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Let G be a solvable Lie group endowed with right Haar measure. We define andstudy a dense Frechet *-subalgebra S of L1(G), consisting of smooth functionsrapidly-decreasing at infinity on G. When G is nilpotent, we recover theclassical Schwartz algebra introduced by R. Howe and other authors. We developa distribution theory for S, and we generalize the classical Kernel Theorem ofL. Schwartz to our setting.

Schwartz functions, tempered distributions, and Kernel Theorem on solvable Lie groups