Posted by J. Holmer on December 8, 2007, 10:54 pm, in reply to "Exercises #1 & #17"
#1 (a). Fix \eta>0. Then
\int_{|x|>\eta} |K_\delta(x)| = \int_{|x|>\eta/\delta} \phi(x) dx
by a scale change of variable. This -->0 by dominated convergence.
#1(b). Suppose support of \phi is contained in |x|\leq R. Then the bound (iii) in the definition of approximation to the identity is only nontrivial when |x|\leq \delta R. Since \phi is bounded (say by c), we have
|K_\delta(x)| \leq c/\delta^d = (c\delta R^{d+1}) / ((\delta R)^{d+1}) <= c' \delta / |x|^{d+1}
where c' = cR^{d+1}.
#17. Follow the idea in the proof of Theorem 2.1. That is, decompose the region of integration in to dyadic annuli, and on each piece use the assumption (iii) in the definition of approx. to identity.
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