I assume you are referring to the argument in page 313 of Jean's paper http://www.springerlink.com/content/a343g53033872345/ .  The point here is that the bound does not hold for all t, but for a single t (out of k possible choices t_1,...,t_k).  This is a pigeonholing argument, based on the estimation of

$$ \sum_{i=1}^k \| f * P_{\delta t_i} - f * P_{\delta/t_i} \|_{L^2}^2$$

which can be done by Plancherel's theorem and routine computations (if the $t_i$ are lacunary, as noted in Jean's paper).  The use of pigeonholing to turn qualitative results (such as dominated convergence) to quantitative ones (at the cost of losing some control on the parameter for which the bound is attained) is an important trick in the subject; I discuss it at http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/

Incidentally, I found the reading of Jean's papers as a graduate student to be simultaneously extremely frustrating and extremely rewarding.  Decoding an offhand remark or a mysterious step in his paper was often as instructive as reading several pages of arguments by other authors...