Generalized Fractional Maximal Operators on Musielak-Orlicz-Morrey Spaces Over Unbounded Metric Measure Spaces

We are concerned with the boundedness of the generalized fractional maximal operator Mρ ,λ , λ ≥ 1 , on Musielak-Orlicz-Morrey spaces LΦ ,κ ,θ ₁(X) over unbounded metric measure spaces X, where ρ (x,r) is a positive function on X × (0, ∞ ) satisfying certain conditions, as an extension of earlier results. As an important special case, we prove the boundedness of Mρ ,λ in the framework of double phase functionals with variable exponents Φ (x,t) = tᵖ⁽ˣ⁾ + a(x) tˢ⁽ˣ⁾, \ x ∈ X, \ t ≥ 0 , where $$p(x)<s(x)$$ for x∈ X , a(· ) is a non-negative, bounded and Hölder continuous function of order θ ∈ (0,1] . The main novelty is that the underlying space need not be bounded, even in the case of the doubling metric measure space or the case of Φ (x,t) = tᵖ⁽ˣ⁾ ( log (e + t)) q(x) .