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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 $k$ J$possible choices$t_1,\dots,t_k$); t_1,\dots,t_J$); note that Jean crucially refers in the paper to a "suitable" $t$ rather than an arbitrary $t$. This is a pigeonholing argument, based on the estimation of

$$\sum_{i=1}^k sum_{j=1}^J \| f * P_{\delta t_it_j} - f * P_{\delta^{-1} t_it_j} \|_{L^2}^2$$

which can be done by Plancherel's theorem and routine computations (if the $t_i$ t_j$are lacunary, as noted in Jean's paper). The use of pigeonholing to turn qualitative results (such as dominated convergence) to quantitative ones (at the cost of losing some control on the parameter for which the bound is attained) is an important trick in the subject; I discuss it at http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/ . Another key trick displayed here is to always be aware whether one needs to control the worst-case choice of parameter (i.e. uniform bounds), average-case choice of parameter (e.g. integrated or probabilistic bounds), or best-case choice of parameter (e.g. what comes from the pigeonhole principle). In this case, because one only needs the bound for a single t, best-case analysis suffices, and one can use many more tricks in this setting than in worst-case or average-case analysis. Incidentally, I found the reading of Jean's papers as a graduate student to be simultaneously extremely frustrating and extremely rewarding. Decoding an offhand remark or a mysterious step in his paper was often as instructive (and as time-consuming) as reading several pages of arguments by some other authors. (But his papers do become much easier to read once one has internalised enough of his "box of tools"...) 3 added 91 characters in body; added 14 characters in body; added 24 characters in body I assume you are referring to the argument in page 313 of Jean's paper http://www.springerlink.com/content/a343g53033872345/ . The point here is that the bound does not hold for all t,$t$, but for a single t$t$(out of k$k$possible choices t_1,...,t_k).$t_1,\dots,t_k$); note that Jean crucially refers in the paper to a "suitable"$t$rather than an arbitrary$t$. This is a pigeonholing argument, based on the estimation of $$\sum_{i=1}^k \| f * P_{\delta t_i} - f * P_{\delta^{-1} t_i} \|_{L^2}^2$$ 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/ . Another key trick displayed here is to always be aware whether one needs to control the worst-case choice of parameter (i.e. uniform bounds), average-case choice of parameter (e.g. integrated or probabilistic bounds), or best-case choice of parameter (e.g. what comes from the pigeonhole principle). In this case, because one only needs the bound for a single t, best-case analysis suffices, and one can use many more tricks in this setting than in worst-case or average-case analysis. 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 (and as time-consuming) as reading several pages of arguments by some other authors. (But his papers do become much easier to read once one has internalised enough of his "box of tools"...) 2 added 608 characters in body 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 $$\sum_{i=1}^k \| f * P_{\delta t_i} - f * P_{\delta/t_iP_{\delta^{-1} t_i} \|_{L^2}^2$$ 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/ . Another key trick displayed here is to always be aware whether one needs to control the worst-case choice of parameter (i.e. uniform bounds), average-case choice of parameter (e.g. integrated or probabilistic bounds), or best-case choice of parameter (e.g. what comes from the pigeonhole principle). In this case, because one only needs the bound for a single t, best-case analysis suffices, and one can use many more tricks in this setting than in worst-case or average-case analysis.

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 some other authors..authors. (But his papers do become much easier to read once one has internalised enough of his "box of tools"...)

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