Harmonic analysis of SL(2) over a locally compact field

We carry over the pioneer work of Kunze and Stein concerning representation theory and harmonic analysis on SL(2, R) to the group G = SL(2, K), K a locally compact totally disconnected nondiscrete field. The main result is that convolution by an L^p(G) function, 1 ? p < 2, is a bounded operator on L²(G). To accomplish this result we develop the appropriate estimates (which depend upon the work of Sally et al.) that enable us to apply the Kunze and Stein interpolation theory to the Fourier-Laplace transform for the group G. Best possible estimates are obtained.