Estimates for Bessel and Riesz Potentials
Here we interrupt the development of the general theory in order to gain a deeper understanding of some of the aspects of the spaces L α,p . In Section 3.1 we give some simple pointwise estimates of potentials in terms of maximal functions. These are going to be used in several of the following chapters. We apply them here to obtain elementary proofs of certain integral inequalities, among which are the Sobolev inequalities of Theorem 1.2.4. In Section 3.2 we pursue a more special subject; we give a sharp exponential integral estimate in the “borderline case” αp = N. Sections 3.3 and 3.5 are devoted to the question under which circumstances a function T “operates” on functions f in L α,p in the sense that the composite function T o f also belongs to L α,p . This is in part motivated by the desire to prove the equivalence of capacities formulated in Section 2.7. Another consequence is a one-sided approximation theorem, given in Section 3.4, which has turned out to be useful in the theory of nonlinear partial differential equations. Finally, in Section 3.6 we prove an important inequality of B. Muckenhoupt and R. L. Wheeden, comparing Riesz and Bessel potentials with the fractional maximal function M α f, which will have a role to play later.
KeywordsMaximal Function Sobolev Inequality Nonlinear Partial Differential Equation Pointwise Estimate Riesz Potential
Unable to display preview. Download preview PDF.