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Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. (English) Zbl 1261.42034

Summary: Let \(\mathcal X\) be a metric space with doubling measure and \(L\) a one-to-one operator of type \(\omega\) having a bounded \(H_\infty\)-functional calculus in \(L^2(\mathcal X)\) satisfying the reinforced \((p_L, q_L)\) off-diagonal estimates on balls, where \(p_L \in [1, 2)\) and \(q_L \in (2, \infty]\). Let \(\varphi : \mathcal X \times [0, \infty) \rightarrow [0, \infty)\) be a function such that \(\varphi (x, \cdot)\) is an Orlicz function, \(\varphi (\cdot, t) \in \mathbb A_\infty(\mathcal X)\) (the class of uniformly Muckenhoupt weights), \(\varphi(\cdot, t)\) and its uniformly critical upper type index \(l(\varphi) \in (0, 1]\) satisfies the uniformly reverse Hölder inequality of order \((q_L/l(\varphi))'\), where \((q_L/l(\varphi))'\) denotes the conjugate exponent of \(q_L/l(\varphi)\). The authors introduce a Musielak-Orlicz-Hardy space \(H_{\varphi, L}(\mathcal X)\), via the Lusin-area function associated with \(L\), and establish its molecular characterization. In particular, when \(L\) is nonnegative, self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of \(H_{\varphi, L}(\mathcal X)\) is also obtained. Furthermore, a sufficient condition for the equivalence between \(H_{\varphi, L}(\mathbb R^n)\) and the classical Musielak-Orlicz-Hardy space \(H_{\varphi}(\mathbb R^n)\) is given. Moreover, for the Musielak-Orlicz-Hardy space \(H_{\varphi, L}(\mathbb R^n)\) associated with the second-order elliptic operator in divergence form on \(\mathbb R^n\) or the Schrödinger operator \(L := -\Delta + V\) with \(0 \leq V \in L^1_{\text{loc}}(\mathbb R^n)\), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform \(\nabla L^{-1/2}\) is bounded from \(H_{\varphi, L}(\mathbb R^n)\) to the Musielak-Orlicz space \(L^\varphi(\mathbb R^n)\) when \(i(\varphi) \in (0, 1]\), from \(H_{\varphi, L}(\mathbb R^n)\) to \(H_\varphi(\mathbb R^n)\) when \(i(\varphi) \in (\frac{n}{n + 1}, 1]\), and from \(H_{\varphi, L}(\mathbb R^n)\) to the weak Musielak-Orlicz-Hardy space \(WH_\varphi(\mathbb R^n)\) when \(i(\varphi) = \frac{n}{n + 1}\) is attainable and \(\varphi(\cdot, t) \epsilon \mathbb A_1(\mathcal X)\), where \(i(\varphi)\) denotes the uniformly critical lower type index of \(\varphi\).

MSC:

42B35 Function spaces arising in harmonic analysis
42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
35J10 Schrödinger operator, Schrödinger equation
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
30L99 Analysis on metric spaces
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References:

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