In modern times, topological quantum groups are well understood through operator algebraic methods. Moreover, Fourier analysis and non-commutative Lp analysis can be explored on topological quantum groups. In this presentation, we will talk about what kinds of analysis can be studied on quantum groups. The first objective of this presentation is to provide some results on Lp-lq multipliers on compact(or discrete) quantum groups. Those can be regarded as quantum analogues of a theorem of Littlewood on random Fourier series, Hardy-Littlewood inequalities and local Hausdorff-Young inequalities. Then applications of these results to a complete representability problem and an improvement of uncertainty relations under localizations will be discussed. The second objective is to complete our perception on the similarity problem on topological quantum groups. The famous results of Day and Dixmier in 1950 state that every uniformly bounded representation of an amenable group is similar to a unitary representation. In short, we say that amenability implies unitarizability. However, it turns out that amenable quantum groups are not always unitarizable within the framework of topological quantum groups, and this issue will be addressed in detail.