Probabilistic estimates for tensor products of random vectors

We prove some probabilistic estimates for tensor products of random vectors. As an application we obtain embeddings of certain matrix spaces into $L_1$.Comment: 15 page

On the Distribution of Random variables corresponding to Musielak-Orlicz norms

Given a normalized Orlicz function $M$ we provide an easy formula for a distribution such that, if $X$ is a random variable distributed accordingly and $X_1,...,X_n$ are independent copies of $X$, then the expected value of the p-norm of the vector $(x_iX_i)_{i=1}^n$ is of the order $\| x \|_M$ (up to constants dependent on p only). In case $p=2$ we need the function $t\mapsto tM'(t) - M(t)$ to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into $L_1[0,1]$. We also provide a general result replacing the $\ell_p$-norm by an arbitrary $N$-norm. This complements some deep results obtained by Gordon, Litvak, Sch\"utt, and Werner. We also prove a result in the spirit of their work which is of a simpler form and easier to apply. All results are true in the more general setting of Musielak-Orlicz spaces

The acidic sulfonimidation of aryl–oxazolines and the electrophilic fluorination of aryl–oxazolines

Aryl oxazolines have been employed as directing groups for several different coupling reactions. In the exploration of using phenyl oxazoline as a directing group for electrophilic fluorination, it was discovered that acidic sulfonimide nucleophiles including dibenzenesulfonimide, o–benzenesulfonimide, dimethanesulfonimide, and N–(methylsulfonyl)–benzenesulfonamide are discovered to open a variety of alkyl, aryl and heteroaryl–2–oxazoline rings to provide the sulfonimidation products in refluxing 1,4–dioxane. The electron rich 2–oxazoline substrates worked well for the nucleophilic ring opening reactions while no reaction took place for the electron poor 2–oxazoline substrates. 4,4–dimethyl–2–phenyl–2–oxazoline was thus rationally designed inhibit sulfonimidation and to act as a removable ortho directing group for the palladium catalyzed C–H electrophilic fluorination of arenes. Using NFSI as the fluorinating agent, and a Pd(II), Ag(I) catalytic system, electrophilic C(sp2–H) ortho–fluorination took place on a variety of aryl substrates to afford the corresponding mono and di–fluorinated products. The oxazoline directing group was then hydrolyzed unmasking the carboxylic acid moiety, demonstrating the synthetic utility of this reaction