Images of wavelets under the influence of convolution operators

Images of integration operators in weighted function spaces

“…Recall that A s,w pq (R N ) stands for either B s,w pq (R N ) or F s,w pq (R N ) with a weight w from A loc ∞ . We begin with an example for the case N = 2 and the operator I 1 Ω + 1 (see (3) with m 1 = 1) of the form…”

“…Connections between images and pre-images of integration (differentiation) operators in B s,w pq (R N ) and F s,w pq (R N ) have been studied in [27, Theorem 2.3.8], [3], [23,Theorem 2.20], [10, § 4], [29, p. 23]. Let D m stand for derivatives.…”

Norm related inequalities for fractional integrals

“…For simplicity, we assume f (x) ≡ 0 for x ∈ (−∞, a) or x ∈ (a, +∞). 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].…”

“…More precisely, in Theorem 5. 4, we obtain conditions for the validity of embedding inequalities for (sub)spaces of images and pre-images of the operators I α a ± with fractional α > 0 in B s pq (R, w). In particular, for I α 0 ± one of our results reads…”