This paper studies Banach space valued Hausdorff-Young inequalities. The largest part considers ways of changing the underlying group. In particular the possibility to deduce the inequality for open subgroups as well as for quotient groups arising from compact subgroups is secured. A large body of results concerns the classical groups$T$^{$n$},$R$^{$n$}and$Z$_{$k$}. Notions of Fourier type are introduced and they are shown to be equivalent to properties expressed by finite groups alone.

We study the boundedness problem for maximal operators $ \mathcal{M} $ associated with averages along smooth hypersurfaces$S$of finite type in 3-dimensional Euclidean space. For$p$> 2, we prove that if no affine tangent plane to$S$passes through the origin and$S$is analytic, then the associated maximal operator is bounded on $ {L^p}\left( {{\mathbb{R}^3}} \right) $ if and only if$p$>$h$($S$), where$h$($S$) denotes the so-called height of the surface$S$(defined in terms of certain Newton diagrams). For non-analytic$S$we obtain the same statement with the exception of the exponent$p$=$h$($S$). Our notion of height$h$($S$) is closely related to A. N. Varchenko’s notion of height$h$($ϕ$) for functions$ϕ$such that$S$can be locally represented as the graph of$ϕ$after a rotation of coordinates.

We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.

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