This paper surveys a large class of nonlinear extremal problems in Hardy and Bergman spaces. We discuss the general approach to such problems in Hardy spaces developed by S. Ya. Khavinson in the 1960s, but not well known in the West. We also discuss the major difficulties distinguishing the Bergman space setting and formulate some open problems.

We perform a convergence analysis for discretization of Helmholtz scattering and resonance problems obtained by Hardy space infinite elements. Super-algebraic convergence with respect to the number of Hardy space degrees of freedom is achieved. As transparent boundary spheres and piecewise polytopes are considered. The analysis is based on a G̊arding-type inequality and standard operator theoret...

We study the adaptive decomposition of functions in the monogenic Hardy spaces H2 by higher order Szegö kernels under the framework of Clifford algebra and Clifford analysis, in the context of unit ball and half space. This is a sequel and a higher-dimensional generalization of our recent study on the complex Hardy spaces.

We show that some Hardy-type inequalities on the circle can be proved to be true on the real line. Namely, we discuss the idea of getting Hardy inequalities on the real line by the use of corresponding inequalities on the circle. In the last section, we prove the truth of a certain open problem under some restrictions.

We study the Hardy field associated with an o-minimal expansion of the real numbers. If the set of analytic germs is dense in the Hardy field, then we can definably analytic separate sets in R2, and we can definably analytic approximate definable continuous unary functions.

It is nowadays well-known that Hardy’s inequality (like many other inequalities) follows directly from Jensen’s inequality. Most of the development of Hardy type inequalities has not used this simple fact, which obviously was unknown by Hardy himself and many others. Here we report on some results obtained in this way mostly after 2002 by mainly using this fundamental idea.

Despite a true antipathy to the subject Hardy contributed deeply to modern probability. His work with Ramanujan begat probabilistic number theory. His work on Tauberian theorems and divergent series has probabilistic proofs and interpretations. Finally Hardy spaces are a central ingredient in stochastic calculus. This paper reviews his prejudices and accomplishments through these examples. ∗Dep...

In this paper, we consider the [Formula: see text]-Hardy inequalities on the sphere. By the divergence theorem, we establish the [Formula: see text]-Hardy inequalities on the sphere. Furthermore, we also obtain their best constants. Our results can be regarded as the extension of Xiao's (J. Math. Inequal. 10:793-805, 2016).

Scales of equivalent weight characterizations for the Hardy type inequality with general measures are proved. The conditions are valid in the case of indices 0 < q < p <∞, p > 1. We also include a reduction theorem for transferring a three-measure Hardy inequality to the case with two measures. © 2007 Elsevier Inc. All rights reserved.

A new proof is given of the atomic decomposition of Hardy spaces Hp, 0 < p ≤ 1, in the classical setting on Rn. The new method can be used to establish atomic decomposition of maximal Hardy spaces in general and nonclassical settings.