RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 97 (2), 2003, pp. 243–255 An´ alisis Matem´ atico / Mathematical Analysis (Ultra)distributions of L p -growth as Boundary Values of Holomorphic Functions C. Fern ´ andez, A. Galbis and M.C. G ´ omez-Collado To the memory of I. Cioranescu and K. Floret Abstract. We study the representation of distributions (and ultradistributions of Beurling type) of Lp-growth, 1 ≤ p ≤∞, on R N as boundary values of holomorphic functions on (C \ R) N . (Ultra)distribuciones de crecimiento L p como valor frontera de funciones holomorfas Resumen. Estudiamos la representaci´ on de distribuciones (y ultradistribuciones de tipo Beurling) en R N con crecimiento Lp, 1 ≤ p ≤∞, como valor frontera de funciones holomorfas en (C \ R) N . 1. Introduction Shortly after Schwartz introduced his theory of distributions, K¨ othe represented distributions on the unit circle as boundary values of holomorphic functions on its complementary and his results were generalized by Tillmann. Since then, many authors have been concerned with the problem of representing several classes of distributions and ultradistributions as boundary values of holomorphic functions. Let us mention the work of Bengel [1], Carmichael [4], Luszczki and Zielezny [13], Meise [14], Petzsche and Vogt [17], Tillmann [20] and Vogt [22]. See also the section 4 of the recent paper [12]. In 1994, Carmichael and Pilipovi´ c [6] represented each ultradistribution of L p -growth (1 <p< ∞) in R N as the boundary value of a holomorphic function satisfying appropriate estimates and conversely, every such a function is shown to have an ultradistribution of L p -growth as boundary value. The boundary value problem for distributions and ultradistributions of L ∞ or L 1 - growth is more involved. In fact, for p =1 or p = ∞, Carmichael and Pilipovi´ c only obtained partial results and their methods did not permit to prove the surjectivity of the corresponding boundary value operators. They worked in the context of ultradistributions as they were defined by Komatsu [11]. In [8] the authors completely solved the problem of representing bounded distributions and ultradistri- butions on R as boundary values of holomorphic functions in C \ R. The case of bounded distributions had not been treated previously in the literature. The lack of nice topological properties of the involved spaces does not permit us to apply the tensor techniques as in [16, 3.6] in order to extend the results obtained in [8] to the several variables setting. Presentado por Jos´ e Bonet. Recibido: 5 de Febrero de 2003. Aceptado: 31 de Octubre de 2003. Palabras clave / Keywords: Distributions, ultradistributions, boundary values. Mathematics Subject Classifications: 46F05, 46F20. c  2003 Real Academia de Ciencias, Espa ˜ na. 243