Embeddings of Decomposition Spaces

Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an embedding between the two? A decomposition space D(Q, L-p, Y) is determined by a covering Q = (Q(i))(i is an element of I) of the frequency domain, an integrability exponent p, and a sequence space Y subset of C-I. Given these ingredients, the decomposition space norm of a distribution g is defined as parallel to g parallel to(D(Q,Lp,Y)) = parallel to (parallel to F-1 (phi(i) center dot (g) over cap)parallel to(Lp))(i is an element of I)parallel to(Y), where (phi(i))(i is an element of I) is a suitable partition of unity for Q. We establish readily verifiable criteria which ensure the existence of a continuous inclusion (an embedding) D(Q, L-p1, Y) (sic) D(P, L-p2, Z), mostly concentrating on the case where Y = l(w)(q1) (I) and Z = l(v)(q2) (J). Under suitable assumptions on Q, P, we will see that the relevant sufficient conditions are p(1) <= p(2) and finiteness of a nested norm of the form parallel to(parallel to alpha(i)beta(j) center dot v(j)/w(i))(i is an element of Ij) parallel to(lt))(j is an element of J) parallel to(ls), with I-j = {i is an element of I : Q(i) boolean AND P-j not equal (sic)} for j is an element of J. Like the sets I-j, the exponents t, s and the weights alpha, beta only depend on the quantities used to define the decomposition spaces. In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings. These sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of p(1), p(2), our criteria yield a complete characterization for the existence of the embedding. The same holds for arbitrary values of p(1), p(2) under more strict assumptions on the coverings. We also prove a rigidity result, namely that-for (p(1), q(1)) not equal (2, 2)-two decomposition spaces D(Q, L-p1, l(w)(q1)) and D(P, L-p2, l(v)(q1)) can only coincide if their ingredients are equivalent, that is, if p(1) = p(2) and q(1) = q(2) and if the coverings Q, P and the weights w, v are equivalent in a suitable sense. The resulting embedding theory is illustrated by applications to alpha-modulation and Besov spaces. All known embedding results for these spaces are special cases of our approach; often, we improve considerably upon the state of the art.

Automated summary

This paper investigates smoothness spaces in harmonic analysis and raises the question of whether there is an embedding between two such spaces. By defining the decomposition space norm and establishing verifiable conditions, sufficient criteria for the existence of an embedding are presented. Furthermore, it is proven that two decomposition spaces can only coincide if their ingredients are equivalent. The resulting embedding theory has made significant progress in practical applications.