Internat. J. Math. & Math. Sci. VOL. 13 NO. 2 (1990) 243-246 243 A NOTE ON SOME SPACES L OF DISTRIBUTIONS WITH LAPLACE TRANSFORM SALVADOR PEREZ ESTEVA Instituto de Matemticas Universidad Nacional Autnoma de Mxico Mxico, D.F. 04510 M6xico (Received March 2, 1989) ABSTRACT. In this paper we calculate the dual of the spaces of distributions L introduced in [I ]. Then we prove that L is the dual of a subspace of C OR). KEY WORDS AND PHRASES. Convolution, Laplace Transform, Strict Inductive Limit. 1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. Primary 44A35, Secondary 44AI0 I. INTRODUCTION Let D’ and S’ be the classical Schwartz’s spaces of distributions in and denote by L the Laplace transformation. In (Prez-Esteva [I ]) were introduced spaces L a as follows: PY La is the subspace of L OR) of functions f with supp f C [a,) and oy loc e f L2OR), where e (x)=e-Tx. L a is a Hilbert space with the inner product -Y -y oy I (f,g) e_2fg dx then we define [a DP[a Dp DY oY where is the distributional derivative of order p. Since Dp La a is bijective, we can copy the Hilbert space structure of L a oy py oY L a on We have the continuous inclusions PY L a C L b for a>b pY py’ L a C L a PY qy, if p < q Hence for p {0,I,...} the strict inductive limit L ind lim L a PY a PY makes sense. Then L ind lim L ind lim L -p Y p PY p PY is also well defined. In[l] it was studied the spaces of distributions g for which the convolution f f,g: [ L Y is continuous.