Question

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.

Answer 1

$L^p$ boundedness (for $p\in(1,+\infty)$) of your discrete Riesz transform can be derived applying De Leeuw's theorem (Theorem 3.8 in Chapter VII of Introduction to Fourier Analysis on Euclidean Spaces by E. Stein). The multiplier $m(\xi):=\frac{\xi_i}{|\xi|}$ gives rise to a bounded operator on $L^p(\mathbb{R}^n)$ ($p$ as above) by classical singular integral theory (or the Mihlin-Hormander multiplier theorem) and hence $b:=m_{|\mathbb{Z}^n}$ gives a bounded operator on $L^p(\mathbb{T}^n)$. De Leeuw's theorem is a particular instance of the phenomenon of transference (see http://www.amazon.com/Transference-Analysis-Regional-Conference-Mathematics/dp/0821816810 for this).

Answer 2

In my site, I give a proof of the Mihlin-Hormander multiplier theorem on $L^{p}(\mathbb{R}^{d})$ which was a consequence of a more general theorem, see nucaltiado.wordpress.com.