Locally explicit fundamental principle for homogeneous convolution equations Текст научной статьи по специальности «Математика»

УДК 517.55

Locally Explicit Fundamental Principle for Homogeneous Convolution Equations

Alekos Vidras*

Department of Mathematics and Statistics University of Cyprus POB 20537, Nicosia 1678 Cyprus

Received 13.03.2019, received in revised form 20.05.2019, accepted 26.05.2019 In the present paper a locally explicit version of Ehrenpreis's Fundamental Principle for a system of homogeneous convolution equations f * fj =0, j = 1,. .. ,m, f £ E(Rn), fj G E'(Rn), is derived, when the Fourier Transforms jlj, j = 1,... ,m are slowly decreasing entire functions that form a complete intersection in Cn.

Keywords: fundamental principle, division formula. DOI: 10.17516/1997-1397-2019-12-4-466-474.

1. Fundamental principle for homogeneous convolution equations

Probably, the monograph [1] was the first in the field that illustrated that both, residues and integral representation formulas in several complex variable, are powerful tools allowing to provide solutions to seemingly untractable otherwise mathematical problems. In the present paper weighted integral representation formulas and different than in [1] realization of residues allow to approach from local point of view the Fundamental principle of convolution equations.

Recall that a pluri-subharmonic function p(z) on Cn is called a weight function ([3]) if it is satisfying the following conditions: i) p(z) > 0, ii) log(1 + ||z||) = O(p(z)), iii) if \\z — Z|| < 1, then p(Z) < Aip(z) + A2 for some constants Ai and A2.

Given a weight function p(z) we consider the corresponding subspace of vector space of entire functions A(Cn):

Ap = {f e A(Cn) : 3Af, Bf > 0 : f (z)| < AfeBfp(z)}.

If (fi,..., fm) are m entire functions on Cn, then L denotes the family of m-dimensional affine subspaces L of Cn, such that

ULe£ D{z e Cn : fi(z) =0, 1 < i < n}.

Following [3] we recall the following definition

Definition 1.1. The family (fi,..., fm) of m entire functions is slowly decreasing with respect to L if and only if there exist positive constants ei, Ci, Ki, K2 such that

1) for each L e L the set

O = {z e L : fi^ < ei exp(—Cip(z)), 1 < i < m} (1.1)

has relatively compact components,

2) If O as in (1.1) and z,Z belong to the same component of O, then _p(Z) < Kip(z) + K2.

*msvidras@ucy.ac.cy © Siberian Federal University. All rights reserved

Given a slowly decreasing family of functions (fi, f2,..., fm) with respect to the family L and the weight function p(z) one defines for a component G of O the open set

QG = {z G Cn : there exists Z G G so that \z — Z\ < e2 exp (—C2p(Z))}, (1.2)

for some positive constants e2,C2. Such an open set is called good. Keeping the values of the parameters ei,Ci from Definition 1.1 and the values of the parameters e2,C2 from (1.2) fixed, one obtains the family of open sets I = {QG}GcO. The family I is called good family. If both parameters ei and e2 decrease, while both parameters Ci and C2 increase, then the good family I' so produced is called a good refinement of I. A naturally defined refinement map p : I' —> I corresponds to any open set Q'G' G I' associated with the component G' of an open subset O' of a certain line (m-plane) L G L, the open set Q associated with the unique component G of the open set O such that G' c G. Thus, it is natural to consider the following definition ([3]).

Definition 1.2. A good family I is said to be almost parallel if and only if there exists its good refinement I' such that whenever Q0, Qi G I' and Q0 fl Qi = then Qo U Qi c p(Q0) f p(Qi), where p is the natural refinement map defined above.

We continue by recalling another necessary definition from ([3]).

Definition 1.3. We say that L is an analytic family of lines (m-planes) if and only if there is good family I associated to L with the following property: given Q G I with the associated line (m-plane) L G L there exist local analytic coordinates (s,t) on Q such that Qfl{(s, t) : t = 0} = Qf L and Q f {(s, t) : t = const} = Q f Lt for some Lt G L.

Furthermore, recall that for m ^ n, an m-tuple of holomorphic functions (fi,..., fm), fi : Cn —> C, 1 ^ i ^ m, defines a complete intersection in Cn if and only if the complex dimension of the analytic set of common zeroes Z of the functions fi is equal to n — m, that is,

dime Z = dime(ni^i<mZ/.) = dime ( fi^m {z G Cn : fi(z) = 0}) = n — m.

The following variation of Fundamental Principle for homogeneous system of convolution equation is formulated and proved in [3]

Theorem 1.1. Assume that ¡ij G E '(Rn), for j = 1,...,m, are slowly decreasing and form a complete intersection. That is, for p(z) = \9z\ +log(1 + \z\), z G Cn there exists an analytic , almost parallel family of lines such that fij for j = 1,... ,m, are slowly decreasing with respect to this family in Ap(Cn). Then, there exists a locally finite family of closed Vj, j G J and a partition of the index set J into finite subsets Jk together with partial differential operators df in z with analytic coefficients on the regular points of the set V = {z G Cn : fj (z) =0, j = 1,... ,m} satisfying: 1) UVk C V, 2) each function x —>• df (e-ixz), with z G V, is a solution to f * fj =0, j = 1,... ,m, where f (x) = f (—x), x G Rn, 3) to each solution f G E (Rn) of the system f * fj =0, j = 1,... ,m, there corresponds a family of Borel measures Vj, whose support is contained in the sets Vj and such that the series

f (x) = > A>. I di (e-lxz)dv3 (z)| (1.3)

is convergent in the space E(Rn).

2. Integral representation formula depending on parameter

For Ro > 0, we define the sequence closed balls with doubling radius property

Kl = B(0, 2 lR0) = {x G Rn : ||x|| < 2 lR0}, l G N

Then {Kl}leN is an increasing sequence of compact convex set s satisfying Kl c intKl+l and UKl = 1". Let also te}^ be a sequence of elements from D(1") such that suppxl C Kl+1, Xi = 1 in some neighborhood of Kl, l G N. The set U of all continuous functions on C" of the form

T(z) = sup (Si exp(l ln(2 + ||z||2) + HKl (Zz))),

leN

where {Sl}leN is a sequence of positive constants, is a LAU structure for the set E'(1") ( [9,10]).

We now turn to the localization of the solution f to the system f * Hj =0, j = 1,... ,m. Our purpose is to describe f explicitly in intKl, l G N. In order to do that we first test f against u G D(C"), with suppu c intKl for some l G N. Using the definition of the characteristic function xl we get from Plancherel theorem that

f (t)u(t)dt = (fxi)(-0H№-

(2.1)

The starting point is a weighted Koppelman integral representation formula for the holo-morphic function u0(R), where 0 G D(C") so that 0 = 1 on an open neighborhood of B(0,1) and suppO c B(0, 2). This weighted integral representation formula is constructed following the approach developed in [2,7]. Using it, we will produce a division formula involving the functions jij, j = 1,2,... ,m, [6]. In C" we have the following Heffer functions

k="

fij (z) - fij (Z) = gj,k (z,Z)(zk - Zk), j = 1,... ,n,

k=i

dßj

9j,k(z,z) = (Z + t(z - z))dt

dZk

and the corresponding Heffer (1,0)-form

k=n

gj

(z,z) = J2gjk(z,z)dZk, j = i,2,...:

k=i

Furthermore, following [6], within the spirit of constructions in [2,7], we introduce three pairs (Qi,Gi), (Q2,G2), (Q3,G3) of auxiliary functions defined as follows

Qi(z,z) = (Qn,Qi2,...,Qin)(z,z): D x D

1 j=m

m

Cn

Qii(z,z) = m E I j (z )I2A ^rzf = 1, m jj (z)

Gi(t)

-1 ]T (mt - j),

m! J- J-

j=m-1

j=0

where D c C" is a bounded domain with C2 boundary and A a complex parameter with sufficiently large positive real part, that is, KA >> 0. Similarly, we have

Q2(z,Z) = (Q21 ,Q22,...,Q2")(z,Z): D X D

Cn

where wl(z)

Q2i(z,z,i) G2(t)

(2+ liz\\2)1

Q2i(z,l) = 2 „ - , i

dzi

1, 2,...,n

+N

l G N, N = max(ord(Qj,s)) + n + 1, Qj,s-being the differential

operators describing the action of the residue currents, al are suitably chosen constants. Finally, the third pair of auxiliary functions is defined by

u

n

Qs(z,Z) = (Qsi,Qs2,...,Qsn)(z,Z): D x D Cn,

n t t n n tr n od(HKjp)(z)) . 1 0

Q3i(z,Z,l) = Q3i(Z,l) = 2-dZ-, i = 1,2,...,n

G3(t) = exp(t - 1),

where p is a Cfunction supported in the unit ball, while having mass equal to 1 there. The function HKl is the usual support function for compact convex sets Kl introduced in the previous section. These leads us to define the (1,0)-forms

i=n

qj(z,Z) = E Qji(z,Z)dZi, j = 1, 2, 3.

i=i

In order to derive the Koppelman integral representation formula with a complex parameter A we have to consider the Ci map

S : D x D Cn,

satisfying for every compact K contained in D the estimates a) ||S(z,Z)|| ^ CKHz - Z||, b) (S(z, Z), z - Z) ^ CKHz - Z||2 and the corresponding (1,0)-form

i=n

s(z,Z )=E Si(z,Z )dZi.

i=1

The positive constants CK, CK above depend on K. In order to simplify the notation, we put

(z,Z) = <Qj,z - Z >, j = 1, 2, 3, daG

Ga(t) = v\t=*j(z^ j = 12,3.

Direct application of the results from [2, 7] leads to the following

Proposition 2.1. The function u(£)Q(R), holomorphic in a neighborhood the closed complex ball Be(0, R), satisfies the following Koppelman integral representation formula for £ G Be(0, R):

u(e) = u(oeR = ^ (/u(Z wQmx H/ u(Z )de(R)K^Z ■)), (2.2)

whenever the value of the complex parameter having large enough positive part ^A >> 0 is fixed. The kernel P\(Z,Z) is the (n,n)-form

k=m

Px(£,Z) = E G[k\$i(Z,Z))Bk(Z,Z) A (dqi)k, (2.3)

k=o

G(a2) G(a3)

Bk(£,Z) = E 2 , 3 (dq2)a2 A (dq3)a3, k = 0, 1,...,n

a.2 +a3=n-k

and kernel Z) is the (n,n — 1)-form defined by

k=min{m,n-1}

rT1

k=0

K\(Z,Z) = E G(k)($i(e,Z))Ak(£,Z) a (dqi)k, (2.4)

= ^ G2a2)G3a3) s A (ds)a° A (dq2)a2 A (dq3)a

S,z - Z >2(ao+i) '

ao + a2+a3=n-k—i

whenever k = 0,1,..., min{m,n - 1}. Furthermore, the right side of (2.2) has holomorphic extension into the half-plane ^A > -5, 5 > 0.

Proof. The proof of the first claim is straightforward. First, we apply the method from [2,7] to construct the representation formula (2.2), with summation up to n, where the kernels P\(t, Z) and K\(£, Z) are defined by the relations (2.3) and (2.4) correspondingly, with only difference that summation is up to n in the first case and up to n — 1 in the second one. Then the observation that G1(t) is a polynomial of degree m leads to the desired conclusion. Furthermore, we observe that

1 j=rni=n , r (Z) |2A

Z) = 1+ <Qi(U),t — Z>=1 + jTTgji(z, Z)(b — Zi) =

m j=1 i=1 11 j )

= 1 + mE jT (t) — ij(Z)) = mE jTij® — m!> — \ij(Z)\2X)- (2.5) m ij(Z) m ij(Z) m

Also, elementary computations imply that

j=m i=n

bc qi(t,Z) = ^EI ij (Z)\2(x-1)d< k (Z)J2 gv (t,Z)dZi = m

3 = 1 i=1

. j=m

= Iij(Z)\2{x-1)d<kj(Z)gj(t,Z).

m j=1

Hence, simplifying the notation, we get the (m, m)-form

(dq1)m = ( — 1) ^ — \t\2<±-m A g,

m

where \ i\ = \ i1 \... \ im\, g = g1 A ■ ■ ■ A gm, X = (X,...,X), and 1 = (1,..., 1). Now, looking

"--' "--'

m-times m-times

at every term of the kernels P\, K\ described in (2.3), (2.4) one observes that for every k the integral of the corresponding terms in (n, n) or (n,n — 1) forms can be continued, as function of X, holomorphically in a neighborhood of X = 0. To be more specific, the extensions of distribution valued functions

X —^ G^Ak A (3q1)k X G^Bk A (dq1 )k

defining, for every value of X, terms in the kernels P\ and K\ are holomorphic in the neighborhood of X = 0. This follows from Prop. 3.6 in ([5]) when X1 = X2 = ■ ■ ■ = Xm. We claim that the value of holomorphic extension of the above functions at X = 0 is equal to zero, whenever k < m — 1. That is, the only terms that have a nonzero contribution in the first integral of (2.2) at X = 0 are the term of the kernel P\ that corresponds to k = m and k = m — 1. Similarly, the only terms that have a nonzero contribution in the second integral of (2.2) at X = 0 are the terms of the kernel Kx that corresponds to k = m and k = m — 1. Actually, the terms in question are (n, n) or (n,n — 1) forms, whose coefficients contain factors (or powers of such factors) of the form

Xk n , n(1 — \ j(Z)\2X), n \ j(Z)\2(x-1) A kijd,

iei \ ijl (Z) / iEi1 deJ deJ

where \I\ + \/1\ = n—k or \I\ + \I1\ = n—k — 1, \J\ = k and the subsets of indices are mutually disjoint. The vanishing of the corresponding integrals at X = 0 follows from the application of Prop. 1.5 from ([5]) after the application of Hironaka's de-singularization theorem on f1f2... fm = 0. The only other interesting cases that remain to be seen are those that correspond to the cases when k = m — 1 or k = m. This completes the proof of the proposition. □

3. The division formula

Keeping the notations from previous sections we formulate the following proposition

Proposition 3.1. Assume that J G Ap(Cn), i = 1,... ,m, p(z) = ||9z|| + log(1 + ||z||) form a complete intersection and are slowly decreasing with respect to L. Then for £ G Cn the following equality holds

u(£) = E "i (OU (£) + < d J- (■)... d—(-),û(-)g1(£, ■) A---A gm(£, ■) A Bi(£, ■) >, (3.1) j=i " "

Ml Pm

where Uj (£) is a Fourier transform of distributions with compact supports contained in Kl = B(0, 2lR0) and Bl(■) is a (n — m,n — m) differential form given by

(N )

W ™„ ( ^ ox/u , ^ e /■ ^ \(„(C

m! E exP ( < 2d (Hk, HO * P)(Z U - Z>) Z )f * ^ ß & Z ),

ft+ft =n-m '

where q(£, Z) =< 2^^, £ — Z> +1 and

^2(£,Z) = (2^)n-m (dB(RKlm * p(z)a (daiog(Wi(z)))ft.

Proof. Let us begin with the discussion of the terms in the forms P\ and K\ of degree k = m and k = m — 1. When k = m, we have the terms that are residual

G{r)Bm(dql)m = const.Xm\jf\2(--1)~d]f A g, (3.2)

G(m)Am(dq1)m = const.Am\f\2(^-1)df A gw, (3.3)

where w = Am = —--1——,——T = —-—1—-—^. In this case we have forms, whose

<S,z — Z >2(a0+1) <S,z — Z >2

coefficients, near the set of common zeroes, have growth growth estimates — (z)d f < (1+

f

+ ||z||)-1 exp( — ||z||), because of the slowly decreasing assumption for the entire functions fj, j = 1,... ,m. In the case k = m — 1, we have

G(1m-1)Bm-1(dq1)m-1 = (n — m + 1,n — m + 1)-form, (3.4)

G(1m-1)Am-1(dq1)m-1 = (n — m + 1,n — m + 1)-form, (3.5)

whose support depends on the radius of the ball Ki and whose terms contain, as coefficients, reciprocals of slowly decreasing functions. Thus, since we want to get our division formula to hold over Cn it is enough to see the convergence of these terms while R —> The forms

g2«2)g<«3) s A (ds)a°

2 3 (dq2)a2 A (0q3)a3, ( )

a2!a3! ' 12 v 13 ' <S,z — Z >2(ao+1)

are factors in forms Am, Bm, Am-1, Bm-1, when a2+a3 = n—m or when a0+a2+a3 = n—m+1. Thus, these terms require estimates for the behavior of G2°2), g3°3) and (Oqi)ai, i = 1, 2 and of their products, when R —> Furthermore the same type of estimate is needed for 66(R). First, we observe that since u G D(H), its Fourier transform satisfies for every k G N the estimate

\U(Z)\ < Ck (1+1|Z exp (Hk, (Q(Z))), Z G Cn.

On the other hand, the function (HKi (Q(^)) * p) (Z) is convex, since for a compact convex set K the support function HK is convex also. Therefore we deduce that

k <d (hKi ($(■)) * p) (Z), t — Z X (Hk№■)) * p) (t) — (Hk(*(■)) * p) (Z),

whenever t,Z G Cn. Hence

\e<d(HKim-))*p)(o,i-c>\ ^ g^<9(Hkim-))*p)(o¿-o ^ €(Hk1 (^(■)]*P)(€)-(Hk1 m-))*p)(o^ Thus

D -ce<a(HKi (^))*P)(Z) *-z>\ < Ci(z,a)(1 + ||Z\\)ae-(HKi OOMCO,

where a G Nn, ||Z — t|| ^ 1 and \a\ = ^ a(j) = sum of orders of j-directional derivatives. For the positive constant Cl(t, a) the following estimate holds

\Ci(t,a)\ < d(a)Yt ° 1 exp(4Yi )(1 + UH) 1 a 1 e-(HKi WH^,

For the estimate of d0(-R), when t G Cn is fixed and Z G Cn is such that ||t — Z|| ^ 1, one has that

DaoR \ < Ti(t,k,a)(1 + ||ZH)-ke-(HKi w^Yo,

where k > 0 and a G Nn and rl(t, k, a) is a positive constant. By letting ¿51 (■) = d-1^) A ...

i i1

■ ■ ■ A d-—(■), the above estimates imply the relation (3.1) when R —> to. Pm

4. Locally explicit version of fundamental principle

Keeping the notation from above, we begin the present section by obtaining localized explicit solutions to a system of homogeneous convolution equations. Namely, we have the following proposition

Proposition 4.1. Let f *¡ij = 0, j = 1,..., m be a homogeneous system of convolution equations in E (Rn), where f G E (Rn) and ¡j G E '(Rn), so that the entire functions fa,..., im form a complete intersection in Cn. Let also l be any positive integer so that suppij C Kl = {x G Rn : ||x|| ^ 2lRo} for every j = 1,... ,m. Then, for every t G intKi = {x G Rn : ||x|| < 2lRo} the solution f (t) is represented by

f(t) = ^ < B1(Z),e-i<t,z> J fxi(—t)g(t,Z)Bmi(t,Z^(t)) >, (4.1)

J

where the form Bm,l(t, Z) is the restriction of the form Bm(t, Z) to Kl.

Proof. Since xl+1 = 1 on Kl+1 and suppxi1 C intKl+1 we have that (fxi+1) * ¡j = f * ¡j = 0 holds on Kl = ~B(p,¥R0o) = Bp^y+R—R) for every j = 1,...,m. Hence

(fX i+0 * ¡j = (fXi+1) ■ Mj = 0 j = 1,...,m. Therefore, letting g(t, Z) = g1 (t,Z) A ■ ■ ■ A gm(t, Z), the Plancherel formula becomes

(2n)n / f(t)u(t)dt = f {fX)(-OH№ =

J Rn J Rn

= / (fxi)(-o (E Aj(e)ue | de + / (TXTx-o < a-(c),u(c)g(e,z)Bm>l(e,zd >=

-/Rn \ j=l } M

=< d-(z)J Jxi(-e)u(c)g(e,z)Bml(e,zd >.

M J^eRn

Now, recall that U(Z) = / u(t)e-i<Z't>dt for a test function u G V(intKi). If we assume also

Ki

that the interior of the set suppu C intKl is not empty and that u = 1, then Fubini's Theorem implies that

(2n)n Jf(t)u(t)dt = <o-(z),u(z) J fXi(-e)g(e,z)BmJ(e,z)de>=

Ki £eRn

J u(t)e-i<z't^ < d-(0, y JXi(-Ofxi(-Og(OC)BmM,C)dZ>)dt.

Kl

Note that one can use Fubini's Theorem because the action of the residue current d— = d— A

M Ml

d— A ••• A d— is independent of e-l<z,t>, because this function does not contribute to the M2 Mm

set of common zeroes of the functions Aj, j = 1,... ,m. We now apply Lebesgue' Differentiation Theorem to deduce that for almost all t G int(suppu) one has that (4.1) holds. This concludes the proof of the proposition. □

We now formulate the main result of the paper.

Theorem 4.1. Let f * Mj = 0, j = 1,... ,m be a homogeneous system of convolution equations in E(Rn), where f G E(Rn) and Mj G E'(Rn), so that the entire functions , • • •, Am are slowly decreasing with respect to L and form a complete intersection in Cn. Then, there exists an (n, n — m) differential form

*(z)=/ fxi(-e)g(e,z)Bi(e,z)de,

J^e Rn

whose coefficients are in the space L1(Cn,T), where t is some element in LA U structure, so that

f (t) = (2nn < d—(Z),e-i<t'z>^(z) >, t G Rn. (4.2)

(2n)n m

Proof. For the sequence of compact closed balls

K = {x G R" : ||x|| < 2lR0} ,

where R0 > 0 and l G N exhausting Rn, we describe the LAU structure U. Let pl = e4^l(l^l)Naf be a sequence of positive constants, where the constant y1 depends on Kl and the constant al depends on the solution f that we want to express explicitly. For example y = = max{||t||, t G Kl} = 2lR0 and N is the order of differential operator involved in computation residual term d— A d— A ••• A d—. Let t be some function in LAU structure that

Ml M2 Mm

dominates all the functions

sup (e-471 (Iyi)-Na-2(ul)peHKi (9())) , p = 1,...,N + 1. leN ^ '

From the preceding proposition we know that for every l G N there exists a differential (n, n — m) form 1il(Z) such that

f(t) = jAn <B~(Z),e-i<t'C>*i(Z) >.

(2n)n ¡i

Applying Ascoli's Theorem, we extract a subsequence of (n, n — m) forms from the sequence on {^l} of (n,n — m) forms that converge in the space of differential forms with continuous coefficients in the space L1(Cn,r). □

References

[1] L.A.Aizenberg, A.P.Yuzhakov Integral Representations in Multidimensional Complex Analysis, Transl. AMS 58, 1980.

[2] M.Andersson, M.Passare, A shortcut to weighted representation formulas for holomorphic functions. Ark. Mat., 26(1988), no. 1, 1-12.

[3] C.A.Berenstein, B.A.Taylor, Interpolation problems in Cn with applications to harmonic analysis, J. Analyse Math., 38(1980), 188-254.

[4] C.A.Berenstein, R.Gay, Complex Analysis and Special Topics in Harmonic Analysis, Springer-Verlag, 1991.

[5] C.A.Berenstein, R.Gay, A.Vidras, A.Yger, Residue currents and Bezout identities, Progress in Mathematics 114, Birkhauser, 1993.

[6] C.A.Berenstein, A.Yger, About Ehrenpreis' Fundamental Principle, Geom. and alg. aspects in several complex variables, C.A. Berenstein, D.C.Struppa (ed.), Editel, Rende, 1991, 47-61.

[7] B.Berndtsson, M.Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble), 32(1982), no. 3, 91-110.

[8] B.Berndtsson, M.Passare, Integral formulas and an explicit version of the fundamental principle, J. Fund. Anal., 84(1989), no. 2, 358-372.

[9] L.Ehrenpreis, Fourier Analysis in Several Complex variables, Wiley, New York, 1970.

[10] S.Hansen, Localizable analytically uniform spaces and the fundamental principle, Tran. of AMS, 264(1981), no. 1, 235-250.

Локально явный фундаментальный принцип для однородных уравнений в свертках

Алекос Видрас

Кафедра математики и статистики Университет Кипра POB 20537, Никосия 1678 Кипр

В настоящей статье локально явная версия основополагающего принципа Эренпрейса для системы однородных уравнений в свертках f * fj = 0, j = 1,..., m, f £ E (Rn), fj £ E'(Rn), получается, когда преобразования Фурье jlj, j = 1,. .. ,m — медленно убывающие входные функции, которые образуют полное пересечение в Cn.

Ключевые слова: фундаментальный принцип, формула деления.