Let S be a set of integers and denote the characteristic function of S as \chi_{S}(n)$\chi_{S}(n)$. Define an operator on the space of trig functions by the relation \hat{Tf}(n) = \chi_{S}(n) \hat{f}(n)$\hat{Tf}(n) = \chi_{S}(n) \hat{f}(n)$. Here \hat{f}(n)$\hat{f}(n)$ is the n-th Fourier coefficient of f.
For p>=2$p\geq 2$ we'll call S a L^{p}$L^p$ multiplier set (or just L^{p}$L^p$ multiplier) if there is an inequality of the form ||Tf||{p} \leq c ||f||{p}$\Vert Tf\Vert\_{p} \leq c \Vert f\Vert\_p$. If this inequality holds for some p but fails for p+\epsilon$p+\epsilon$ for every epsilon>0$\epsilon>0$, we'll say that S is a strict L^{p}$L^p$ multiplier.
Note that every set is a L^2$L^2$ multiplier and if S is a L^{p}$L^p$ multiplier for some p then it is a L^{q}$L^q$ multiplier for 2 <= q <= p$2 \leq q \leq p$. Moreover, it follows from a result of Zygmund that almost every (in the obvious sense) set is a strict L^{2}$L^2$ multiplier. (I also think you can prove this via Khintchine's inequality, but I haven't checked this argument.)
Do strict L^{p}$L^p$ multiplier sets exist for every p>2$p>2$? Note that this is similar to the \Lambda(p)$\Lambda(p)$ problem, however, I don't see how to transform a strict \Lambda(p)$\Lambda(p)$ set into a strict L^{p}$L^p$ multiplier set.
