I'm currently studying a particular Fourier multiplier and I came across the following question.
The cubic $d$-dimensional Dirichlet kernel is \begin{equation} D_n(x)=\prod_{i=1}^d D_n^1(x_i), \end{equation} where $D_n^1(y)=\sum_{j=-n}^n e^{ijx}$, and it has $L^1(\mathbb T^d)$-norm that goes as $(\log n)^d$ with respect to the parameter $n$.
Does the same result hold for the spherical $d$-dimensional Dirichlet kernel (recorded below)? \begin{equation} \sum_{k \in \mathbb Z^d, |k|_2\le n} e^{i k\cdot x} \end{equation}
Does it hold with respect to any real and positive value of $n$?
I thank anyone who can suggest a possible proof or any related reference.
This is a great question, and not one that is particularly easy to research and find fast answers for online. The terms to search for here are "Lebesgue constants for spherical partial sums," and there is a nice short paper of Shapiro's by the same title with a proof that $$ \int_{\mathbb T^d}|\sum_{|k|_2<N}e^{ik\cdot x}|\,dx \sim_d N^{(d-1)/2}, $$ whenever $d\ge 2$. For two positive quantities $A,B$, the notation $A\sim_d B$ means that $c_dA\le B\le C_dA$ for absolute dimensional constants $c_d,C_d$. This is a huge contrast with the Lebesgue constants for the cubical partial sums, which as you note are $\sim_d (\log N)^d$.
There is a very readable account of the $N^{(d-1)/2}$ estimate of the Lebesgue constants for spherical partial sums in Section 1 of Lifyland's survey "Lebesgue Constants of Multiple Fourier Series."
What is essentially at play here is that the boundary of the region $\{\xi\in\mathbb R^d:|\xi|_2<N\}$ is a sphere of radius $N$, which has curvature, as compared with the cubical boundary $\{\xi\in\mathbb R^d:|\xi|_\infty<N\}$, which does not have curvature. As a consequence, the Fourier transform of the indicator function of a round ball has a power-law decay.
The interplay of curvature and the Fourier transform is one of the central themes of modern Fourier analysis. For an introduction to this, including the famous "Fourier restriction problem," you could look at the last chapter of Stein and Shakarchi's Functional analysis: introduction to further topics in analysis.