It has been recognized by many authors that contractive inequalities involving norms of $H^p$ spaces can be particularly useful when the objects in question (like the norms and/or an underlying operator) lift in a multiplicative way from one (or few) to several (or infinitely many) variables. This has been my main motivation for looking more systematically at various contractive inequalities in the context of Hardy spaces on the $d$-dimensional torus. I will discuss results from recent studies of Hardy--Littlewood inequalities, Riesz projections, idempotent Fourier multipliers, and Hilbert points (which in one variable is another word for inner functions). We will see interesting phenomena occurring both in the transition from low to high dimension and from low to infinite dimension. The talk builds on joint work with Sergei Konyagin, Herve Queffelec, and Eero Saksman and with Ole Fredrik Brevig and Joaquim Ortega-Cerda.