Ph.D. (Waterloo), M.Sc. (Western Ontario), B.Sc. (Toronto)
The Henstock-Kurzweil integral is an integral with a simple definition in terms Riemann sums but it includes the Riemann, Lebesgue, improper Riemann and Cauchy-Lebesgue integrals. It has the following advantages over Riemann and Lebesgue integrals:
A convenient way to define an integral is through properties of its primitive. The primitive is a function whose derivative is in some sense equal to the integrand. For example, in Lebesgue integration the primitives are the absolutely continuous functions. A function f on the real line is integrable in the Lebesgue sense if and only if there is an absolutely continuous function F such that F'=f almost everywhere. For integration on the entire real line, the primitive must also be of bounded variation. Primitives for Riemann integrals have recently been categorised by Brian Thomson (Characterization of an indefinite Riemann integral, Real Analysis Exchange 35 (2009/2010), 491-496). The class of primitives is also known for Henstock-Kurzweil integrals. However, it is more complicated than the absolutely continuous functions. A problem with the Henstock-Kurzweil integral is that the space of integrable functions is not a Banach space. By taking the primitives as continuous functions and using the distributional derivative we obtain the continuous primitive integral. This includes the Lebesgue and Henstock-Kurzweil integrals. Under the Alexiewicz norm, the space of distributions integrable in this sense is a Banach space. It is the completion of the Lebesgue and Henstock-Kurzweil integrable functions. Primitives can also be taken as regulated functions, i.e., those that have a left and a right limit at each point. Or they can be taken as Lp functions.
Because of their oscillatory kernel, it is natural to treat Fourier transforms as Henstock-Kurzweil integrals. This provides an extension of the Lebesgue theory. Some results are similar to the absolutely convergent case, such as an inversion theorem and convolution. But there are new phenomena such as arbitrarily large growth of the transform and the failure of the transform to exist on countable sets.
The Poisson integral solves the classical Dirichlet problem for the Laplace equation in a half space but existence of the integral imposes certain growth restrictions on the boundary data. It is possible to form a modified Poisson integral by subtracting terms from the Taylor/Fourier expansion of the Poisson kernel. This then lets us write the solution of the Dirichlet problem for arbitrary locally integrable data. I have been working on obtaining the best pointwise and norm estimates of these modified Poisson integrals. Poisson integrals have been considered in the Henstock-Kurzweil sense on the circle.
An elliptic partial differential equation in a bounded domain will have a unique solution if boundary data is specified, provided the coefficients, boundary and boundary data are reasonably well behaved. For an unbounded domain we need some sort of growth condition at infinity to be imposed in order to have a unique solution. I am interested in Phragmen-Lindelof principles that allow the solution to blow up at the boundary but still yield uniqueness.