We resolve a long-standing question on completeness of the nonorthogonal Mexican hat wavelet system, in L for 1 < p < 2 and in the Hardy space H for 2/3 < p ≤ 1. Tools include the discrete Calderón condition, a generalization of the Daubechies frame criterion to a weighted L space, and imbeddings of that weighted space into L and Hardy spaces.

We show that a big Hankel operator on the standard Hardy space of the polydisk D, n > 1, cannot be compact unless it is the zero operator. We also show that this result can be generalized to certain Hankel operators defined on Hardy-Sobolev spaces of the polydisk.

We give a simple proof of the λ = d - 2 cases of the sharp Hardy-Littlewood-Sobolev inequality for d≥3, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for d = 2 via a monotone flow governed by the fast diffusion equation.

Marcinkiewicz multipliers are L bounded for 1 < p < ∞ on the Heisenberg group H ≃ C × R (D. Muller, F. Ricci and E. M. Stein [25], [26]). This is surprising in that this class of multipliers is invariant under a two parameter group of dilations on C × R, while there is no two parameter group of automorphic dilations on H. This lack of automorphic dilations underlies the inability of classical o...

Two related approximation problems are formulated and solved in Hardy spaces of the disc and annulus. With practical applications in mind, truncated versions of these problems are analysed, where the solutions are chosen to lie in finite-dimensional spaces of polynomials or rational functions, and are expressed in terms of truncated Toeplitz operators. The results are illustrated by numerical e...

We give a short summary of Varopoulos’ generalised Hardy-LittlewoodSobolev inequality for self-adjoint C0 semigroups and give a new probabilistic representation of the classical fractional integral operators on Rn as projections of martingale transforms. Using this formula we derive a new proof of the classical Hardy-LittlewoodSobolev inequality based on Burkholder-Gundy and Doob’s inequalities...

Some classical results due to Marcinkiewicz, Littlewood and Paley are proved for the Ciesielski-Fourier series. The Marcinkiewicz multiplier theorem is obtained for Lp spaces and extended to Hardy spaces. The boundedness of the Sunouchi operator on Lp and Hardy spaces is also investigated. 2000 AMS subject classifications: Primary 41A15, 42A45, 42B25, Secondary 42C10, 42B30.

We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators Lpu := −∇ ∗ L(|∇Lu| ∇Lu). If φ is a positive weight such that −Lpφ ≥ 0, then the Hardy type inequality c ∫ Ω |u| φp |∇Lφ| p dξ ≤ ∫

As new applications of Schrödinger type inequalities obtained by Jiang (J. Inequal. Appl. 2016: Article ID 247, 2016) in the Schrödingerean Hardy space, we not only obtain the representation of Schrödingerean harmonic functions but also give a sufficient and necessary condition between the Schrödingerean distributional function and its derivative in the Schrödingerean Hardy space.

Techniques of constrained approximation are used to recover solutions to elliptic partial differential equations from incomplete and corrupted boundary data. The approach involves the use of generalized Hardy spaces of functions whose real and imaginary parts are related by formulae similar to the Cauchy–Riemann equations. A prime motivation for this is the modelling of plasma confinement in a ...