Hello,

I am searching for a reference of an " easy " sufficient condition insuring that a bounded sequence $(b_{\mathbf{n}\in\mathbb{Z}^d})\in\ell^\infty$ defines a bounded operator from $L^p(\mathbb{T}^d)$ to itself for all finite $p>1$ (*via* the multiplication of the Fourier coefficients of course).

In fact my real goal is to check that the sequence $b_{\mathbf{n}}=n_i/|\mathbf{n}|$ (the discrete version of the Riesz transform) is indeed an $L^p(\mathbb{T}^d)$ multiplier and hence that the (periodic) Leray projection $\mathbb{P}:L^2(\mathbb{T}^d)^d \rightarrow L^2_{\text{div}}(\mathbb{T}^d)^d$ is a bounded operator from $L^p(\mathbb{T}^d)$ to itself for all finite $p>1$.

I am quite sure that the previous result is true since it's the case in $\mathbb{R}^d$ case *via* singular operator theory, but it should be easier in the compact case ! I am only searching for a simple, if possible modern, reference. This seems absent in the Grafakos book, at least in the form that I had just stated.

Thanks for your help !

Ayman.