A multiplier theorem for Fourier series in several variables
Tom 106 / 2006
Colloquium Mathematicum 106 (2006), 221-230 MSC: Primary 42B15; Secondary 60G42. DOI: 10.4064/cm106-2-4
We define a new type of multiplier operators on $L^p(\mathbb T^N)$, where $\mathbb T^N$ is the $N$-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension $N$. Our construction is motivated by the conjugate function operator on $L^p(\mathbb T^N)$, to which the theorem applies as a particular example.