Smooth molecular decompositions of functions and singular integral operators

Bilinear operators with non-smooth symbol, I

“…As norm estimates involving smoothness and decay are needed frequently, it is convenient to work within the setting of test functions and molecules as in [9], for instance. In this terminology a function φ = φ(x) is said to be an M µ -test function when it has continuous derivatives up to order [µ] such that…”

Continuous Wavelets and Frames on Stratified Lie Groups I

“…If G = R n , the fact that T = S φ,ψ,b satisfies the conclusions of Theorem 6.8 has a long history. For instance, in Lemma 9.1.5 of [5], condition (ii) of Theorem 6.8 (b) is verified for this T , if G = R. If G = R n , Theorem 6.8 is verified for this T , and more general operators T , in [17], sections 2.1-2.3.…”

“…In section 8 we shall examine wavelet frame expansions in other Banach spaces (besides L 2 ). Again such questions have been discussed in the earlier works we have cited ( [9], [10], [18], [17], and [27]) where again one requires a to be close to 1. (In particular, in [27], Maggioni addresses such questions on stratified groups.)…”

Triebel–Lizorkin spaces with non-doubling measures

“…More precisely, Sawyer and the first author [14] first introduced a test function space M(X), which is also called smooth molecular space in [11], and approximations to the identity {S k } ∞ k=−∞ whose kernels satisfy all size and regularity conditions as in Coifman's construction, and furthermore, a second difference smoothness condition. They then proved that if N is large enough, R N is bounded on M(X) with operator norm less than 1.…”