A critical growth rate for the pluricomplex Green function

Momm, Siegfried

doi:10.1215/s0012-7094-93-07218-3

Cited by 7 publications

(8 citation statements)

References 12 publications

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“…Our results on the existence of right inverses extend results of Schwerdtfeger [33], Taylor [34], Meise and Taylor [16] for Q C N , of [20], [23], [22], [17], Korobe! nik and Melikhov [8] for open or compact convex sets in C N , and of Langenbruch [12] and Korobe!…”

Section: Introductionsupporting

confidence: 88%

“…2.7, and get a partial differential operator PD of infinite order, such that the zeros of P are positive multiples of L a a. Since PD admits a continuous linear right inverse on AQ, as in [22], Lemma 2.5 (applying the proof of Proposition 2.6), i.e. as in Theorem 3.9, we see that condition UfagY Q is fulfilled.…”

Section: Proof (I) a (Ii): Putmentioning

confidence: 85%

“…As in [22], Lemma 2.5, it follows that the condition USY Q is fulfilled with analytic solutions of convolution equations on... ( In the first case, we have to show that Q Q H is bounded with nonempty interior. First we show that Q is bounded: Assume that Q is not bounded.…”

Section: Proof (I) a (Ii): Putmentioning

confidence: 95%

“…Let L a a denote the complex hyperplane which is contained in this real one. We now proceed as in [22], Thm. 2.7, and get a partial differential operator PD of infinite order, such that the zeros of P are positive multiples of L a a.…”

Section: Proof (I) a (Ii): Putmentioning

confidence: 99%

“…For the results on intervals and polygons, ad hoc arguments have been used which are limited to the special situation (in fact, part of the proof is a reduction to corresponding problems for inscribed or circumscribed open or closed polyhedra or discs). Our approach consists in extending the technique used in [22] for open Q and in [17] for compact Q, which (for the constructive direction of the results) applies a continuous linear right inverse of the d-operator acting on certain auxiliary spaces.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Analytic Solutions of Convolution Equations on Convex Sets with an Obstacle in the Boundary

Мелихов¹,

Momm²

2000

MATH. SCAND.

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Section: Introductionsupporting

confidence: 88%

Section: Proof (I) a (Ii): Putmentioning

confidence: 85%

Section: Proof (I) a (Ii): Putmentioning

confidence: 95%

Section: Proof (I) a (Ii): Putmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analytic Solutions of Convolution Equations on Convex Sets with an Obstacle in the Boundary

Мелихов¹,

Momm²

2000

MATH. SCAND.

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On Holomorphic Perturbations of Infinite Order Differential Equations

Vidras

1997

Mathematische Nachrichten

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It is proven in the present article that the solutions of infinite order differential equation with holomorphic parameter w E U C a! depend holornorphically on w in the neighborhood of characterisitic points of certain directions. Preliminaries and statement of the main theoremsIt was shown in [7, 9, 131 that, roughly speaking, the solutions of the system of differential equations cannot change holomorphically with respect to the parameters w = (wl,. . . , wn), even when the coefficients of the system depend holomorphically on w . In the present note we study this phenomena for infinite order differential equations with holomorphic parameters and determine when such a problem has a solution. More precisely, we construct a fundamental solution, holomorphic in w near characteristic points of certain directions. However a basic gap remains, since the method presented here excludes all the "real" directions when n > 1, [7]. Our approach has its origins in [4]. Let U be a connected open set in C . Here we will consider a special class of operators of infinite order P with holomorphic coefficients. These are operators that can be written in the form P = P ( w , D ) = c a,(w)DQ Q with D = (D1, . . . , D,) = ( G , 8 . . . , &) and a, holomorphic functions on U satisfying the following estimates: For each compact set K C C U and for each h > 0 there 1991 Mathematics Subject Classification. Primary 32A15; Secondary 35R50, Keywords and phmses. Entire functions, characteristic point.As usual a subset R of U x 6" is called conic if for all t > 0, (w,tC) E R whenever (w, C) E R. For simplicity we will assume that an open conic neighborhood of (w, 5)in U x 6" is a set of the form G x V, where G is an open connected subset of U andV is an open, convex cone in C ". w E u, P(w,C) = P W ( 0 . Definition 1.2. ([l, 81.) A point (wg,co) E U x C" is called characteristic for the operator P(w,D,) if for every open conic neighborhood R = G x V of (wo,C0) and for every T > 0 there exists (w',C') E Cl n { (w,C) E U x C" : ICl > r } so that P(w'l<') = 0. Vidras, On Holomorphic Perturbations 269 Since we view the symbol P(w,C) &a a family {Pw(C)},E~ of entire functions of exponential type zero, with holomorphic parameter 20, we have the following equivalent formulation of the Definition 1.2. Definition 1.3. A point (200, ( 0 ) E U x C " is called characteristic for the family of operators {Pw(Dz)}wE~ if for every open neighborhood G, , of W O , for every open, convex cone Vc, containing Co and for any r > 0 there exist w t E G, , and C' in Vc, n (5 E 6" : ICI > r } so that P,t(C') = 0. We denote the set of all characteristic points by CharP. Example 1.4. In the above Example 1.1 the point (0,l) is characteristic for the operator Consider the unit sphere S2"-l = {& C # O , < € a " } .-By adding the above sphere at "infinity" to C " we obtain the compact space 6 ", equiped with the usual topology. To indicate the fact that we consider the unit sphere at infinity we denote it by S2-l and by = the direction of C E C", C # 0. Definition 1.5. We w...

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Elliptic partial differential equationsfor real analytic functions

Momm

2000

Math Z

Self Cite

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A critical growth rate for the pluricomplex Green function

Cited by 7 publications

References 12 publications

Analytic Solutions of Convolution Equations on Convex Sets with an Obstacle in the Boundary

Analytic Solutions of Convolution Equations on Convex Sets with an Obstacle in the Boundary

On Holomorphic Perturbations of Infinite Order Differential Equations

Elliptic partial differential equationsfor real analytic functions

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