Some remarks on convolution equations

LEMMA 1. - For any fixed p E% ,5 > 0 and a G S"-1, let x* = x*(s, a) G 9Hp(s) be an arbitrary point such that <^,a>=^(^;a). // Tr(x*,s , a) denotes the linear variety {(x , y) C R^ : <x - x * , a> = 0, y = Np + s} , then there exists a hyperplane V(p , x*, a) which is tangent to 9[p] (cf.(2) at the point (x* , Np + s) and contains 7r(;c*, s , a). For each p G ®(S2), the so-called normal mapping v of the convex surface 9[p] can be defined as follows (cf. [I], [2]), Fwx € n, ( A) Cf. [9] where sim

LEMMA 2. — Given ft as above and constants N real and 6 > 0, there exists a function p € % (ft) such that N- = N and Cp(s)>- (s>\\ (3) o This lemma has an interesting consequence which will be used in Section 2. Let p be the function of Lemma 2 and;^ = x*(s^ , a) as above (s^ > 1). By Lemma 1, there exists a vector Vp(x^) G 31p(x^) such that Vp(x^) = \\v-(^)H.a.Hence, for any x^ G ft and ^ = p(x^\ the convexity of p implies < (x, - x,, p(x^) - p(x,)), (Vp(x,) , - 1)> < 0. (4) ( l) Cf. (!), p. 55

LEMMA 3. - Assume that Sl' C n. Ifp is a function in %(ft') satisfying condition (6), then for each SQ > 0, there exists a function p G a(ft) also satisfying (6) and such that p = p ' on Hy ( p ) and p < p ' everywhere. Let e be the class of all functions X which are concave, increasing, continuously differentiable on [0 , °°) and such that 0 < 2(s 4- 1) \'(s) < 1 for all s > 0.

LEMMA 4.(1) - Let \ be a function in (° and p its inverse. Then for any a > 0 and b > 1, / 00 00 . max (S ^(^-^ , ^ ^-W5)) (7) ^=0 ^=o / < (3+^) ^(t)- (2)

LEMMA 5. — Given a distribution $ € &' and an integer j > 0, assume that there exist t real, a G S""1 and an open set U in R" such that for any multiindex 7 , 17! < /, the integral ^00= f F e^$(?) rf? (12) • "r^co converges absolutely and locally uniformly in U. Then $ G C^(U). The order of a distribution $ ($ E §') is defined here as the infimum M($) of all numbers M such that, for some C > 0 and R > 0, I^COKCe^^^^ By the Paley-Wiener theorem (or by the previous lemma), sing supp $ = 0 (Le; $

LEMMA 6. — Given e> 0 and g analytic in A(ZQ ; 2e), then for each k = 0 , 1 , . . . ,

PROPOSITION 1. — For every $ € &' and arbitrary a, sup inf sup Ap(...; a) == h^W (p e S ($ ; a ; 5)} , (22) 6 p s

THEOREM 1. — For every distribution $ E S,an arbitrary convex region R containing the support of <&, and any function a(a) > 1, A$ ; S2 ; a ; a) == ^($ ; a ; a) == h^a) (Va). (26) Proo/ — First we shall prove the inequality ^(<&;a;a)>^^(a). (27) By (25) we can obviously assume that sing supp $ ^ 0 ; and, that aE S""1 is fixed so that ^($;a;a)<A,^(a). (On the other hand, the case ^?*($ ; a ; a.) = — o° is not excluded). Let A be any number > — oo such that /?*($; a; a) < A</^(a). Chooses > 0, £ <$

PROPOSITION 2. — Gn^w a distribution <&€§', r/^ following conditions are equivalent : (i) rt^r^ ^c^ constants r > 0 , C > 0 and A real such that for every entire function f, ©($;c;A;m/0)| < IWADI, (v? e c") ; (46) (ii) condition (i) "with f = 1 only ; (iii) condition (i) w^YA 1. . .ly replaced in (46) &y [...],. ; (iv) condition (iii) w^A / = 1 o^fy.$

Lemma 6. DEFINITION. — A distribution $ with compact support is said to be of class <%, provided $ satisfies one of the conditions (i) — (iv) in Proposition 2. <% will denote the class {$ : $ E (R}. The simplest examples of distributions of class <% are linear partial differential operators with constant coefficients ; in other words, all polynomials are in the class <%. This was proved by Malgrange in [14], and a similar statement appears in Ehrenpreis [10]. Actually, in this case one can show

PROPOSITION 3. - Let P(?) be an exponential polynomial, i.e. P(D= S P,(?)e<^>, fc=i ^here the P^s are polynomials and a^ = (c^ ,. . . , c^) E C" ore ^ so-called frequencies of P. 5^ Ap(?) = max Re<c^ , ?>. Then, for each € > 0, there exists a constant C = C(e,P) such that, iff is an analytic function in the polydisk A(? ;£), then !/(?) I e^ <C [/(?)?(?)],. (47)

PROPOSITION 4. — For each <!»€<%, cv. supp 4> = cv. sing supp $. (48)$

THEOREM 2. — (The Titchmarsh-Lions formula for singular sup- ports). For each $ e <% and ^ e 8', cv. sing supp ($ * ^) = cv. sing supp $ + cv. sing supp ^. (52) /Voo/ — It suffices to show that cv. sing supp ($ * ^) D cv. sing supp $ -h cv. sing supp ^ (53) By Proposition 4, one has to prove, ^.^W - ^(cO > ^.^(a)- (54) Let a be fixed. If e is any positive number, then by Theorem 1, for all 5 small and s large, there is a function q satisfying (50) with $ replaced by $ * ^. Hence by Lemma 4 and i$