Absolute regularity for convergent integrals

Nonabsolutely convergent Poisson integrals

If a function f has finite Henstock integral on the boundary of the unit disk of R 2 then its Poisson integral exists for |z| < 1 and is o((1 − |z|) −1) as |z| → 1 −. It is shown that this is the best possible uniform pointwise estimate. For an L 1 measure the best estimate is O((1 − |z|) −1). In this paper we consider estimates of Poisson integrals on the unit circle with respect to Alexiewicz...

Boundary regularity results for variational integrals

We prove partial Hölder continuity for the gradient of minimizers u ∈W (Ω,R ), Ω ⊂ R a bounded domain, of variational integrals of the form

Non-absolutely Convergent Integrals in Metric Spaces

Gaussian integrals involving absolute value functions

Evaluating the Gaussian integrals (expectation, moments, etc.) involving the absolute value function has been playing important roles in various contents. For example, in [KN08] and [LW09], the expected number of zeros of random harmonic functions, which is also the average number of images of certain gravitational lensing system, was associated with the expectation of absolute value of certain...

A simple partial regularity proof for minimizers of variational integrals

We consider multi-dimensional variational integrals F [u] := Ω f (·, u, Du) dx where the integrand f is a strictly convex function of its last argument. We give an elementary proof for the partial C 1,α-regularity of minimizers of F. Our approach is based on the method of A-harmonic approximation, avoids the use of Gehring's lemma, and establishes partial regularity with the optimal Hölder expo...