Images of integration operators in weighted function spaces

Abstract

Images of integration operators of natural orders are considered as elements of Besov and Triebel-Lizorkin spaces with local Muckenhoupt weights on R N . The results connect entropy and approximation numbers of embedding operators with the same characteristics of the integration operators.

Norm related inequalities for fractional integrals

“…Norm related inequalities in B s pq (R, w) for integrals (1) and (2) of natural orders α were considered in [31]. This work continues the study of the same problem by extending the results obtained in [31,Theorems 5.1,5.2] to fractional α > 0.…”

“…(i) We need to introduce two spline wavelet systems with orders suitable for decomposing norms in the both sides of (54). To this end we determine J for B s pq (R, w) and define δ, M > 0 and N ≥ − (31). Therefore, by following the idea from Example 5.1, α 0 and α * must be taken in such a way to comply the condition α 0 + α = α * .…”

“…Connections between images and pre-images of operators in B s pq (R, w) have been studied in [23,Theorem 2.3.8], [4], [20,Theorem 2.20], [9, § 4], [25, p. 23]. Norm related inequalities in B s pq (R, w) for integrals (1) and (2) of natural orders α were considered in [31]. This work continues the study of the same problem by extending the results obtained in [31,Theorems 5.1,5.2] to fractional α > 0.…”