Carleman estimates and boundedness of associated multiplier operators

Let P(D) be the Laplacian (Formula presented.) or the wave operator □. The following type of Carleman estimate is known to be true on a certain range of p, q: (Formula presented.) with C independent of (Formula presented.) The estimates are consequences of the uniform Sobolev type estimates for second order differential operators due to Kenig-Ruiz-Sogge [1] and Jeong-Kwon-Lee [2]. 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 (Formula presented.) □, 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.