We will show both original and known results on Harmonic Analysis for functions defined on the infinite-dimensional torus, which is the topological compact group consisting of the Cartesian product of countably infinite many copies of the one-dimensional torus, with its corresponding Haar measure. Such results will include:

- Fourier Analysis on $\mathbb{T}^{\omega}$: basic properties, absolutely divergent series, Calder'on-Zygmund decomposition and differentiation of integrals.
- Mixed norm $L^{\bar{p}}(\mathbb{T}^{\omega})$ spaces, $\bar{p}=(p_1,p_2,\ldots)$: definition, properties, duality, interpolation and weak spaces.
- M. Riesz Theorems on $\mathbb{T}^{\omega}$, a.e. convergence, rectangular partial sums and $L^{\bar{p}}$ convergence.

Several open problems and other questions will be considered. Some of the results presented are joint work with Emilio Fernandez (Universidad de La Rioja, Spain).

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Last updated: 19 Sep 2024