Let S be a set of integers and denote the characteristic function of S as $\chi\_{S}(n)$. Define an operator on the space of trig functions by the relation $\hat{Tf}(n) = \chi\_{S}(n) \hat{f}(n)$. Here $\hat{f}(n)$ is the n-th Fourier coefficient of f.


For $p\geq 2$ we'll call S a $L^p$ multiplier set (or just $L^p$ multiplier) if there is an inequality of the form $\Vert Tf\Vert\\_{p} \leq c \Vert f\Vert\\_p$. If this inequality holds for some p but fails for $p+\epsilon$ for every $\epsilon>0$, we'll say that S is a strict $L^p$ multiplier.


Note that every set is a $L^2$ multiplier and if S is a $L^p$ multiplier for some p then it is a $L^q$ multiplier for $2 \leq q \leq p$. Moreover, it follows from a result of Zygmund that almost every (in the obvious sense) set is a strict $L^2$ multiplier. (I also think you can prove this via Khintchine's inequality, but I haven't checked this argument.) 


Do strict $L^p$ multiplier sets exist for every $p>2$? Note that this is similar to the $\Lambda(p)$ problem, however, I don't see how to transform a strict $\Lambda(p)$ set into a strict $L^p$ multiplier set.



L^{p} multiplier sets
Let M\_p be the class of L^p muliplier sets, as considered in this equation. It is known: 


1. This is an algebra of sets, but not a sigma-algebra. [Not sure of the reference.]
2. The inclusion M\_p \subsetneq M\_q is strict for 2\le p


MR1728363 (2001g:42013)
Mockenhaupt, Gerd(5-NSW-SM); Ricker, Werner J.(5-NSW-SM)
Idempotent multipliers for $L^p(\bf R)$. 


