HK-Integral and HK-Spaces

This chapter introduces the Henstock–Kurzweil integral (HK). This is the easiest to learn and best known of those integrals, which integrate nonabsolutely integrable functions and extend the Lebesgue integral. Section 3.1 provides a fairly detailed account of the HK-integral and its properties in both the one- and n-dimensional case. Section 3.2 discusses a new class of Banach spaces (KS p spaces), which are for nonabsolutely integrable functions as the L p spaces are for Lebesgue integrable functions. These spaces contain the L p spaces as continuous dense and compact embeddings. Section 3.3 covers some additional classes of Banach spaces associated with nonabsolutely integrable functions that may have future interest. First, we introduce the natural parallel class of Sobolev type spaces for the HK-integral. In the following section, we define a more important class of spaces S D p [ ℝ n ] , 1 ≤ p ≤ ∞ $$SD^{p}[\mathbb{R}^{n}],\;1\leqslant p\leqslant \infty $$ . These spaces contain the test functions of Schwartz (Théorie des Destributions. Hermann, Paris, 1966) 𝒟 [ ℝ n ] $$\mathcal{D}[\mathbb{R}^{n}]$$ , as a dense continuous embedding. In addition, they have the remarkable property that for any multi-index α , D α u S D = u S D $$\alpha,\;\left \|D^{\alpha }\mathbf{u}\right \|_{SD} = \left \|\mathbf{u}\right \|_{SD}$$ , where D is the distributional derivative. We call them the Jones strong distribution Banach spaces. As an application, we obtain a nice a priori estimate for the nonlinear term of the classical Navier–Stokes initial-value problem. In Sect. 3.4, we introduce a class of spaces in honor of our deceased colleague Woodford W. Zachary. These spaces all extend the class of functions of bounded mean oscillation to include the HK-integrable functions.