Characterization of the Linear Partial Differential Operators with Constant Coefficients Which are Surjective on Non‐quasianalytic Classes of Roumieu Type on ℝ^N

Abstract

After it was observed by DE GIORGI and CATTABRIGA [I41 and PICCININI [31] that the heat operator is not surjective on the space of all real-analytic functions on R3, HORMANDER [ 181 characterized the linear partial differential operators with constant coefficients which are surjective on all real-analytic functions on a given convex open subset of RN. On the other hand, CATTABRIGA [Ill, [12], showed that the heat operator is not surjective on the classical Gevrey classes Tcd1(lR3) for 1 < d < 2. Sufficient conditions for the surjectivity of linear partial differential operators with constant coefficients were given by CATTABRIGA [Ill and ZAMPIERI [34], [35], where the latter used methods of HORMANDER [18].In the present paper we characterize the surjectivity of linear partial differential operators with constant coefficients on the non-quasianalytic classes gcw,(IRN) which extend the Gevrey classes ridl(IRN) for d > 1 (see PETZSCHE and VOGT [30] and BRAUN, MEISE and TAYLOR [S]). More precisely, we show that for a complex polynomial P i n N variables the differential operator P(D): &{wl@2N) 4 E {~~( I R~) is surjective if and only if the zero variety V ( P ) : = ( z E C N : P ( -Z ) = 0)of P satisfies the following equivalent Phragmtn-Lindelof conditions: PL(o):There exist 6 > 0 and A 2 1 such that for each B 2 A and E > 0 there exists C > 0 such that each plurisubharmonic function u on V ( P ) which satisfies u(z) < IIm zI + 6o(z) and u(z) 5 B IIm zI for all z E V also satisfies u(z) I A IIm zI + EO(Z) + C for all z E V .There exist R > 0 and A 2 1 such that for each B 2 A and E > 0 there exists C > 0 such that, for each B E V(P) with 181 2 R, each plurisubharmonic function u on V ( P ) which satisfies u(z) I w(B) and u(z) I B (Im Z( for all z E V ( P ) , )Z -el < o(0) also satisfies u(0) I A /Im 81 + m ( 8 ) + C. PL,(w):') Research partially supported by Deutsche Forschungsgemeinschaft. 2'

Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

“…It is worth noting that there is an extensive theory of surjectivity of convolution operators on A(Ω) and related spaces (see [5], [7], [14], [22], [23]). …”

Local radial Phragmén-Lindelöf estimates for plurisubharmonic functions on analytic varieties

“…His characterization was given in terms of global and also of local conditions of Phragmén-Lindelöf type for plurisubharmonic functions on the zero variety of the symbol P . Since then, it was shown in a number of papers that similar Phragmén-Lindelöf conditions on algebraic varieties can be used to characterize other properties of (systems of) such operators (see, e.g., Andreotti and Nacinovich [1], Boiti and Nacinovich [3], Braun, Meise, and Vogt [9], Franken and Meise [11], Kaneko [13], Meise, Taylor, and Vogt [15], Momm [18], Palamodov [20], Zampieri [24]). …”

“…The present authors, along with D. Vogt, have studied this question in [4], [5], [6], [8], [9], [14], [15], [16], [17]. The main result of this paper, Theorem 10, gives a local geometric condition on an analytic variety near a real point ξ which guarantees that any plurisubharmonic function u on the variety that vanishes on its real points can grow only linearly, u(z) = O(|z − ξ|), near ξ.…”

Untitled

“…From several results in recent years, starting with Hörmander's characterization of the constant coefficient partial differential equations P (D)u = f that have a real analytic solution u for every real analytic function f , it has become clear that certain properties of the partial differential operator P (D) are equivalent to estimates of Phragmén-Lindelöf type for plurisubharmonic functions on the algebraic variety V = V (P ) := z ∈ C n : P (z) = 0 (see Hörmander [15], Zampieri [32], Kaneko [16], [17], Braun, Meise and Vogt [7], [8], Braun [3], [4], Meise, Taylor and Vogt [19], [20], [21], [23], [25], Palamodov [30] and Momm [28]). There are several different such estimates that are of interest, corresponding to different properties of the operator, e.g.…”

“…(8) Next note that for l with µ < l ≤ m there is j with ν < j ≤ m so that (5) holds. Hence (4) implies that (α l (ζw), ζw) ∈ Γ(ξ 0 , δ/2, R) if R is sufficiently large enough.…”