isomorphism

R n`1 Q x 1 Þ ÝÑ p0, x 1 q P R 1,n`1 .
Denote by S n the unit sphere of R n`1 . The map

S n Q x 1 Þ ÝÑ R ˚p1, x 1 q
yields an isomorphism of S n with Ξ{R ˚; the inverse isomorphism being described by

Ξ{R ˚Q R ˚x Þ ÝÑ R ˚x X tx 0 " 1u.
Let G " SO 0 p1, n `1q be the connected component of the identity in the group of isometries for the Lorentzian form on R 1,n`1 . Then G acts on Ξ and commutes with the action of R ˚on R 1,n`1 , so that G acts on Ξ{R ˚and yielding an action of G on S n .

Let us give more details on the action of G on the unit sphere S n . For x 1 " px 1 1 , . . . , x 1 n`1 q P S n and g P G, observe that pgp1, x 1 qq 0 ą 0 and define gpx 1 q P S n by p1, gpx 1 qq " pgp1, x 1 qq ´1 0 gp1, x 1 q.

For g P G and x 1 P S n , set cpg, x 1 q " pgp1, x 1 qq ´1 0 .

Clearly cpg, x 1 q is a smooth and strictly positive function on G ˆSn . Moreover, the function c satisfies the cocycle property cpg 1 g 2 , x 1 q " c `g1 , g 2 px 1 q ˘cpg 2 , x 1 q, g 1 , g 2 P G, x 1 P S n .

This action turns out to be conformal on S n , i.e. for any g P G, x 1 P S n and arbitrary ξ P T x S n , the differential Dgpx 1 q satisfies }Dgpx 1 qξ} " cpg, x 1 q}ξ}, and the term cpg, x 1 q is called the conformal factor of g at x 1 . Let e n`1 " p0, 0, . . . , 0, 1q, and let

κ : R n ÝÑ S n t´e n`1 u defined by px 1 , . . . , x n q Þ ÝÑ ˆ2x 1 1 `|x| 2 , ¨¨¨, 2x n 1 `|x| 2 , 1 ´|x| 2 1 `|x| 2
ḃe the inverse map of the stereographic projection. The action of G on S n can be transferred to a rational action (not everywhere defined) on R n , for which we still use the notation

G ˆRn Q pg, xq Þ ÝÑ gpxq P R n .
The map κ is conformal and hence, the rational action of G on R n transferred from its action on S n is conformal. For g P G defined at x P R n , denote by Ωpg, xq the corresponding conformal factor. Below, among other things, we will find an expression for Ωpg, xq.

where n " LiepN q. Now let x P R n and let g P G be defined at x. Since gpxq " gn x p0q, it follows from (4.2) that Dgpxq " Dpgn x qp0q " e ´tpgnxq mpgn x q. (4.3) As a consequence, we obtain Ωpg, xq " e ´tpgnxq , g P G, x P R n . (4.4)

We close this paragraph by the following standard result.

Lemma 4.1. Let x, y P R n and let g P G be defined at x and y. Then

}gpxq ´gpyq} 2 " Ωpg, xq }x ´y} 2 Ωpg, yq. (4.5)
5. The principal series representations of SO 0 p1, n `1q on the space of differential forms Let E k pS n q be the space of differential k-forms on the unit sphere S n (0 ď k ď n). For λ P C, let ρ k λ be the representation of G " SO 0 p1, n `1q on E k pS n q given by ρ k λ pgqω px 1 q " cpg ´1,

x 1 q λ `Lg ´1 ω ˘px 1 q, g P G, ω P E k pS n q,
where L g is the diffeomorphism x 1 Þ ÝÑ gpx 1 q on S n and L g is the induced action on differential forms. Here cpg ´1, x 1 q is conformal factor given by (4.1). Below we will describe the noncompact model for this series of representations, obtained from the present model through the stereographic projection.

Denote by DE k pR n q the space of k-forms represented as in (2.6) with the complexvalued coefficients ω i 1 ,...,i k in DpR n q.

Now, for g P G and ω P E k pR n q let

π k λ pgqωpxq " Ωpg ´1, xq λ L g´1 ωpxq, (5.1)
where Ωpg ´1, xq is given by (4.4). This formula defines formally a representation of G. As it stands, the representation is not globally defined. In what follows, it will be enough to observe that for a relatively compact open subset U of R n , there exists a small neighborhood V of the neutral element in G (depending on U ) such that for any g P V , g ´1 is defined on U . Hence, for any smooth differential k-form ω with Supppωq Ă U , the object π k λ pgq ω is well defined, it belongs to E k pR n q and has a compact support. This allows us to define the corresponding infinitesimal representation dπ k λ of the Lie algebra g " LiepGq by

dπ k λ pXqω :" d dt π k λ pexpptXqqω ˇˇt"0 , X P g.
The expression is well defined when ω P DE k pR n q. The operator dπ k λ pXq is a first order differential operator with polynomial coefficients and hence can be extended to SE k pR n q.

The representations π k λ can also be viewed as principal series representations. Indeed, rewrite (5.1) as π k λ pgqωpxq " Ωpg ´1, xq λ ωpg ´1pxqq ˝Dg ´1pxq,

where ωpg ´1pxqq ˝Dg ´1pxq is the k-form given by ωpg ´1pxqq ˝Dg ´1pxqpv 1 , . . . , v k q " ωpg ´1pxqqpDg ´1pxqv 1 , . . . , Dg ´1pxqv k q. Now using (4.3) and (4.4) we get π k λ pgqωpxq " e ´pλ`kqtpg ´1nxq σ k `mpg ´1n x q ˘´1 ωpg ´1pxqq, (

where σ k is the representation of M " SOpnq on Λ k " Λ k pR n q b C. The presentation (5.2) is just the noncompact realization of a principal series representation (cf [START_REF] Knapp | Representation Theory of Semisimple Groups[END_REF]). This yields the identification

π k λ » Ind G P pσ k b χ λ`k b 1q, (5.3)
where, for λ P C, we denote by χ λ the character of A giver by χ λ pa t q " e λt . We pin down that the representation σ k is an irreducible representation of SOpnq, except for the case where n is even and k " n 2 (see [START_REF] Fischmann | Conformal symmetry breaking differential operators on differential forms[END_REF][START_REF] Ikeda | Spectra of eigenforms of the Laplacian on S n and P n pCq[END_REF]), but for our purpose, this makes no difference.

The intertwining Knapp-Stein operators play a crucial role in semi-simple harmonic analysis. In the present situation they are given as follows (see [START_REF] Fischmann | A family of Riesz distributions for differential forms on Euclidean space[END_REF]),

I k λ ωpxq " ż R n R k ´2n`2λ px ´yq ωpyqdy, ω P SE k pR n q
where R k ´2n`2λ is the tempered distribution defined by (3.3). In the notations of the previous section, I k λ is nothing but the convolution operator J k s , defined in (3.12), with s " ´2n `2λ. The operators I k λ , defined first for n 2 ă Re λ ă n so that the integral converges for ω P SE k pR n q, can be analytically continued to the complex λ-plane as a meromorphic family of convolution operators by tempered distributions, thus mapping SE k pR n q into S 1 E k pR n q. The following (a priori formal) relation holds for any g P G :

I k λ ˝πk
λ pgq " π k n´λ pgq ˝Ik λ .

(5.4)

The relation is first proved when n 2 ă Re λ ă n, and shown (using the covariance property (4.5) of }x ´y} 2 ) to be valid for forms in DE k pR n q and g in an appropriate small neighborhood of the neutral element of G. The corresponding infinitesimal form of the intertwining relation (5.4) is

I k λ ˝dπ k λ pXq " dπ k n´λ pXq ˝Ik λ , (5.5)
for X P g and valid on SE k pR n q. By analytic continuation, it is then extended meromorphically in λ.

The following property will be required later on.

Proposition 5.1. For generic λ, the operator I k λ is injective on the space SE k pR n q. Proof. Since I k λ is the convolution product with the tempered distribution R k ´2n`2λ , then saying that I k λ is injective is equivalent to prove that (for generic λ) the multiplication operator by the Fourier transform F `Rk ´2n`2λ ˘is injective on SE k pR n q. Recall from (3.4) that, generically in λ, we have F `Rk ´2n`2λ ˘pxq " 2}x} n´2λ´2 ´pn ´k ´λqι x ε x `pλ ´kqε x ι x ¯.

Recall also from Section 2 that pι x ε x q 2 " }x} 2 ι x ε x , pε x ι x q 2 " }x} 2 ε x ι x , pι x ε x qpε x ι x q " 0, pι x ε x q 2 `pε x ι x q 2 " }x} 4 Id Λ .

(5.6)

If we assume, in addition, that n ´k ´λ ‰ 0 and λ ´k ‰ 0, then the identities (5.6) imply that for x ‰ 0 the operator pn ´k ´λqι x ε x `pλ ´kqε x ι x is invertible. Let ω P SE k pR n q and λ as above so that FpR k ´2n`2λ qpxqωpxq " 0. As FpR k ´2n`2λ qpxq is invertible for x ‰ 0, it follows that ωpxq " 0 for x ‰ 0, and therefore ω " 0 on R n .

The covariance property of the source operator

We can now start to give the conformal interpretation of Theorem 3.3. On one hand, we saw in the previous section that the convolution operators J k s are related to the Knapp-Stein intertwining operators. So it remains to understand the conformal property of the multiplication by }x ´y} 2 .

The group G " SO 0 p1, n `1q acts rationally on the space R n ˆRn by the diagonal extension of its action on R n , and hence on E k, pR n ˆRn q, giving a realization of the tensor product representation π k λ b π µ . More explicitly π k λ b π µ pgq ωpx, yq " Ωpg ´1, xq λ Ωpg ´1, yq µ L g´1 ω px, yq. Define the multiplication operator M : E k, pR n ˆRn q ÝÑ E k, pR n ˆRn q by M ωpx, yq " }x ´y} 2 ωpx, yq.

The covariance property (4.5) of }x ´y} 2 immediately implies the following result. Here again the relation is valid when applied to differential forms in DE k, pR n ˆRn q and g in a small enough neighborhood of the neutral element of G. The rigorous infinitesimal version reads as follows : For every X P g, we have

M ˝d`π k λ b π µ ˘pXq " d `πk λ´1 b π µ´1 ˘pXq
k, λ,µ " 16}x ´y} 2 r l k,λ b r l ,µ ´32 n ÿ j"1 px j ´yj q ! p2λ ´n `2q r ∇ k,λ,j b r l ,µ ´p2µ ´n `2q r l k,λ b r ∇ ,µ,j ) ´32p2λ ´n `2qp2µ ´n `2q n ÿ j"1 r ∇ k,λ,j b r ∇ ,µ,j ´32p2µ ´n `2qpµ `1qpµ ´ qpµ ´n ` q r l k,λ b Id E ´32p2λ ´n `2qpλ `1qpλ ´kqpλ ´n `kq Id E k b r l ,µ
, where r l k,λ " pλ ´n `kqpλ ´k `1q δ d `pλ ´n `k `1qpλ ´kq d δ r ∇ k,λ,j " pλ ´n `kqpλ ´k `1qB x j ´pλ ´kqε j δ ´pλ ´n `kq δ ι j , and similarly for r l ,µ and r ∇ ,µ,j . We will call F k, λ,µ the source operator.

Theorem 3.3 can now be reformulated as follows.

Theorem 6.2. The differential operator F k, λ,µ acts on E k, pR n ˆRn q and satisfies κ

k, λ,µ M ˝`I k λ b I µ ˘" `Ik λ`1 b I µ`1 ˘˝F k, λ,µ
, where κ k, λ,µ " 16pλ ´k `1qpλ ´n `k `1qpµ ´ `1qpµ ´n ` `1q.

Moreover, we have the following covariance property of the source operator F k, λ,µ . Theorem 6.3. For all λ, µ P C and for any X P g, we have

F k, λ,µ ˝d`π k λ b π µ ˘pXq " d `πk λ`1 b π µ`1 ˘pXq ˝F k, λ,µ . Proof.
In the light of Theorem 6.2, Proposition 6.1 and the identity (5.5), we have

`Ik λ`1 b I µ`1 ˘˝F k, λ,µ ˝d`π k λ b π µ ˘pXq " κ k, λ,µ M ˝`I k λ b I µ ˘˝d `πk λ b π µ ˘pXq " κ k, λ,µ M ˝d`π k n´λ b π n´µ ˘pXq ˝`I k λ b I µ " κ k, λ,µ d `πk n´λ´1 b π n´µ´1 ˘pXq ˝M ˝`I k λ b I µ " d `πk n´λ´1 b π n´µ´1 ˘pXq ˝`I k λ`1 b I µ`1 ˘˝F k, λ,µ " `Ik λ`1 b I µ`1 ˘˝d `πk λ`1 b π µ`1 ˘pXq ˝F k, λ,µ . Now, use Proposition 5.1 to finish the proof.
As G is connected, the infinitesimal covariance property of the operator F k, λ,µ implies first its covariance under the group G, that is

F k, λ,µ ˝`π k λ b π µ ˘pgqω " `πk λ`1 b π µ`1 ˘pgq ˝F k, λ,µ ω,
for ω P DE k, pR n ˆRn q and g P G is such that g ´1 is defined on a neighborhood of the support of ω.

Remark 6.4. Let us mention that improving on our results, it is possible to construct a differential operator r F k, λ,µ on S n ˆSn , which admits F k, λ,µ as its local expression on S n tp0, 0, . . . , 0, ´1qu ˆSn tp0, 0, . . . , 0, ´1qu » R n ˆRn and which is covariant for G with respect to pρ k λ b ρ µ , ρ k λ`1 b ρ µ`1 q. We skip the proof as this corresponds to general standard results. See for instance Section 8.2 in [START_REF] Ben Saïd | Conformally covariant bi-differential operators on a simple real Jordan algebra[END_REF] or Fact 3.3 in [START_REF] Kobayashi | Conformal symmetry breaking operators for differential forms on spheres[END_REF].

It is possible to compose the source operators, to yield more covariant differential operators. Indeed, for arbitrary integer m ě 1, we set

F k, λ,µ;m :" F k, λ`m´1,µ`m´1 ˝¨¨¨˝F k, λ`1,µ`1 ˝F k, λ,µ . Then F k, λ,µ;m intertwines the representations π k λ b π µ and π k λ`m b π µ`m .
The following statement gives another approach to these operators.

Conformally covariant bi-differential operators on differential forms

From the source operators F k, λ,µ , one can construct covariant bi-differential operators under the action of G " SO 0 p1, n `1q. First, introduce the restriction map res : E k, pR n ˆRn q ÝÑ C 8 pR n , Λ k b Λ q defined by pres ωqpxq " ωpx, xq, where C 8 pR n , Λ k b Λ q denotes the space of complex-valued smooth functions on R n with values in Λ k bΛ . Let G acts on E k, pR n ˆRn q by π k λ bπ µ . Using the realization of π k λ and π µ as principal series representations (see (5.3)), the following result is immediate. As a representation of M " SOpnq, the representation σ k b σ is in general not irreducible. Let Γ be a minimal invariant subspace of Λ k b Λ under the action of SOpnq. Let σ Γ be the corresponding irreducible representation of SOpnq on Γ and let p Γ be the orthogonal projection on Γ. Define the map res Γ by res Γ " p Γ ˝res .

We can refine the previous proposition as follows. In some cases, it is possible to give an explicit expression for these covariant bidifferential operators. For instance, assume that 0 ď k ` ď n, then the representation Λ k` appears in the decomposition of the tensor product Λ k b Λ with multiplicity one and the projection (up to a normalization factor) is given by p Λ k` pω b ηq " ω ^η.

For m " 1, the bi-differential operator B k, ;Γ λ,µ;1 is given by B k, ;Γ λ,µ;1 pω b ηqpxq " ´32 ! p2µ ´n `2qpµ `1qpµ ´ qpµ ´n ` q r l k,λ ωpxq ^ηpxq `p2λ ´n `2qp2µ ´n `2q n ÿ j"1 r ∇ k,λ,j ωpxq ^r ∇ ,µ,j ηpxq `p2λ ´n `2qpλ `1qpλ ´kqpλ ´n `kqωpxq ^r l ,µ ηpxq

) (7.1)
If in addition k " " 0, i.e. ω, η P C 8 pR n q, then B 0,0;C λ,µ;1 pω b ηqpxq " ´64pλ `1qpλ ´nqpµ `1qpµ ´nq

!

Proposition 6 . 1 .

 61 The operator M satisfies M ˝`π k λ b π µ ˘pgq " `πk λ´1 b π µ´1 ˘pgq ˝M .

Proposition 6 . 5 .

 65 For any integer m ě 1, we haveκ k, λ,µ;m M m ˝`I k λ b I µ ˘" `Ik λ`m b I µ`m ˘˝F k, λ,µ;m , where κ k, λ,µ;m " κ k, λ`m´1,µ`m´1 ¨¨¨κ k, λ`1,µ`1 κ k, λ,µ and M m " M ˝¨¨¨˝M , m-times. Proof. For m " 1, this is Theorem 6.2. Assume m ě 2. By induction on m we have κ k, λ,µ;m M m ˝`I k λ b I µ ˘" κ k, λ,µ;m M ˝M m´1 ˝`I k λ b I µ " κ k, λ`m´1,µ`m´1 M ˝pI k λ`m´1 b I µ`m´1 q ˝F k, λ,µ;m´1 " `Ik λ`m b I µ`m ˘˝F k, λ`m´1,µ`m´1 ˝F k, λ,µ;m´1" `Ik λ`m b I µ`m ˘˝F k, λ,µ;m .

Proposition 7 . 1 .

 71 For any pλ, µq the map res intertwines the representations π k λ b π µ and Ind G P `pσ k b σ q b χ λ`µ`k` b 1 ˘.

Proposition 7 . 2 .

 72 For any pλ, µq the map res Γ intertwines the representations π k λ b π µ and Ind G P `σΓ b χ λ`µ`k` b 1 ˘. Now define the bi-differential operators B k, ;Γ λ,µ;m : E k, pR n ˆRn q ÝÑ C 8 pR n , Γq by B k, ;Γ λ,µ;m :" res Γ ˝F k, λ,µ;m , where C 8 pR n , Γq denotes the space of smooth functions on R n with values in Γ Ă Λ k bΛ . Theorem 7.3. The operator B k, ;Γ λ,µ;m is a bi-differential operator covariant with respect to π k λ b π µ and Ind G P pσ Γ b χ λ`µ`k` `2m b 1q.

  ,s dι j

									¯b ´α	4
									,t δd `β ,t dδ
	n ÿ j"1 Bx Finally, the third term of the operator E k, B ´sα k,s s,t in (3.19) can be rewritten as:
	2	n ÿ j"1	px j ´yj q ´αk,s δd `βk,s dδ	¯b ´tα ,t	By j B	`γ ,t ε j δ `δ ,t dι j	4
	n ÿ j"1 By It is worthwhile noting that by the anti-commutator laws in (2.7) we may rewrite the ´´2pn ´2kqpε j δ ´dι j q `αk,s B Bx j B ¯b ´tα ,t
	fourth term of the operator E k, s,t in (3.19) as:
		2	n ÿ j"1	´sα k,s	B Bx j	`γk,s ε j δ `δk,s dι j	¯b ´tα ,t	B By j	`γ ,t ε j δ `δ ,t dι j

j `γk,s ε j δ `δk,s dι j ¯b ´´2pn ´2 qpε j δ ´dι j q `α ,t B By j 2`n sκ k,s `pn ´kqγ k,s `kδ k,s ˘Id E b ´α ,t δd `β ,t dδ ¯. j `γ ,t ε j δ `δ ,t dι j 2`n tκ ,t `pn ´ qγ ,t ` δ ,t ˘´α k,s δd `βk,s dδ ¯b Id E .

Choose e n`1 " p0, 0, . . . , 0, 1q as origin on the sphere S n , and let rp1, e n`1 qs be the corresponding isotropic ray. The stabilizer of rp1, e n`1 qs is a parabolic subgroup P of G, which has the Langlands decomposition P " M AN with

Denote by N the opposite nilpotent subgroup,

The origin e n`1 on the sphere S n corresponds to the point 0 " p0, 0, . . . , 0q in R n , and hence the parabolic subgroup P is the stabilizer of 0. The group M » SOpnq acts on R n by its natural action and A acts on R n by a t pxq " e ´tx, a t P A.

The group N acts on R n by translations, n y pxq " x `y, n y P N .

The explicit action of N on R n (which is rational) will not be needed, but it is easily verified that Dn y p0q " Id n , n y P N.

Up to a closed subset of null Haar measure, the group G is equal to N ˆM ˆA ˆN . The corresponding decomposition of g P G is g " npgqmpgqa tpgq npgq.

An elementary computation gives

Dgp0q " Ad mpgqp0q |n ˝Ad a t pgq |n " e ´tpgq mpgq, (4.2)


Robert Halm

Report on RF Properties of a New Commercially Available Superhydrophobic Coating

. We performed an experiment to characterize the RF properties of this coating, in order to begin assessment of this product's suitability for use on various radar antennas. We measured the RF characteristics of a sample radome using a near-field test range both before and after it was coated with Neverwet, and compared the results in order to study the properties of the coating. Our results suggest that Neverwet may be almost perfectly transparent to RF, even at higher frequencies, and may cause no significant loss of energy or change in gain pattern when it is applied to a radome. However, the size of our experiment was very small due to funding and practical limitations, and we interpret these results as an impetus to conduct more thorough and narrowly aimed testing rather than as a basis on which to adopt this coating on any radar system.

Introduction:

Corrosion is a major technical issue for the US Navy, due largely to the constant exposure of many systems to a moist saltwater environment, and radar systems are no exception. One approach to the prevention of corrosion in marine and other highhumidity environments is the use of superhydrophobic coatings. These coatings drastically reduce the adhesion of water and some subset of other liquids to the surface which they are used to treat. This can not only reduce standing liquid water on these surfaces from rain, spray, or condensation, due to its increased tendency to bead up and roll off, but in fact it can be effective in preventing the chemical wetting of the treated surface; even when a droplet of water is in physical contact or near-contact with a treated surface, the superhydrophobic coating can prevent the chemical interaction of the surface with the water or the solutes it carries, hence preventing corrosion. As an additional bonus, this treatment can also substantially inhibit icing, which may be of use on a variety of systems.

To date, coatings with very strong superhydrophobic properties have been produced and tested in lab settings, but these coatings have suffered the drawback of being very physically delicate and easily destroyed or removed from their surface, while more robust superhydrophobic coatings which are commercially available and practical for use in non-lab applications (automotive waxes and polishes, for example) have much weaker hydrophobic properties [START_REF] Vinnod | Highly Durable, Superhydrophobic and Oleophobic Coatings[END_REF][START_REF] Hosono | Superhydrophobic Perpendicular Nanopin Film by Bottom-Up Process[END_REF][START_REF] John | Polymer and Rubber Preservative[END_REF]. Neverwet is a new superhydrophobic coating produced by Ross Nanotechnologies of Leole, PA, which is both strongly superhydrophobic and durable as a surface coating. This product will (as of 2012) soon be available commercially, and we have been asked to evaluate it for possible use on radar systems.

In this experiment, we set out to evaluate the RF properties of Neverwet.

Obviously, if this product is highly absorptive of RF, or causes dramatic phase shifts, or has other unusual RF properties, we would be limited in its applicability to some components and component surfaces of radar systems, whereas if it is essentially RFtransparent in thin layers, it can be applied broadly on most or all exterior or interior surfaces of a system where corrosion has been a problem, without regard to RF impact.

We note that other properties, such as the coating's resistance to heat or extreme cold or its color, do need to be taken into account in determining its overall applicability, but Ross Nanotech is currently (as of 2012) performing research into many of these areas, so pending their results, we concluded that the most logical topic for us to approach first would be Neverwet's RF behavior.

Procedure:

We procured a sample of radome, as well as a 4.5-10.0 GHz wide band horn, and constructed a simple wooden fixture to support our radome sample. We then set up the radome on its fixture and the wide band horn on a tripod immediately behind it (less than 0.5" of clearance between the horn and the radome) so that the horn was directed toward the scan plane, through the radome. With an OEWG WR112 probe equipped on the scanner, we then took scans of the horn's near-field gain pattern with and without the radome in place (but with the radome fixture in place in both cases). We partially covered the tripod and radome fixture with anechoic foam, in order to minimize reflection, but this was mostly a measure to err on the side of caution, since neither of these was anywhere near the main beam of the horn, nor could they be in the signal path without at least two reflections off of the anechoic walls of the range chamber in which the testing was conducted, each of which should reduce the signal strength by several orders of magnitude. We took our initial scans with and without the uncoated radome on September 14 th , then shipped the radome to Ross Nanotechnology for coating. We received the coated radome back on November 9 th , and retested it in the same way, on the 16 th and 18 th , also taking scans without the radome in place on each of these two days. On each day, the setup of the horn and its supporting tripod, and the radome and its fixture, along with the alignment of the horn and the rest of the setup, were done according to a written procedure, which is attached as Appendix A, in order to ensure that positioning of the elements remained as constant as possible from one scan to the next.

The data were collected using a proprietary software program developed by NSI (the manufacturer of the measurement apparatus), which is installed on the controlling machine in the near-field range, then exported to .txt formatted data files. These files were then analyzed MATLAB using code. This code is attached as Appendix C. As a reasonability check, plots produced by this code were compared to plots produced by NSI's program on the near-field controlling machine, and the values for data within MATLAB were checked against values of data manually read from the .txt data files.

Analysis:

Each scan produced a .txt data file the main body of which is a 201-by-201-by-4 matrix. The matrix records the phase, amplitude, and angular coordinates of a 120degree by 120-degree angular volume of the far field, which the NSI program computes from the measured near-field data. This far field is normalized to have its main peak at zero decibels, and amplitude is represented in decibels (dB). The actual, unnormalized amplitude of the main peak is recorded in the file header of each data file. For each scan, this far-field data covers the same angular volume relative to the horn and radome. All those data are imported into our MATLAB code, but we focused on the normalized amplitude data in the matrix, and on the beam peak magnitude.

For each day, we calculated the difference in actual beam peak magnitude, and also calculated the difference in amplitude at each point in the far field between the scans with and without radome. These differences form a matrix which represents the effect of the radome on far-field amplitude (hereafter the "difference matrix"). This difference matrix can be viewed as a 3-dimensional plot, or 2-dimensional cuts can be plotted. In addition to examining plots of this difference matrix, we did several calculations to create metrics of the matrix, such as average value, and average absolute value. By using both of these approachesvisual inspection of plots and mathematical comparison of calculated valueswe can make a well-rounded evaluation of the impact the radome had on the far field on each day: visual inspection of plots allows us to look for anomalies or problematic features like sidelobes or new patterns in the far field, while the use of metric values allows us to make a very precise evaluation of the overall, average effect of the radome.

We used a total of four metric values calculated for each day:  Average Value: the average of all points of the difference matrix.

 Average Absolute Value: the average of the absolute values of each point of the difference matrix.

 Cropped Average Value: the average of each point of the difference matrix within fifteen degrees of azimuth and fifteen degrees of elevation of the main beam peak.

 Cropped Average Absolute Value: the average of the absolute values of each point of the difference matrix within fifteen degrees of azimuth and fifteen degrees of elevation of the main beam peak. This cropping of the matrix to within fifteen degrees of the beam peak helps to focus on the portion of the data which is not only more essential to antenna performance, but, more importantly, which is received at a higher amplitude, and hence is less obscured by electromagnetic noise and other forms of experimental error. In separate scans, both with and without the actuating motors of the scanner in operation, we measured the noise floor of the chamber at our operating frequency of 9.5 GHz, and found it to be in the region of -70 dB to -80 dB. We note that this means that well away from the beam peak, where the measured amplitude is in the region of -50 dB, on the order of one percent error may be attributable to background and instrumental noise.

The use of cropped and uncropped metrics alongside one another allows us to ensure that we can accurately evaluate the least noisy and most relevant data, while also looking out for the possibility of systematic biases near the edges of the gain pattern. Also, the use of beam peak magnitude, normalized average, and normalized average absolute value allow us to look for changes in total loss, changes in pattern shape, and changes in the noisiness of the pattern, respectively. Between these three values (actually five, due to the independent use of cropped and uncropped data to calculate separate values) we should be able to see any significant impact which the coating has had on the RF characteristics of the radome. Table 1, below, shows each of these five values for each of the three days on which we performed scans. Table 1 also shows the average apparent effect of the Neverwet coating on each metricthe difference between the impact of the radome before it was treated and the average impact it had after being treated. We will discuss the significance of each of these values in more depth below, but first, we will discuss the interpretation of the plots shown immediately below as figures 1 through 6.

Discussion:

These figures show amplitude of the calculated far field in horizontal and vertical cuts through zero degrees at 9.5 GHz, with each plot showing a red curve taken from the pattern when the radome was in place, and a blue curve from the pattern which was measured without the radome. We must keep in mind that all these far fields have been normalized to have the main beam peak define zero decibels, so changes in overall signal strength will not appear on these plots. This gives us the advantage, however, of making it very easy to see changes in the shape of the pattern (and the actual change in overall strength is recorded in Table 1 below, so this information is not lost). In each plot, the disparity between the two curves represents the impact of the radome, so comparing that impact between the days before and after the radome was coated should show the effect of the coating. These days are plotted on separate axes due to day-to-day drift in the equipment and environment.

Our first observation is that the radome has very little effect on the pattern at all.

There is often some change toward the edges of the cut, particularly in the vertical cuts, but these differences occur at amplitudes more than 30 dB below the main peak, i.e., the power measured where these changes are occurring is less than 0.1% of the power at the main beam peak. Such weak signals are likely to cause greater experimental error, so we should not necessarily interpret these changes as significant alteration of the pattern due to the radome. One possibly genuine impact of the radome is seen at the edges of the horizontal cut, where the amplitude measured with the radome in place is consistently higher by one to three decibels than the amplitude measured without the radome in place.

This may be due to horizontal scattering caused by the edges of the radome, since the radome sample we used was only slightly wider than the horn in azimuth, and hence diffraction and other edge effects at the extremes of azimuth are possible. We note, however, that this is constant whether or not the radome has been coated with Neverwet, so these effects are not of interest to us. This very low impact on the pattern due to the radome is helpful to us, because it will likely make it easier for us to observe or detect any changes in the pattern due to the Neverwet layer.

Our next observation is that not only the horizontal edge effects mentioned above, but the impact of the radome on the pattern overall is very consistent across the three days, and does not show any appreciable difference between the days during which the radome had already been coated and those on which it was uncoated. At first, we might be slightly concerned about the vertical cut comparison on November the eighteenth, as the difference between the curves with and without radome seems to be greater here than in most of the other comparisons, but if we look closely we can see that the entire curve with the radome has been shifted slightly to the right (i.e., toward positive elevation) as compared to the curve without radome, and that if this slide is corrected, the patterns are very similar. This is most likely due to a slight misalignment of the horn in elevation on one of the two scans taken that day, and not due to any effect of the radome. By examining these curves, we can see that there is no particular anomaly or specific effect on the shape of the far field pattern caused by coating the radome with Neverwet, so we can now continue on to examining the metrics we discussed above, and which are shown below in Table 1.

First, let us examine the difference in the actual, unnormalized amplitude of the beam peak. Since we have seen from the plots above that the shape of the pattern has not changed from one scan to another, either due to the presence of the radome or from day to day, by any significant amount, we can see that this value is in fact a proxy for the signal strength across the whole far field pattern; if one of the beam peaks were four decibels higher, for example, we would know that the entire pattern was about four decibels stronger, since its shape has remained the same. We can see that the changes in amplitude are fairly random at first glanceon the eighteenth, the coating seemed to make the radome more lossy (a negative number of this value indicates a lower amplitude with the radome in place than when it was absent, while a positive value indicates the opposite), while on the sixteenth the coating actually seemed to make the radome less lossy. If we consider two things, however, the significance of these values will come into focus. First, the actual differences in amplitude caused by the radome, either with or without the coating, are around one tenth of a decibel, which is a very small amount. Secondly, on two out of the three days, the calculated far field actually has a higher beam peak amplitude with the radome in place than without it, albeit slightly, which is physically very unlikely (especially since the thickness of the radome is about 140% of a quarter wavelength). Looking at these facts together, we can conclude with some security that all the observed variability, that is, the calculated difference in beam peak amplitude with and without the radome, is so small that it is in the same range as the total sensitivity of the measurement and calculation process. In short, no effect on beam peak amplitude can be detected through the influence of measurement and calculation error. We can also say that the total sensitivity, which is on the order of a few standard deviations, or a few tenths of a decibel, is sensitive enough that it would easily be able to pick up changes as large as just one decibel, so we can say with a high probability that the coating does not increase the lossiness of the radome it is coating by more than a few tenths of a decibel.

Next, let us consider the cropped and uncropped averages of the difference matrices. A significant effect on this value would mean that adding the superhydrophobic coating caused the radome to change the shape of the far field pattern in a specific way: Since the main beam is always normalized to zero decibels in the data matrices from which these values are calculated, the average value of the difference matrix is driven by how much the rest of the pattern increased or decreased on average as compared to the beam peak. That is to say, a negative average value of the difference matrix means that the radome is lowering the rest of the pattern compared to the peakmaking the peak sharper or higher. By contrast, a positive value would mean that the radome was flattening the pattern or raising it compared to the peakmaking the peak lower or broader. Likewise, if we see a negative impact on this value due to the Neverwet coating, it means that the coating is causing the radome to have an increased sharpening effect on the pattern, while the opposite would mean that the coating is causing the radome to have an increased flattening effect on the pattern. What this value measures, then, very precisely, is the overall shape or slope of the pattern. When we look at the values for this metric, we see a somewhat muddled picture.

Both with and without cropping, we see that the coating, on average, seems to have a slight tendency to increase the sharpening of the pattern due to the radome (it tends to decrease the value of this metric), and in each case we see that this effect seems to be on the order of ten percent of the measured flattening which the radome tends to exert on the pattern by itself. However, although we seem to see substantial sharpening due to Neverwet on the 18 th , the 16 th actually seems to show a slight flattening as compared to the uncoated radome. Finally, we must note that the actual amplitude we are talking about is on the order of one percent of one decibel in the main beam, and perhaps half a decibel in the more error-prone fringes of the pattern. Even considering that this number is an average over a number of data points, such a value is small compared to the error of the measurement and calculation process. So, as with the beam peak magnitude, it seems most likely that the coating does not have any effect on the flatness or sharpness of the pattern which we would call significant, and certainly no impact that approaches even one decibel on average, at least anywhere near the main beam.

Finally, let us consider the average absolute value of the difference matrix, again calculated separately with and without cropping. If there were no measured impact on average value, the previous metric we discussed, then this metric would be a measurement of noisiness introduced by the radome, since if the difference varies randomly around zero across the pattern, this will show no change in the average but will increase the average of the absolute value. However, we must keep in mind that, because the average value was calculated to change by a trivial amount, this metric is slightly biased toward the positive. (This metric can only be biased toward the positivebecause it is an absolute value, both negative and positive changes in the average will cause it to increase.)

At first glance, again, there appears to be a significant effect, even taking the biasing into account, because the increases in average absolute value for each method of calculation are greater than the change in average value given for that method.

However, if we look at the values from the individual days, we will see that what is shown is not a very consistent pattern of impact. When considering the cropped values, both the sixteenth and the eighteenth seem to show similar increases over the baseline value from the fourteenth of September, however, the uncropped values appear very different. According to the values calculated without cropping, the sixteenth showed a value which was actually slightly lower than the baseline in September, while the eighteenth showed a very substantial increase by comparison. Finally, once again, we can see that the effect of the coating is on the order of only a few percent of a decibel in the main peak, and around half a decibel including the whole pattern. Therefore, the most likely cause of this apparent difference is error, and we can say with some confidence that an effect as large as noisiness with a variance on the order of a decibel in the region of the beam peak would have been detected.

Conclusions:

Based on both visual examination of the curves taken from the calculated far field of each scan, and on an analysis of the metrics we calculated, there seems to be no substantial impact of any kind on the far field gain pattern of our test setup which could be securely attributed to the Neverwet coating on our test radome. In all likelihood, this means that there was no non-trivial effect of any kind on the RF energy passing through it; random variations or non-uniform biases on the phase or amplitude of the signal would have manifested changes in the far field. Furthermore, since we used a high test frequency, which should bring out both lossiness and phase impacts of transmission through the coating, we can be even more confident that no such effect will appear at any lower frequency. This being said, the quantity of data, and hence our confidence in our conclusions, was substantially limited in this experiment. Our next step should be to consider RF testing of a more thorough scale, with a specific radar system in mind.

Additionally, we should assess our requirements for physical and environmental durability for such a coating in different applications, and compare these requirements against the research conducted by Ross Nanotechnology in order to determine what additional testing, if any, should be conducted to address weatherability concerns before this coating is put into trial or full use.

Update -Related Research 2012-2018:

In the roughly six years since the original publication of this paper, published research into the topic of superhydrophobic coatings suitable for radar applications has been unfortunately minimal. Wang et al. [START_REF] Wang | A simple, one-step hydrothermal approach to durable and robust superparamagnetic, superhydrophobic and electromagnetic wave-absorbing wood[END_REF] have examined a possible treatment for wood or other organic substances as potential composite matrices which imparts a degree of superhydrophobicity, while also measuring the effect on RF absorption, but this obviously does not pertain directly to radar applications, and the present authors are unaware of any more directly relevant research in the same interval. Accordingly, we believe the research described in this paper still has relevance to the field.

adelta_1118=abs(H_1118-C_1118);adelta_1118=adelta_1118(:,:,3); adelta_1118a=abs(H_1118a-C_1118a);adelta_1118a=adelta_1118a(:,:,3); %These four lines calculate the absolute value of the difference %matrix.

ameandelta_0914=mean(mean(adelta_0914)) ameandelta_1116=mean(mean(adelta_1116)) ameandelta_1118=mean(mean(adelta_1118)) ameandelta_1118a=mean(mean(adelta_1118a)); %These four lines calculate the average of the absolute value calculated %above.

wdelta_0914=(R_0914(:,:,3)-C_0914(:,:,3))./((R_0914(:,:,3)+C_0914(:,:,3))./2); wdelta_1116=(H_1116(:,:,3)-C_1116(:,:,3))./((H_1116(:,:,3)+C_1116(:,:,3))./2); wdelta_1118=(H_1118(:,:,3)-C_1118(:,:,3))./((H_1118(:,:,3)+C_1118(:,:,3))./2); wdelta_1118a=(H_1118a(:,:,3)-C_1118a(:,:,3))./((H_1118a(:,:,3)+C_1118a(:,:,3))./2); %These four lines calculate the amplitude-weighted difference in amplitude %at each point in the normalized calculated far field. The difference at %each point is divided by the mean of the values of the two scans for that %point in the far field. This helps to illustrate which differences caused %by the interposition of the radome into the beam are significant, i.e. %occur in the main beam, and which are not, i.e. occur well away from the %main beam and many decibels down, and hence are more subject to noise and %experimental error.

meanwdelta_0914=mean(mean(wdelta_0914)); meanwdelta_1116=mean(mean(wdelta_1116)); meanwdelta_1118=mean(mean(wdelta_1118)); meanwdelta_1118a=mean(mean(wdelta_1118a)); %These four lines compute the mean value of the amplitude-weighted %differences. This value will be our primary metric of the effect of the %interposition of the radome into the beam on each day.

ameanwdelta_0914=mean(mean(abs(wdelta_0914))); ameanwdelta_1116=mean(mean(abs(wdelta_1116))); ameanwdelta_1118=mean(mean(abs(wdelta_1118))); ameanwdelta_1118a=mean(mean(abs(wdelta_1118a))); %These four lines compute the mean of the absolute value of the %amplitude-weighted differences. Between this value and meanwdelta, any %substantial change in the pattern is likely to be captured.

  meandelta_0914c=mean(mean(delta_0914(75:125,75:125))) meandelta_1116c=mean(mean(delta_1116(75:125,75:125))) meandelta_1118c=mean(mean(delta_1118(75:125,75:125))) meanwdelta_0914c=mean(mean(wdelta_0914(75:125,75:125))); meanwdelta_1116c=mean(mean(wdelta_1116(75:125,75:125))); meanwdelta_1118c=mean(mean(wdelta_1118(75:125,75:125))); ameandelta_0914c=mean(mean(abs(delta_0914(75:125,75:125)))) ameandelta_1116c=mean(mean(abs(delta_1116(75:125,75:125)))) ameandelta_1118c=mean(mean(abs(delta_1118(75:125,75:125))))%These nine lines compute equivalents to the meanwdelta and ameanwdelta %values, but limited to the part of the far field within 15 degrees of %azimuth and 15 degrees of elevation of the main beam, as well as a value %meandelta_xxxxc, which calculates the average value of the change between %scans with and without radome without weighting by division.

TABLE 1

 1

		Difference of
		Unnormalized
		Amplitudes		Average		Cropped Average
		at the Beam	Average Value	Absolute Value	Cropped Average	Absolute Value
		Peak	"meandelta_xxxx"	"ameandelta_xxxx"	"meandelta_xxxxc"	"ameandelta_xxxxc"
	September 14th	0.1019	-0.1703	0.7939	0.1104	0.1707
	November 16th	0.2722	-0.0803	0.6864	0.1604	0.2580
	November 18th	-0.0069	-1.3287	2.2570	0.0274	0.2306
	Apparent Effect of
	Neverwet	0.0308	-0.5342	0.6778	-0.0166	0.0736

  designed to convert the formatted .txt data file output %by the small nearfield chamber into a matrix containing the phase and %magnitude, organized into the angular geometry of the predicted farfield %which these data represent. This function calls digitread.m, findany.m, %and locate.m directly. designed to convert the formatted .txt data file output %by the small nearfield chamber into a matrix containing the phase and %magnitude, organized into the angular geometry of the predicted farfield %which these data represent. This function calls digitread.m, findany.m, %and locate.m directly. used to convert a string representing a written number %into its numeric value. It can accomodate up to four digits. This %function does not call any other functions, either directly or indirectly.

		snfdata(j,i,1)=digitread(currentline(2:9)); for j=1:xpoints maxfarfield=digitread(currentline(coord:coord+8));
	snfdata(j,i,2)=digitread(currentline(12:19)); currentline=fgetl(fid); elseif v2==1
			snfdata(j,i,3)=digitread(currentline(22:29)); snfdata(j,i,1)=digitread(currentline(2:9)); t=1;
	snfdata(j,i,4)=digitread(currentline(32:39)); snfdata(j,i,2)=digitread(currentline(12:19)); elseif v3==1 && v4<2
		end v4=v4+1; snfdata(j,i,3)=digitread(currentline(22:29));
	end		snfdata(j,i,4)=digitread(currentline(32:39)); numbpoints=locate(currentline,'#pts')+7;
	fclose(fid); end snfimport_C_0914.m end xpoints=digitread(currentline(numbpoints:numbpoints+3));
	function [snfdata,maxfarfield]=snfimport_C_0914 elseif v3==1 && v4<3
	end fclose(fid); v4=v4+1;
	%README: % numbpoints=locate(currentline,'#pts')+7; snfimport_C_1116.m end ypoints=digitread(currentline(numbpoints:numbpoints+3));
	%This function is designed to convert the formatted .txt data file output end
	%by the small nearfield chamber into a matrix containing the phase and end
	%magnitude, organized into the angular geometry of the predicted farfield function [snfdata,maxfarfield]=snfimport_C_1116
	%which these data represent. This function calls digitread.m, findany.m, snfdata=zeros(xpoints,ypoints,4);
	%and locate.m directly. %README: % currentline=0; snfimport_H_1116.m for i=1:ypoints
	fullname=fullfile('D:\Documents and Settings\robert.halm\Desktop\Projects\PolyPaint\Neverwet\Matlab\Data'... ,'without superhydro no raydom 9-14-11.txt'); fid=fopen(fullname,'r'); %These three lines allow the user to select a %data file to open, which should be a .txt. t=0;u=0; v4=1; while t==0 currentline=fgetl(fid); tmpstr='Max far-field (global)'; v1=findany(currentline,tmpstr); function [snfdata,maxfarfield]=snfimport_H_1116 %README: % for j=1:xpoints currentline=fgetl(fid); snfdata(j,i,1)=digitread(currentline(2:9)); snfdata(j,i,2)=digitread(currentline(12:19)); snfdata(j,i,3)=digitread(currentline(22:29)); snfdata(j,i,4)=digitread(currentline(32:39)); %This function is fullname=fullfile('D:\Documents and Settings\robert.halm\Desktop\Projects\PolyPaint\Neverwet\Matlab\Data'... ,'R without superhydro without raydom 11-16-11.txt'); fid=fopen(fullname,'r'); %These three lines allow the user to select a %data file to open, which should be a .txt. t=0;u=0; end end fclose(fid); %This function is fullname=fullfile('D:\Documents and end
	tmpstr='Data starts on next line:'; v4=1; Settings\robert.halm\Desktop\Projects\PolyPaint\Neverwet\Matlab\Data'...
	v2=findany(currentline,tmpstr); ,'R with superhydro with raydom 11-16-11.txt');
	tmpstr='#pts'; while t==0 fid=fopen(fullname,'r');	%These three lines allow the user to select a
	v3=findany(currentline,tmpstr); if v1==1 currentline=fgetl(fid); tmpstr='Max far-field (global)'; %data file to open, which should be a .txt. digitread.m
	coord=locate(currentline,'Max far-field (global)')+25; v1=findany(currentline,tmpstr); t=0;u=0;
	maxfarfield=digitread(currentline(coord:coord+8)); tmpstr='Data starts on next line:'; v4=1;
	elseif v2==1 v2=findany(currentline,tmpstr); function [numb]=digitread(Astring)
	t=1; tmpstr='#pts'; while t==0
	elseif v3==1 && v4<2 v3=findany(currentline,tmpstr); currentline=fgetl(fid); %FUNCTION README:
	v4=v4+1; numbpoints=locate(currentline,'#pts')+7; xpoints=digitread(currentline(numbpoints:numbpoints+3)); elseif v3==1 && v4<3 if v1==1 coord=locate(currentline,'Max far-field (global)')+25; maxfarfield=digitread(currentline(coord:coord+8)); elseif v2==1 t=1; tmpstr='Max far-field (global)'; v1=findany(currentline,tmpstr); tmpstr='Data starts on next line:'; v2=findany(currentline,tmpstr); tmpstr='#pts'; %This function is digitreadx=zeros(1,9);
	v4=v4+1; elseif v3==1 && v4<2 v3=findany(currentline,tmpstr); sign=1;
	numbpoints=locate(currentline,'#pts')+7; v4=v4+1; if v1==1 digitreadpointfound=0;
	ypoints=digitread(currentline(numbpoints:numbpoints+3)); numbpoints=locate(currentline,'#pts')+7; coord=locate(currentline,'Max far-field (global)')+25; digitready=0;
	end	xpoints=digitread(currentline(numbpoints:numbpoints+3)); maxfarfield=digitread(currentline(coord:coord+8));
	end elseif v2==1 digitreadsize=size(Astring);digitreadsize=digitreadsize(2);
	elseif v3==1 && v4<3 t=1; if digitreadsize>8
	snfdata=zeros(xpoints,ypoints,4); v4=v4+1; elseif v3==1 && v4<2 digitreadlimit=8;
	currentline=0; numbpoints=locate(currentline,'#pts')+7; v4=v4+1; digitreadint=1;
	for i=1:ypoints ypoints=digitread(currentline(numbpoints:numbpoints+3)); numbpoints=locate(currentline,'#pts')+7; elseif digitreadsize>4
	for j=1:xpoints end xpoints=digitread(currentline(numbpoints:numbpoints+3)); digitreadlimit=digitreadsize;
	end	currentline=fgetl(fid); digitreadint=1;
	snfdata(j,i,1)=digitread(currentline(2:9)); elseif v3==1 && v4<3 else
	snfdata(j,i,2)=digitread(currentline(12:19)); snfdata=zeros(xpoints,ypoints,4); v4=v4+1; digitreadlimit=digitreadsize;
	snfdata(j,i,3)=digitread(currentline(22:29)); currentline=0; numbpoints=locate(currentline,'#pts')+7; if digitreadsize<=4
	snfdata(j,i,4)=digitread(currentline(32:39)); for i=1:ypoints ypoints=digitread(currentline(numbpoints:numbpoints+3)); digitreadint=5-digitreadsize;

Appendix A: PROCEDURE FOR RADOME TEST FOR THE EVALUATION OF "NEVERWET" SUPERHYDROPHOBIC COATING [OMITTED]

Appendix B: