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March 1, 2026Results in Mathematics0 citationsOpen Access

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

TOTakao OhnoOita University TSTetsu ShimomuraHiroshima University

Key Points

The aim is to investigate the boundedness of generalized fractional maximal operators on Musielak-Orlicz-Morrey spaces over unbounded metric measure spaces.
Examined the generalized fractional maximal operator $$M_{\rho ,\lambda }$$ with $$\lambda \ge 1$$.
Analyzed Musielak-Orlicz-Morrey spaces $$L^{\Phi ,\kappa ,\theta _1}(X)$$.
Considered conditions on the positive function $$\rho (x,r)$$ across the domain.
Proved boundedness specifically for double phase functionals with variable exponents.
Established conditions under which the maximal operator is bounded on the specified spaces.
Showed applicability even when the underlying space is unbounded.
Demonstrated boundedness for both doubling metric measure spaces and specific functional forms like $$\Phi (x,t) = t^{p(x)} \left( \log (e + t)\right) ^{q(x)}$$.

Abstract

We are concerned with the boundedness of the generalized fractional maximal operator M, , 1, on Musielak-Orlicz-Morrey spaces L^, , ₁ (X) over unbounded metric measure spaces X, where (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 M, in the framework of double phase functionals with variable exponents (x, t) = t^p (x) + a (x) t^s (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 (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 (x, t) = t^p (x) ( (e + t) ) ^q (x).

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