The key tool of paradifferential operator calculus is developed in this chapter, beginning with Meyer’s ingenious formula for F(u) as M(x, D)u + R where F is smooth in its argument (s), u belongs to a Hölder or Sobolev space, M(x, D) is a pseudodifferential operator of type 1,1, and R is smooth. From there, one applies symbol smoothing to M(x, ξ) and makes use of results established in Chapter 2. The tool that arises is quite powerful in nonlinear analysis. The first glimpse we give of this is that it automatically encompasses some important Moser estimates. We re-derive elliptic regularity results established in Chapter 2, after establishing some microlocal regularity results. In §3.3 we do this using symbol smoothing with δ < 1; in §3.4 we present some results of Bony and Meyer dealing with the δ = 1 case, the case of genuine paradifferential operators.
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