||Let P(D) be the Laplacian Delta, or the wave operator square. The following type of Carleman estimate is known to be true on a certain range of p, q: parallel to e(v.x)u parallel to(Lq(Rd)) <= C parallel to e(v.x)P(D)u parallel to(Lp(Rd)) with C independent of v is an element of R-d. The estimates are consequences of the uniform Sobolev type estimates for second order differential operators due to Kenig-Ruiz-Sogge  and Jeong-Kwon-Lee . The range of p, q for which the uniform Sobolev type estimates hold was completely characterized for the second order differential operators with nondegenerate principal part. But the optimal range of p, q for which the Carleman estimate holds has not been clarified before. When P(D) = Delta, square, or the heat operator, we obtain a complete characterization of the admissible p, q for the aforementioned type of Carleman estimate. For this purpose we investigate L-p-L-q boundedness of related multiplier operators. As applications, we also obtain some unique continuation results.