Function Space and Operator Theory for Nonlinear Analysis
This chapter examines a number of analytical techiques, which will be applied to diverse nonlinear problems in the remaining chapters. For example, we study Sobolev spaces based on L p , rather than just L 2. Sections 1 and 2 discuss the definition of Sobolev spaces H k, p , for k ∈ Z+, and inclusions of the form H k, p ⊂ L q . Estimates based on such inclusions have refined forms, due to E. Gagliardo and L. Nirenberg. We discuss these in §3, together with results of J. Moser on estimates on nonlinear functions of an element of a Sobolev space, and on commutators of differential operators and multiplication operators. In §4 we establish some integral estimates of N. Trudinger, on functions in Sobolev spaces for which L∞-bounds just fail. In these sections we use such basic tools as Hölder’s inequality and integration by parts.
KeywordsFunction Space Operator Theory Nonlinear Analysis Hardy Space Compact Manifold
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