Caltech/UCLA Joint Analysis Seminar

The classical triangle inequality in Lp estimates the norm of the sum of two functions in terms of the sums of the norms of these functions. Perhaps one drawback of this estimate is that it does not see how "orthogonal" these functions are. For example, if f and g are not identically zero and they have disjoint supports then the triangle inequality is pretty strict (say for p>1). Motivated by the L2 case, where one has a trivial inequality ||f+g||2 ≤ ||f||2 + ||g||2 + 2 |fg|1, one can think about the quantity |fg|1 as measuring the "overlap" between f and g. What is the correct analog of this estimate in Lp for p different than 2? My talk will be based on a joint work with Carlen, Frank and Lieb where we obtain one extension of this estimate in Lp, thereby proving and improving the suggested possible estimates by Carbery, and another work with Mooney where we further refine these estimates. The estimates will be provided for all real p's.