We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and uniqueness results, as well as pointwise saturation of the constraint, are established. We also derive a critical point equation which gives rise to a dual formu...

Differential forms are interesting and important generalizations of real functions and distributions. Many interesting results and applications of differential forms have recently been found in some fields. As an important tool the Hardy-Littlewood inequality have been playing critical roles in many mathematics, including potential analysis, partial differential equations and the theory of elas...

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight r for functions in R. The exact Hardy constant cb = cb(n) is found and generalized minimizers are given. The constant cb vanishes on a countable set of b, which extends the known case n = 2, b = 0 which corresponds to the trivial Hardy inequality in R. Analogous inequalities are proved in the ca...