... y |)( 1 + 2 j|y − z |) ≥ (1 + 2 j|x − y |), (20 ) |ψk∗ f (z)| |ψk∗ f(z)|r[ψ∗k,af(x )] 1−r(1 + 2 k|x − z |) a(1−r),and(1 + 2 k|x − z |) a(1−r)(1 + 2 j|x − z |) a 2 (k−j)a(1 + 2 k|x ... estimates (1 7) and (1 8) into (1 6), we obtain that for allf ∈S(R n ), y∈ R n , and j ∈ N, |ψj∗ f (y)| C N ∞k=j 2 jn 2 (j−k )N |ψk∗ f(z)|(1 + 2 j|y − z |) adz. (1 9) For any r ∈ (0, 1], dividing ... z |) ar,we obtain that for all f ∈S(R n ), x∈ R n and j ∈ N, ψ∗j,af(x) C N ∞k=j 2 (j−k )N 2 kn|ψk∗ f (z)|r(1 + 2 k|x − z |) ardz[ψ∗k,af(x )] 1−r (21 ) A Characterization of...