The most beautiful mathematics to Godfrey Harold Hardy was that which had no application. For Hardy, mathematics was purely for intellectual challenge. He justified the pursuit of pure mathematics with the argument that its very “uselessness” meant that it could not be used to cause harm. Hardy went so far as to describe applied mathematics as “ugly”, “trivial”, and “dull” [1]. Despite Hardy’s ...

Extension by integer translates of compactly supported function for multiplier spaces on periodic Hardy spaces to multiplier spaces on Hardy spaces is given . Shannon sampling theorem is extende d to Hardy spaces. 1 . Introduction and statement of results The purpose of this paper is to establish a natural extension from multiplier spaces M(p) on periodic Hardy spaces HP (T) to multiplie r spac...

We consider approximation of L p functions by Hardy functions on subsets of the circle. We rst derive some properties of traces of Hardy classes on such subsets, and then turn to a generalization of classical extremal problems involving norm constraints on the complementary subset. Approximation dans l'espace de Hardy de fonctions L p sur un sous{ensemble du cercle R esum e : Nous etudions l'ap...

Let Ω be a strongly Lipschitz domain of Rn. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the non-tangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L1. Under suitable assumptions on L, we identify this maximal Hardy space with atomic Hardy spaces, namely with...

We give a systematic study on the Hardy spaces of functions with values in the non-commutative L-spaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the non-commutative martingale inequalities. Our non-com...

Let P be a linear, elliptic second order symmetric operator, with an associated quadratic form q, and let W be a potential such that the Hardy inequality λ0 ∫ Ω Wu 2 ≤ q(u) holds with (non-negative) best constant λ0. We give sufficient conditions so that the spectrum of the operator 1 W P is [λ0,∞). In particular, we apply this to several well-known Hardy inequalities: (improved) Hardy inequali...

The main purpose of this paper is to briefly review the earlier works of multiparameter Hardy space theory and boundedness of singular integral operators on such spaces defined on product of Euclidean spaces, and to describe some recent developments in this direction. These recent works include discrete multiparameter Calderón reproducing formulas and Littlewood-Paley theory in the framework of...