Poincaré–Sobolev and isoperimetric inequalities, maximal functions, and half-space estimates for the gradient
We investigate whether the L¹-algebra of polynomial hypergroups has non-zero bounded point derivations. We show that the existence of such point derivations heavily depends on growth properties of the Haar weights. Many examples are studied in detail. We can thus demonstrate that the L¹-algebras of hypergroups have properties (connected with amenability) that are very different from those of groups.
We prove that for each linear contraction T : X → X (∥T∥ ≤ 1), the subspace F = {x ∈ X : Tx = x} of fixed points is 1-complemented, where X is a suitable subspace of L¹(E*) and E* is a separable dual space such that the weak and weak* topologies coincide on the unit sphere. We also prove some related fixed point results.
In this note we exhibit points of weak*-norm continuity in the dual unit ball of the injective tensor product of two Banach spaces when one of them is a G-space.
We study classes of operators represented as a pointwise absolutely convergent series of simpler ones, starting with rank 1 operators. In this short note we address the question, how far the repetition of this procedure can lead.