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

We improve our results on boundedness of the Riesz potential in the central Morrey–Orlicz spaces and the corresponding weak-type version. We also present two new properties of the central Morrey–Orlicz spaces: nontriviality and inclusion property.

In the following lemma and later, B(x 0 , r 0 ) will denote an open ball with the center at for any t > 0.
Proof of this lemma can be found in [3, Lemma 1].

Riesz potential in the central Morrey-Orlicz spaces
We will work with the central Morrey-Orlicz spaces, defined by the Orlicz functions.A function Φ : [0, ∞) → [0, ∞) is called an Orlicz function, if it is a strictly increasing continuous and convex function with Φ(0) = 0. Let f : R n → R be a Lebesgue measurable function and α ∈ (0, n).The Riesz potential is defined as The linear operator I α plays an important role in various branches of analysis, including potential theory, harmonic analysis, Sobolev spaces, partial differential equations and can be treated as a special singular integral.That is why it is important to study its boundedness between different spaces.Many authors investigated boundedness of I α in Morrey, Orlicz and Morrey-Orlicz spaces.We present here our main theorem on the boundedness of the Riesz potential in the central Morrey-Orlicz spaces.
In order to prove our result we will use estimate from [11] for the Hardy-Littlewood maximal operator in central Morrey-Orlicz spaces.The Hardy-Littlewood maximal operator M or centred maximal function Mf of a function f defined on R n is defined at each For any Orlicz function Φ and 0 ≤ λ ≤ 1, maximal operator M is bounded on M Φ,λ (0), provided Φ * ∈ ∆ 2 , and then there exists a constant C 0 > 1 such that (see [11,Theorem 6(i)]).Moreover, the maximal operator M is bounded from M Φ,λ (0) to W M Φ,λ (0), that is, there exists a constant Furthermore, in the proof of the main result we will use Hedberg's pointwise estimate from [6, p. 506].
Lemma 2 (Hedberg).If f : R n → R is a Lebesgue measurable function and α ∈ (0, n), then for all x ∈ R n and r > 0 For the sake of completeness, we include its proof, taking care about the constant C H in the estimate.For any x ∈ R n and r > 0 THEOREM 1.Let 0 < α < n, Φ, Ψ be Orlicz functions and either 0 < λ, µ < 1, λ = µ or λ = 0 and 0 ≤ µ < 1. Assume that there exist constants and (ii) The operator I α is bounded from M Φ,λ (0) to W M Ψ,µ (0), that is, there exists a constant In our earlier paper [3, Theorem 3] it was proved result under conditions ( 1) and ( 3), and the latter means that The condition ( 3) is stronger than the assumption (2) because 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.
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].
Let us comment on what we can get when the numbers λ and µ come from "boundaries".
Remark 1.If λ = µ = 0 we come to the same conclusion as in [3, Remark 4], that is, condition (1) is sufficient for the boundedness of Note that in this case condition (2) follows from (1).
and therefore I α is not bounded from M Φ,λ (0) to L Ψ (R n ).
Proof of Theorem 1. (i) For any x ∈ B r and f ∈ M Φ,λ (0) we consider two disjoint subsets We estimate the Riesz potential I α f (x) by a sum of two integrals By Hedberg's pointwise estimate, given in Lemma 2, we obtain This implies, for x ∈ B 1 r , that On the other hand, for any u > 0. Thus, applying assumption (1) we obtain To estimate the second integral I 2 f (x), first note that when x ∈ B 1 r and |y| > 2r we have |x| < r < |y|/2 and |y − x| ≥ |y| − |x| > |y|/2, and so |x − y| α−n < 2 n−α |y| α−n .Thus, following Hedberg's method, as in [3, pp.18-20], we obtain Then, from Lemma 1, it follows that Applying assumption (1) we get Thus, for x ∈ B 1 r , we obtain where Let now x ∈ B 2 r .We can write I α f (x) as follows where δ is defined in the following way Hedberg's pointwise estimate from Lemma 2 to I 3 f (x) gives and from the assumption (2) we get Next, since equality (4) holds it follows that Applying again Hedberg's method for I 4 f (x) we obtain where B |x|+2 k+1 δ is the smallest ball with the centre at origin containing B(x, 2 k+1 δ).From Lemma 1, using the fact that B |x|+2 k+1 δ ∩ B(x, 2 k+1 δ) = B(x, 2 k+1 δ), we get Since |x| ≤ r and 2 k δ ≤ ( t vn ) ) n ≤ 4 n max{|B r |, t}.So using the concavity of Φ −1 , we get where Based on the assumptions of (1), (2) and the fact that |B δ | < |B r | we get Thus, for x ∈ B 2 r we obtain Finally, since B r = B 1 r ∪ B 2 r and the last two sets are disjoint, and by the convexity of Ψ it follows that where (ii) Similarly to the previous case, we will present B r as a union of two disjoint subsets B r = B 1 r ∪ B 2 r , where B 1 r and B 2 r are defined in the same way as in the first part of the proof with respect to the constant c 0 , that is, Then we get where . We follow the same calculations as in the proof of Theorem 3(ii) in [3] and we get , u ≤ d(I 5 , u) + d(I 6 , u) where we used the property Ψ(u) d(g, u) = v d(g, Ψ −1 (v)) = v d(Ψ(g), v) for any u > 0 with v = Ψ(u).
From the first part of the proof of this theorem for any r > 0 we have For I 6 from the first part of the proof of this theorem we obtain where δ is defined as in ( 4) with respect to c 0 , that is, Thus, and doing the same calculations as in the proof of Theorem 3(ii) in [3] we get Hence, .
Below we present examples for our Theorem 1.In our earlier paper [3] we have shown that Example 1 holds under conditions (1) and (3), which clearly means that it also holds under conditions (1) and (2) of Theorem 1.