If we have a polynomial $\sum a_n e^{inx}$, its analytic projection is $\sum_{n>0} a_n e^{inx}$.
M. Riesz theorem tells us that the operator T that sends polynomials into its analytic projections, that is
$$T\left(\sum a_n e^{inx}\right)=\sum_{n>0} a_n e^{inx}$$
is bounded in $L^p$ for $1<p<\infty$. And $\|\sum_{n>0} a_n e^{inx}\|_p\leq K_p \|\sum a_n e^{inx}\|_p$, where $K_p$ is a constant that only depends on $p$.
Is there a version of this result for polynomials of the type $\sum_{}a_{k,j}e^{ikx}e^{ijy}$?