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Characterization of the boundedness of fractional maximal operator and its commutators in Orlicz and generalized Orlicz-Morrey spaces on spaces of homogeneous type. (English) Zbl 1473.42015

In this paper the authors find necessary and sufficient conditions for the boundedness of the fractional maximal operator \(M_{\alpha}\) and the fractional maximal commutator \(M_{b,\alpha}\) (where \(b\) is a function in \(BMO(X)\)) in the Orlicz spaces \(L^{\Phi}(X)\) and generalized Orlicz-Morrey spaces \(\mathcal{M}^{\Phi, \phi}(X)\) on spaces of homogeneous type \(X=(X,d,\mu)\) in the sense of R. R. Coifman and G. Weiss [Analyse harmonique non-commutative sur certains espaces homogènes. Etude de certaines intégrales singulières. (Non-commutative harmonic analysis on certain homogeneous spaces. Study of certain singular integrals.). Lect. Notes Math. 242 (1971; Zbl 0224.43006)]. Here \(d\) is a quasi-metric, \(\mu\) a doubling measure, and is assumed that \(\mu(X)=\infty\), compactly supported continuous functions are dense in \(L^1(X,\mu)\) and \(X\) is \(Q\)-homogeneous, i.e. \(\mu(B(x,r)) \sim r^Q\).
This paper generalizes results when \(X=\mathbb{R}^n\) of F. Deringoz et al. [Positivity 22, No. 1, 141–158 (2018; Zbl 1388.42057); Anal. Math. Phys. 9, No. 1, 165–179 (2019; Zbl 1416.42019)].
For a \(Q\)-homogeneous space \((X,d,\mu)\) and \(0\leq \alpha<Q\), the maximal fractional function \(M_{\alpha}\) (when \(\alpha=0\), \(M_0\) is denoted \(M\)) is defined for each locally integrable functions \(f\) in \(X\) by \[ M_{\alpha} f(x) = \sup_{r>0} \mu(B(x,r))^{-1+\frac{\alpha}{Q}}\int_{B(x,r)} |f(y)|\,d\mu(y), \] where \(B(x,r)\) is the ball in \(X\) centered at \(x\) and of radius \(r>0\).
Similarly in this setting and for \(b\in BMO(X)\), the operator \(M_{b,\alpha}\) for \(0\leq \alpha<Q\) is defined by \[ M_{b,\alpha} f(x) = \sup_{r>0} \mu(B(x,r))^{-1+\frac{\alpha}{Q}}\int_{B(x,r)} |b(x)-b(y)|\,|f(y)|\,d\mu(y). \]
For a Young function \(\Phi\) the Orlicz (\(L^\Phi(X)\)) and weak Orlicz spaces (\(WL^{\Phi}(X)\)) on spaces of homogeneous type \((X,d,\mu)\) are defined in the paper, and it is noted that \(\|f\|_{WL^{\Phi}}\leq \|f\|_{L^{\Phi}}\). A Young function \(\Phi\in \nabla_2\) if \(\Phi(r)\leq \Phi (kr)/2k\) for all \(r\geq 0\) and for some \(k>1\), and \(\Phi\in\Delta_2\) if \(\Phi(2r)\leq k\Phi (r)\) for all \(r\geq 0\) and for some \(k>1\).
The authors show that for a \(Q\)-homogeneous space \((X,d,\mu)\), \(0< \alpha<Q\) and \(\Phi\), \(\Psi\) Young functions, then \(M_{\alpha}\) is bounded from \(L^{\Phi}(X)\) to \(WL^{\Psi}(X)\) if and only if for all \(r>0\), \[ r^{\alpha}\lesssim \Psi^{-1}(r^{-Q}) / \Phi^{-1}(r^{-Q}). \tag{\(\ast\)} \] Furthermore if \(\Phi\in \nabla_2\), then \(M_{\alpha}\) is bounded from \(L^{\Phi}(X)\) to \(L^{\Psi}(X)\) if an only if (\(\ast\)) holds. The key estimate is, under the hypothesis of the theorem and assuming (\(\ast\)), that for any \(C_0>0\), there is \(C_1>0\) such that for all non-zero \(f\in L^{\Phi}(X)\) \[ M_{\alpha}f(x) \leq C_1 \|f\|_{L^{\Phi}} (\Psi^{-1} \circ \Phi)\left (\frac {Mf(x)}{C_0\|f\|_{L^{\Phi}}}\right ). \]
The authors show that for a \(Q\)-homogeneous space \((X,d,\mu)\), \(0< \alpha<Q\), \(b\in BMO(X)\), and Young functions \(\Phi\in \Delta_2\cap \nabla_2\), \(\Psi\in \Delta_2\) such that for all \(r>0\) \[ \sup_{r<t<\infty} \Big (1+\ln{\frac{t}{r}}\Big )\Phi^{-1}(t^{-Q})t^{\alpha}\lesssim r^{\alpha}\Phi^{-1}(r^{-Q})\] then \(M_{b, \alpha}\) is bounded from \(L^{\Phi}(X)\) to \(L^{\Psi}(X)\) if and only if (\(\ast\)) holds.
The authors also show that if the reverse condition to (\(*\)) holds for the Young functions \(\Psi\), and \(\Phi\), namely \[ \Psi^{-1}(r^{-Q}) / \Phi^{-1}(r^{-Q}) \lesssim r^{\alpha},\] then \(b\in BMO(X)\) is necessary for the boundedness of \(M_{b,\alpha}\) from \(L^{\Phi}(X)\) to \(WL^{\Psi}(X)\).
These results are then extended to the generalized Orlicz-Morrey spaces \(\mathcal{M}^{\Phi, \phi}(X)\) on spaces of homogeneous type \(X=(X,d,\mu)\), of which the Orlicz spaces are a particular case.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
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