Boundedness of the Riesz potential in central Morrey–Orlicz spaces

Abstract

Boundedness of the maximal operator and the Calderón–Zygmund singular integral operators in central Morrey–Orlicz spaces were proved in papers (Maligranda et al. in Colloq Math 138:165–181, 2015; Maligranda et al. in Tohoku Math J 72:235–259, 2020) by the second and third authors. The weak-type estimates have also been proven. Here we show boundedness of the Riesz potential in central Morrey–Orlicz spaces and the corresponding weak-type version.

A new result on boundedness of the Riesz potential in central Morrey-Orlicz spaces

“…Only later, on the Examples 2 and 3, we will see that the conditions (1) and ( 2) hold but estimate (3) fails, which shows that our Theorem 1 improves Theorem 3 in [3].…”

“…Remark 2. If λ = 0 and 0 < µ < 1, then the condition (3) is not satisfied, as we already mentioned in [3,Remark 3] and therefore the result proved in [3] does not include boundedness of the Riesz potential in this case. On the other hand, in this case, assumption (1) is stronger than (2).…”

“…Ψ −1 (t) = ∞, then by Theorem 2 in [3] the Riesz potential I α is not bounded from M Φ,λ (0) to M Ψ,λ (0). In particular, I α is not bounded from M p,λ (0) to M q,λ (0) for any 1 ≤ p, q < ∞ (see also [8]).…”

“…and clearly the integral in ( 2) is smaller than the integral in (3). This improvement provides us with larger classes of Orlicz functions Φ and Ψ, defining central Morrey-Orlicz spaces where the operator I α is bounded.…”

“…The properties of these spaces can be found in [3]. If Φ(u) = u p , 1 ≤ p < ∞ and λ ∈ R, then M Φ,λ (0) = M p,λ (0) and W M Φ,λ (0) = W M p,λ (0) are classical central and weak central Morrey spaces.…”