Carleman estimates and boundedness of associated multiplier operators

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 [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 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.

Automated summary

The paper discusses the range of Carleman estimates, particularly focusing on the p, q ranges when dealing with Laplacian Delta, wave operator square, and heat operator. By utilizing uniform Sobolev type estimates and investigating the L-p-L-q boundedness of related multiplier operators, they were able to infer the applicable range of Carleman estimates and obtain some unique continuation results.