Some remarks on convolution equations

Lemma 1. – For any fixed $p \in \mathcal{Q}$ , $s \ge 0$ and $\alpha \in \mathbb{S}^{n-1}$ , let $x^ = x^(s, \alpha) \in \partial H_p(s)$ be an arbitrary point such that $$\langle x^*, \alpha \rangle = h_p(s; \alpha).$$ If $\pi(x^*, s, \alpha)$ denotes the linear variety $$\{(x, y) \in \mathbb{R}^{n+1} : \langle x - x^*, \alpha \rangle = 0, y = \mathbb{N}_p + s \},$$ then there exists a hyperplane $\widetilde{\pi}(p, x^, \alpha)$ which is tangent to $\partial[p]$ (cf.(2) at the point $(x^, N_p + s)$ and contains $\pi(x^*, s, \alpha)$ . For each $p \in \mathfrak{A}(\Omega)$ , the so-called normal mapping $\nu_p$ of the convex surface $\partial[p]$ can be defined as follows (cf. [1], [2]). For $x \in \Omega$ , - (1) Cf. [9] where similar questions are discussed. - (2) The convexity of p means that the set $[p] \stackrel{\text{def}}{=} \{(x, y) \in \mathbb{R}^{n+1} : x \in \Omega, y \ge p(x)\}$ is convex and p is continuous. let $\widetilde{\pi}(p,x)$ be any tangent hyperplane of $\partial[p]$ passing through the point (x,p(x)). If $\alpha=(\alpha_1,\ldots,\alpha_n,\alpha_{n+1})\in S^n$ is the corresponding outer normal vector, then $\alpha\in S_-^n$ . (1). Hence, for some positive $\lambda$ , $\lambda$ , $\alpha_{n+1}=-1$ . Set $\nu_p(x)=\nu_{p,\widetilde{\pi}}(x)=(\lambda\alpha_1,\ldots,\lambda\alpha_n)$ . If x is fixed, $\mathcal{N}_p(x)$ will denote the normal cone of p at x, i.e. the set of all possible vectors $\nu_p(x)$ . Each $\mathcal{N}_p(x)$ is a compact convex subset of $\mathbb{R}^n$ (cf. [2]). The mapping $x\mapsto \nu_p(x)$ , which in general is multivalued, is called the normal mapping of the surface $\partial[p]$ . If the function p has total differential at the point x, then obviously the set $\mathcal{N}(x)$ contains only one vector, namely $$\nu_p(x) = \left(\frac{\partial p}{\partial x_1}, \dots, \frac{\partial p}{\partial x_n}\right)$$ . Hence, by a well-known theorem of K. Reidemeister [16], the normal mapping $\nu_p$ is univalent almost everywhere in $\Omega$ . Furthermore, it is not difficult to see that if, for a fixed $\alpha \in \mathbb{S}^{n-1}$ and $s \ge 0$ , $x^(s, \alpha)$ denotes any point of the sort appearing in Lemma 1, then $\|\nu_p(x^(s, \alpha))\| \nearrow \infty$ when $s \to \infty$ . Moreover, an easy compactness argument shows that for each $p \in \mathcal{R}$ , the function $$C_p(s) = \inf \left\{ \| \nu_p(x) \| : \nu_p(x) \in \mathcal{T}_p(x) , x \in \partial H_p(s) \right\}$$ is such that $$C_n(s) \nearrow \infty$$ for $s \to \infty$ . (2) It is not difficult to construct functions p for which $C_p(s)$ grows arbitrarily fast; this can be done by a procedure called in [9] the convex interpolation. Here we state only a special case which will be used in the sequel.

Lemma 2. – Given $\Omega$ as above and constants N real and $\delta > 0$ , there exists a function $p \in \mathfrak{L}(\Omega)$ such that $N_p = N$ and $$C_p(s) \ge \frac{s}{\delta}$$ $(s \ge 1)$ . (3) This lemma has an interesting consequence which will be used in Section 2. Let p be the function of Lemma 2 and $x_1 = x^*(s_1, \alpha)$ as above $(s_1 > 1)$ . By Lemma 1, there exists a vector $\nu_p(x_1) \in \mathcal{F}_p(x_1)$ such that $\nu_p(x_1) = \|\nu_p(x_1)\|.\alpha$ . Hence, for any $x_2 \in \Omega$ and $s_2 = p(x_2)$ , the convexity of p implies $$\langle (x_2 - x_1, p(x_2) - p(x_1)), (\nu_p(x_1), -1) \rangle \leq 0.$$ (4) (1) Cf. (1), p. 55. whence $$C_p(s_1) \le \frac{s_2 - s_1}{\langle x_2 - x_1, \alpha \rangle}$$ (5) for $\langle x_2 - x_1 \rangle > 0$ . In particular, if $x_2$ is such that $$x_2 = x*(s_2, \alpha) \notin H_n(s_1)$$ then Lemma 2 combined with (5) shows that $$s_1(h_p(s_2, \alpha) - h_p(s_1, \alpha)) \le \delta(s_2 - s_1) \ (\alpha \in \mathbb{S}^{n-1}; s_2 > s_1 > 1)$$ (6) Remark 1. — We have just proved the existence of functions p with property (6). However much more can be said. If $\Omega$ is bounded, then for any $p \in \mathcal{R}(\Omega)$ and $\delta > 0$ , there is a constant $\sigma$ such that the inequality (6) holds for all $\alpha$ and $s_2 > s_1 \ge \sigma$ . For $\Omega$ unbounded, the same statement remains valid, if one assumes that each p grows at infinity faster than indicated in (1). It suffices to suppose, for instance, that $$\lim_{x \to \partial \Omega} \frac{\log p(x)}{\|x\|} = \infty. \tag{1*}$$ The proofs of these statements are omitted since they are rather tedious. However, it will be clear from the context that by using these statements one could simplify the construction in the next section. Finally, the convex interpolation mentioned above easily yields the following lemma about extensions of convex functions:

Lemma 3. — Assume that $\Omega' \subset \Omega$ . If p' is a function in $\mathfrak{R}(\Omega')$ satisfying condition (6), then for each $s_0 > 0$ , there exists a function $p \in \mathfrak{R}(\Omega)$ also satisfying (6) and such that p = p' on $H_{s_0}(p')$ and $p \leq p'$ everywhere. Let $\mathfrak E$ be the class of all functions $\lambda$ which are concave, increasing, continuously differentiable on $[0,\infty)$ and such that $$0 < 2(s+1) \lambda'(s) \le 1$$ for all $s \ge 0$ . LEMMA 4.(1) – Let $\lambda$ be a function in $\mathfrak E$ and p its inverse. Then for any a > 0 and $b \ge 1$ , $$\max \left(\sum_{s=0}^{\infty} e^{a\lambda(s)-bs}, \sum_{s=0}^{\infty} e^{as-bp(s)}\right)$$ (7) $$\leq \left(3 + \frac{a}{2b}\right) e^{a\lambda \left(\frac{a}{b}\right)}.$$ (2)

Lemma 5. – Given a distribution $\Phi \in \mathcal{E}'$ and an integer $j \ge 0$ , assume that there exist t real, $\alpha \in \mathbb{S}^{n-1}$ and an open set U in $\mathbb{R}^n$ such that for any multiindex $\gamma$ , $|\gamma| \le j$ , the integral $$\mathcal{J}_{\gamma}(x) = \int_{\Gamma(t,\alpha)} \zeta^{\gamma} e^{i\langle x,\xi\rangle} \hat{\Phi}(\zeta) d\zeta \qquad (12)$$ converges absolutely and locally uniformly in U. Then $\Phi \in C^{(f)}(U)$ . The order of a distribution $\Phi(\Phi \in \mathcal{E}')$ is defined here as the infimum $M(\Phi)$ of all numbers M such that, for some C > 0 and R > 0, $$|\hat{\Phi}(\xi)| \leq Ce^{M\omega(\xi) + R|\eta|}$$ By the Paley-Wiener theorem (or by the previous lemma), sing supp $\Phi = \emptyset$ (i.e. $\Phi \in C_0^{\infty}(\mathbb{R}^n)$ ) if and only if $M(\Phi) = -\infty$ . Let $\Delta(\zeta; r)$ be the polydisk $\{z \in \mathbb{C}^n : \max_j |\zeta_j - z_j| \le r\}$ and $\dot{\Delta}(\zeta; r)$ its distinguished boundary. If g is analytic in $\Delta(\zeta; r)$ , set $$\begin{cases} |g(\zeta)|_{r} = \max_{z \in \Delta(\zeta; r)} |g(z)| ; \\ [g(\zeta)]_{r} = \frac{1}{(2\pi r)^{n}} \oint_{\dot{\Delta}(\zeta; r)} |g(z)| |dz|. \end{cases}$$ (13)

Lemma 6. – Given $\varepsilon > 0$ and g analytic in $\Delta(z_0; 2\varepsilon)$ , then for each $k = 0, 1, \ldots$ , $$|g^{(k)}(z_0)|_{\varepsilon} \le \frac{k!(k+2)}{2\pi\varepsilon^{k+1}} [g(z_0)]_{2\varepsilon}.$$ (14) The proof is an easy application of the Cauchy formula. One can clearly assume n=1. Fix $r\in(0,\varepsilon]$ and $\zeta$ such that $|\zeta-z_0|=r$ . Then the Cauchy formula and the monotonicity of $G(r)=[g(z_0)]_r$ yield $$\frac{1}{k!} \int_0^{\epsilon} r^{k+1} |g^{(k)}(\zeta)| dr \leq \int_0^{\epsilon} [g(\zeta)]_r r dr$$ $$\leq \iint_{\Delta(z_0; 2\epsilon)} |g(w)| |dw| \leq \int_0^{2\epsilon} [g(z_0)]_{2\epsilon} r dr ,$$ and the estimate (14) easily follows.

Proposition 1. – For every $\Phi \in \mathcal{E}^i$ and arbitrary $\sigma$ , $$\sup_{\delta} \inf_{p} \sup_{s} h_{p}(\ldots; \alpha) \equiv h_{s,\Phi}(\alpha) \quad (p \in \mathcal{R}(\Phi; \sigma; \delta)) , \quad (22)$$ Proof. – As can be easily seen from the Paley-Wiener theorem, for each fixed $\delta$ , $$h_{s,\Phi}(\alpha) = \inf_{p} \sup_{s} \{h_{p}(s;\alpha) : p \in \mathcal{R}_{0}(\Phi;\sigma;\delta)\}, \qquad (23)$$ and formula (22) follows. Remark 3. – Actually, formula (23) leads to a simpler expression for $h_{s,\Phi}$ , $$h_{s,\Phi}(\alpha) = \inf_{p} \sup_{s} \{ h_{p}(s;\alpha) : p \in \mathcal{R}(\Phi) \}. \tag{24}$$ Moreover, by the above proposition, $$\mathcal{L}_{\Phi}^{*} \leqslant h_{s,\Phi}. \tag{25}$$ In view of the last proposition, it is natural to consider the function B^ defined by (20\) and to ask whether the function S^ (or K)* has a similar geometric interpretation. The answer is remarkably simple :

Theorem 1. — For every distribution \$ E S,an arbitrary convex region R containing the support of <&, and any function a(a) > 1, $$\mathcal{L}(\Phi;\Omega;\sigma;\alpha) = \mathcal{L}^*(\Phi;\sigma;\alpha) = h_{ss,\Phi}(\alpha) \qquad (\forall \alpha). \tag{26}$$ Proo/ — First we shall prove the inequality $$\mathcal{L}^*(\Phi;\sigma;\alpha) \geqslant h_{ss,\Phi}(\alpha). \tag{27}$$ By (25) we can obviously assume that sing supp \$ ^ 0 ; and, that aE S""¹ is fixed so that $$\mathcal{L}^*(\Phi;\sigma;\alpha) \leq h_{s,\Phi}(\alpha).$$ (On the other hand, the case ^?\(\$ ; a ; a.)* = — o° is not excluded). Let A be any number > — oo such that $$\mathcal{L}^*(\Phi;\sigma;\alpha) \leqslant A \leqslant h_{s,\Phi}(\alpha).$$ Chooses > 0, £ <A, ^(a) — A. In order to prove (27), it suffices to show that \$ e C^jCo) 0) for all / = 0 , 1 ,. .. and all XQ in the set $$A + 4\varepsilon < \langle x_0, \alpha \rangle < h_{s, \Phi}(\alpha) + \varepsilon. \tag{28}$$ From now on, A ,s ,/ and XQ are fixed. Since sing supp \$ ^= 0, M(\$) > - oo (cf. Sect. 1). Let 6 < £/2 and $$s'_0 > \max \{ \sigma(\alpha), |M(\Phi)| + 1, j + n + 1 \}.$$ By (20\) and (28) there exists peS^,a,8 )* such that, for N=N(p,\$), $$h_p(s_0' + N; \alpha) << x_0, \alpha > -3\varepsilon$$ . (^Cf.C^p.S^ Fix p and $\delta$ . Then the second inequality in (28) combined with (22) shows that, for some integer $s_0 \ge s_0'$ , $$h_p(s_0 + N; \alpha) < < x_0, \alpha > -3\varepsilon \le h_p(s_0 + N + 1; \alpha).$$ (29) If we choose $s_1 \ge s_0 + 1$ so that $$h_n(s_1 - 1 + N; \alpha) \le \langle x_0, \alpha \rangle - 2\varepsilon \langle h_n(s_1 + N; \alpha), (30)$$ then $$s_0 + N > 1$$ ; $s_1 > 3s_0 + 2N > s_0 + 1$ . (31) Indeed, $s_0 + N \ge s_0' + N > |M(\Phi)| + 1 + N > 1$ ; on the other hand, by (29), (30) and (6), $$\varepsilon < h_p(s_1 + N) - h_p(s_0 + N) < \frac{\varepsilon(s_1 - s_0)}{2(s_0 + N)},$$ which gives (31). Similarly, one obtains the inequality $$h_n(s_1 + N; \alpha) < < x_0, \alpha > -\varepsilon.$$ (32) For each s=0, 1,..., let $x^=x^(s,\alpha)$ be as in Lemma 1. Then there exists a hyperplane $\widetilde{\pi}$ which is tangent to $\partial[p]$ at the point $(x^(s_1+N;\alpha), p(x^(s_1+N;\alpha)))$ and such that, for some t>0, $t\alpha=\nu_{p,\widetilde{\pi}}(x^*(s_1+N;\alpha))$ . It will be shown below that, for $$U = \{x \in \mathbb{R}^n : \langle x - x_0, \alpha \rangle > -\varepsilon \}$$ and the variety $\Gamma(t, \alpha)$ , the hypotheses of Lemma 5 are satisfied. This will complete the proof of (12). Let $\gamma$ be any multiindex of length $|\gamma| \le j$ . We claim that the integral $\mathcal{J}_{\gamma}(x)$ in (12) is uniformly and absolutely convergent in U. Since by (12), (11) and (17\*), $$|\mathcal{J}_{\gamma}(x)| \le C \sum_{s=0}^{\infty} \varepsilon_s \int_{\mathbb{R}^n} (2 + |\xi|)^{E_s} d\xi, \qquad (33)$$ where $$E_s(x) = N + j - s + t [h_p(s; \alpha) - \langle x, \alpha \rangle] \quad (s \ge 0), \quad (34)$$ it suffices to prove that, for instance, $$\mathbf{E}_{s}(x) \le -n - 1. \tag{35}$$ Consider first s=0, 1,..., $s_1-1+N$ . Then the inequality (4), applied to $x_1=x^(s_1+N;\alpha), x_2=x^(s_0+N;\alpha), \nu_p(x_1)=t\alpha$ , and the inequalities (29), (30) and (31) give $$t \ge \frac{s_1 - s_0}{h_n(s_1 + N; \alpha) - h_n(s_0 + N; \alpha)} > \frac{2(s_0 + N)}{\varepsilon}$$ (36) Hence by (30) and (31), the function $E_s(x)$ satisfies in U the estimate $$E_s(x) < N + j - s - t\epsilon < N + j - s - 2(s_0 + N) < -n - 1.$$ (37) Now let $s \ge s_1 + N$ . Then, for $x \in U$ , inequalities (4) and (32) yield $$\left\{ E_{s}(x) \leq N + j - s + t \left[ h_{p}(s; \alpha) - h_{p}(s_{1} + N; \alpha) \right] + t \left[ h_{p}(s_{1} + N) - \langle x, \alpha \rangle \right] \leq j - s_{1} \\ < -n - 1. \right\} (38)$$ Thus, inequality (27) is established. By inequality (21), it remains to prove that, for an arbitrary $\Omega \supset \text{supp } \Phi$ , $$\mathcal{L}(\Phi; \Omega; \sigma; \alpha) \leq h_{\bullet \bullet, \Phi}(\alpha). \tag{39}$$ We can assume $\mathcal{L}(\Phi; \Omega; \sigma; \alpha) > -\infty$ . Let $\varepsilon > 0$ be fixed. Then one has $\Phi = \Phi_1 + \Phi_2$ , where $\Phi_2 \in C_0^{\infty}(\mathbb{R}^n)$ and $\Phi_1 \in \mathscr{E}'$ is such that $$h_{s,\Phi_1}(\alpha) < \varepsilon + h_{ss,\Phi}(\alpha).$$ (40) Let $\Omega'$ be the open $\varepsilon$ -neighborhood of the set cv. sing supp $\Phi$ . Choose $p' \in \mathcal{R}_0(\Omega'; 1:; \delta)$ . Then by (40), $$|\hat{\Phi}_1(\zeta)| \le C' e^{N'\omega(\xi)} k(p'; \{\varepsilon_s\}; 1; \zeta)$$ (41) where $N' = N(p'; \Phi)$ . Taking an arbitrary $s_0' \ge |N'|$ and setting $s_0 = s_0' + N'$ in Lemma 3, we obtain a function $$p\in \mathfrak{L}_0(\Omega\;;1\;;\delta)\subset \mathfrak{L}_0(\Omega\;;\sigma\;;\delta).$$ However the Paley-Wiener theorem for $C_0^{\infty}$ -functions shows that for ⁽¹⁾ The Paley-Wiener theorem implies that if p' is chosen so that supp $\Phi_1 \subset \inf H_p(1)$ , then $N' \leq M(\Phi_1) + 1 = M(\Phi) + 1$ . any real M and the same sequence $\{\epsilon_{\rm s}\}$ as in (41), there is a $C_{\rm M} > 0$ such that $$|\hat{\Phi}_{2}(\zeta)| \leq C_{\mathsf{M}} e^{\mathsf{M}\omega(\xi)} k(p; \{\varepsilon_{s}\}; 1; \zeta). \tag{42}$$ Adding the last two inequalities (with $M = N(p'; \Phi)$ in (42)) we get $$|\hat{\Phi}(\zeta)| \le C'' e^{\mathbf{N}(\mathbf{p}'; \Phi)} k(\dots; \zeta). \tag{43}$$ Hence $N = N(p; \Phi) \leq N'$ and $$h_p(s_0' + N) \le h_p(s_0' + N') = h_{p'}(s_0) \le h_{ss,\Phi} + \varepsilon$$ , which proves (39). Remark 4. — It follows from Theorem 1 that $\sigma$ plays only an auxiliary role in the formulae (20) and (20\*). However, Theorem 1 also has the following consequence. It can be shown (cf. [4], Prop. 3 (1)) that, as in the case of the space $\mathcal{O}(\Omega)$ (cf. (16)), one can describe the topology of the space $\mathcal{E}'(\Omega)$ by means of all inequalities (17) with $p \in \mathcal{R}_0(\Omega)$ . But then the equation $\mathcal{E}(\Phi; \Omega; \alpha) \equiv h_{ss.\Phi}(\alpha)$ means that by looking at the inequalities (17) for $p \in \mathcal{R}_0(\Omega)$ only (2) (i.e. at all the seminorms of $\Phi$ in the topology of $\mathcal{E}(\Omega)$ ), one can determine the set cv. sing supp $\Phi$ . This obviously fails for the set cv. supp $\Phi$ . In this case we have to use inequalities (17) for all $p \in \mathcal{R}_0(\Phi)$ . (In other words, the set cv. supp $\Phi$ cannot be described by the function which, in accordance with the above notation, could be denoted by $\mathcal{E}(\Phi; \Omega; \sigma; \alpha)$ .) This shows that the topology of $\mathcal{E}'(\Omega)$ is more closely related to singular supports than to supports$

Proposition 2. – Given a distribution $\Phi \in \mathcal{E}'$ , the following conditions are equivalent: (i) there exist constants r > 0, C > 0 and A real such that for every entire function f. $$\Theta(\Phi; C; A; \zeta) |f(\zeta)| \leq |\hat{\Phi}(\zeta) f(\zeta)|, \qquad (\forall \zeta \in \mathbb{C}^n) ; (46)$$ - (ii) condition (i) with $f \equiv 1$ only; - (iii) condition (i) with |...|, replaced in (46) by [...],; - (iv) condition (iii) with $f \equiv 1$ only. Proof. — Let $\zeta$ be fixed. It suffices to show that (ii) $\Rightarrow$ (i) $\Rightarrow$ (iii). Consider the inner function B(z) corresponding to the function $\hat{\Phi}(z)$ and the polydisk $\Delta(\zeta;r)$ (cf. [17]). Then $\hat{\Phi}(z) = Q(z) B(z)$ where both Q and B are analytic in $\Delta(\zeta;r)$ , $Q(z) \neq 0$ in $\Delta(\zeta;r)$ and |B(z)| = 1 on $\Delta(\zeta;r)$ . The mean value property of harmonic functions combined with condition (ii) gives $$|Q(\zeta)| = |Q(\zeta)|_{r} = |\hat{\Phi}(\zeta)|_{r} \geqslant \Theta(\cdots; \zeta).$$ Hence, if f is an arbitrary entire function, then $$|f(\zeta)| \Theta(\ldots;\zeta) \leq |f(\zeta)Q(\zeta)| \leq |f(\zeta)Q(\zeta)|_{r} = |f(\zeta)\Phi(\zeta)|_{r},$$ and the inequality (46) is established. The rest follows easily from Lemma 6. DEFINITION. — A distribution $\Phi$ with compact support is said to be of class $\mathcal{R}$ , provided $\Phi$ satisfies one of the conditions (i) — (iv) in Proposition 2. $\hat{\mathcal{R}}$ will denote the class $\{\hat{\Phi}: \Phi \in \mathcal{R}\}$ . The simplest examples of distributions of class $\mathcal{R}$ are linear partial differential operators with constant coefficients; in other words, all polynomials are in the class $\hat{\mathcal{R}}$ . This was proved by Malgrange in [14], and a similar statement appears in Ehrenpreis [10]. Actually, in this case one can show that inequality (46) is satisfied for each positive r(1). It is natural to ask for other examples of such distributions. (1) Distributions with this property will be called of class $\mathcal{R}_{n}$ .

Proposition 3. – Let $P(\zeta)$ be an exponential polynomial, i.e. $$P(\zeta) = \sum_{k=1}^{m} P_k(\zeta) e^{\langle \alpha_k, \zeta \rangle},$$ where the $P_k$ 's are polynomials and $\alpha_k = (\alpha_k^1, \ldots, \alpha_k^n) \in C^n$ are the so-called frequencies of P. Set $h_P(\zeta) = \max_k Re < \alpha_k$ , $\zeta > Then$ , for each $\varepsilon > 0$ , there exists a constant $C = C(\varepsilon, P)$ such that, if f is an analytic function in the polydisk $\Delta(\zeta; \varepsilon)$ , then $$|f(\zeta)| e^{h_{p}(\zeta)} \le C[f(\zeta) P(\zeta)]_{\epsilon}. \tag{47}$$ Remark 5. — In particular, if all frequencies are pure imaginary, i.e. $\alpha_k = -ia_k$ , $a_k \in \mathbb{R}^n$ , then P is the Fourier transform of a distribution T with finite support and $h_p(\zeta) = h_{s,T}(\eta)$ . In this case, Proposition 3 implies that $T \in \mathcal{R}_0$ Proof of Proposition 3: We can assume that $$h_{\rm p}(\zeta) = {\rm Re} < \alpha_1, \zeta >.$$ There exists a differential operator $Q_1(D)$ with constant coefficients such that $$Q_1(D) P(\zeta) = C_1 e^{\langle \alpha_1, \xi \rangle} \qquad (C_1 \neq 0).$$ $(Q_1(x_1,\ldots,x_n))$ can be defined as a certain product of factors $(x_j-\alpha_k^j)$ ; hence $C_1$ depends on the coefficients of the polynomial $P_1$ and the frequencies of P). The Cauchy formula then yields $$|C_1 e^{\langle \alpha_1, \xi \rangle}| \leq C_2(\varepsilon, P) [P(\zeta)]_{\varepsilon}$$ As in the proof of Proposition 2 we shall consider a function R(z) which is analytic and $\neq 0$ in $\Delta(\zeta; \varepsilon)$ and such that |R(z)| = |P(z)| on $\dot{\Delta}(\zeta; \varepsilon)$ . (For n = 1, it suffices to divide P by the corresponding Blaschke product; for n > 1, one has to use inner functions, cf. Rudin [17]). Then, $$|R(\zeta)| = [R(\zeta)]_{\varepsilon} \geqslant C_3 e^{h_p(\zeta)}$$ Applying the Cauchy formula to $R(\zeta) f(\zeta)$ completes the proof. Remark 6. – By virtue of this Proposition, the characteristic function of each bounded parallelepiped in $\mathbb{R}^n$ is in the class $\mathbb{R}$ ; but then it is natural to pose the following problem: Given an arbitrary compact convex set K in $\mathbb{R}^n$ , is its characteristic function $\chi_K$ in the class $\mathscr{R}$ (or even $\mathscr{R}_0$ )? We can prove it only when K is a polyhedron. (\*) Remark 7. — If we replace in (47) the exponent $h_P(\zeta)$ by $-Ch_P(\zeta)$ , C>0, we obtain a weaker inequality which was proved by Ehrenpreis in [11] by using mean periodicity. This inequality is still sufficient for the proof of the division problem for exponential polynomials P in the spaces $\mathcal{O}_F'(\mathbb{R}^n)$ , $\mathcal{O}(\mathbb{R}^n)$ , etc. (cf. [10], [11]). However, if we want to prove that, e.g., for each $\psi \in \mathcal{O}'(\Omega)$ , equation (44) (with $\Phi = P$ ) has a solution $\varphi \in \mathcal{O}'(\Omega')$ where $\Omega' - cv$ . supp $P = \Omega$ , then we need in (47) $h_P$ with the "+" sign. More generally, one can show that for distributions of class $\mathcal{O}$ the solution of this division problem is surprisingly simple (cf. [9], Th. 2). To some extent this explains the interest in such distributions. One can expect that these distributions will have some further interesting properties. We shall mention only two of them.

Proposition 4. – For each $$\Phi \in \mathcal{R}$$ , cv. supp $\Phi = \text{cv. sing supp } \Phi$ . (48) Proof. – Inequality (46) with $f \equiv 1$ and estimate (17) with arbitrary $p \in \mathcal{R}_0(\Phi; 1; \delta)$ imply by Lemma 4, (1) $$\exp[h_{s,\Phi}(\eta)] \leq C'(1+|\eta|) \exp[(N_p - A)\omega + h_p(|\eta|/\omega;\eta)]. \tag{49}$$ Fix $\varepsilon \in (0, 1)$ , $\alpha \in \mathbb{S}^{n-1}$ and consider only those $\zeta \in \mathbb{C}^n$ for which $\eta/|\eta| = \alpha$ . Then for each $\delta \leq \delta_{\varepsilon}$ and $s \geq s_{\varepsilon}$ there is a function $q = q_s \in \mathfrak{R}_n(\Phi; 1; \delta)$ such that $$\begin{cases} h_q(s + N_q; \alpha) < \varepsilon + h_{ss,\Phi}(\alpha) \\ N_q \leq M(\Phi) + 1 \quad (cf. (^1), p. 66). \end{cases}$$ (50) $^{(^1)} N_p = N(p; \Phi).$ ^(\*) Added in proofs: If K is a sphere, or more generally, an ellipsoid, then $\chi_K \notin \mathcal{R}$ (cf. our forthcoming paper, On convolution equations I). Taking logarithms on both sides of (49) and dividing by $|\eta|$ we get $$\begin{split} h_{s.\,\Phi}(\alpha) &\leqslant \left[\log\,\mathrm{C}_q'\,+\,(\mathrm{N}_q-\mathrm{A})\omega\,+\,\log(1+|\eta\,|)\right]\,|\eta\,|^{-1} \\ &\quad +\,h_q(|\eta\,|/\omega\,\,;\alpha). \end{split} \label{eq:hspectrum} \tag{51}$$ Setting $|\eta| = (s + N_q)\omega$ and letting $s \rightarrow \infty$ , inequalities (50) and (51) imply $$h_{s,\Phi}(\alpha) \leq h_{ss,\Phi}(\alpha) + \varepsilon$$ , which proves the proposition.

Theorem 2. — (The Titchmarsh-Lions formula for singular supports). For each $\Phi \in \mathcal{R}$ and $\Psi \in \mathcal{B}'$ , cv. sing supp $$(\Phi * \Psi) = \text{cv.}$$ sing supp $\Phi + \text{cv.}$ sing supp $\Psi$ . (52)