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Morrey spaces and fractional operators. (English) Zbl 1193.42095

The purpose of this paper is to study certain estimates related to the fractional integral operator, defined by \[ I_{\alpha}f(x)=\int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n(1-\alpha)}}dy \;\;\;\text{ for}\;\;0 <\alpha <1, \] and the fractional maximal operator, defined by \[ M_{\alpha}f(x)=\sup_{x \in Q} \frac{1}{|Q|^{1-\alpha}}\int_{Q}|f(y)|dy \;\;\;\text{ for}\;\;0 \leq \alpha < 1 \] in the framework of Morrey spaces. Here, the supremum is taken over all cubes \(Q\) in \(\mathbb{R}^n\) containing \(x\) wth sides parallel to the coordinate axes.
Applications to the Fefferman-Phong and the Olsen inequalities are also included.
Let \(0 < p_{1} \leq p_{0} \leq \infty\). For an \(L^{p_{1}}\) locally integrable function \(f\) on \(\mathbb{R}^n\) we set \[ \|f\|_{p_{0},p_{1}}=\sup_{Q}|Q|^{1/p_{0}}(\frac{1}{|Q|} \int_{Q}|f(x)|^{p_{1}})^{1/p_{1}} \] where the supremum is taken over all cubes \(Q\) in \(\mathbb{R}^n\) with sides parallel to the coordinate axes. We call the Morrey space \(\mathcal{M}^{p_{0}}_{p_{1}}\) the set of all \(L^{p_{1}}\) locally integrable functions \(f\) on \(\mathbb{R}^n\) with \(\|f\|_{p_{0},p_{1}} < \infty\). When \(1 \leq p_{2} < \infty\), an \(l^{p_{2}}\)-valued function \((f_{\nu})_{\nu \in \mathbb{N}}\) on \(\mathbb{R}^n\) is said to be measurable if each \(f_{\nu}\) is a measurable function and \(\sum_{\nu}|f_{\nu}(x)|^{p_{2}} < \infty\) almost everywhere. For \(0 < p_{1} \leq p_{0} \leq \infty\) and \(1 \leq p_{2} < \infty\), we define the space \(\mathcal{M}^{p_{0}}_{p_{1}}(l^{p_{2}})\) consisting of all \(l^{p_{2}}\)-valued measurable functions \((f_{\nu})\) such that \[ \|(f_{\nu})\|_{p_{0},p_{1},p_{2}}= \|(\sum_{\nu}|f_{\nu}|^{p_{2}})^{1/p_{2}}\|_{p_{0},p_{1}} < \infty. \] Theorem. Suppose \(0 < \alpha < 1\).
(i) Let \(0 < q_{1} \leq q_{0} < \infty\) and \(0 < r_{1} \leq r_{0} \leq \infty\). Suppose that \(q_{1} \leq 1, q_{1} \leq r_{1}, q_{0} \leq r_{0}\) and \(0 \leq \beta = \alpha-(1/r_{0})<1\). Then, for any locally integrable function \(f\) such that \(\|M_{\beta}f\|_{q_{0},q_{1}}< \infty\) and for any function \(g\) in \(\mathcal{M}^{r_{0}}_{r_{1}}\), \[ \|g\cdot I_{\alpha}f\|_{q_{0},q_{1}}\leq C\|g\|_{r_{0},r_{1}}\|M_{\beta}f\|_{q_{0},q_{1}}. \] (ii) Let \(1 < q_{1} \leq q_{0} < \infty\) and \(1 < r_{1} \leq r_{0} \leq \infty\). Suppose that \(q_{1}, q_{1} \leq r_{1}, q_{0} \leq r_{0}\) and \(0 \leq \beta = \alpha-(1/r_{0})<1\). Then, for any locally integrable functions \((f_{\nu})\) such that \(\|(M_{\beta}f_{\nu})\|_{q_{0}, q_{1},q_{2}} < \infty\) and any functions \((g_{\nu})\) such that \(\sup_{\nu}\|g_{\nu}\|_{r_{0},r_{1}} < \infty\), \[ \|g_{\nu}\cdot I_{\alpha}f_{\nu}\|_{q_{0},q_{1},q_{2}}\leq C\sup_{\mu}\|g_{\mu}\|_{r_{0},r_{1}}\|(M_{\beta}f_{\nu})\|_{q_{0},q_{1},q_{2}}. \] Corollary. Suppose \(0 < \alpha < 1\).
(i) Let \(0 < q_{1} \leq q_{0} < \infty\). Then, for any locally integrable function \(f\) such that \(\|M_{\alpha}f\|_{q_{0},q_{1}}< \infty\), \[ \|I_{\alpha}f\|_{q_{0},q_{1}}\leq C\|M_{\alpha}f\|_{q_{0},q_{1}}. \] (ii) Let \(1 < q_{1} \leq q_{0} < \infty\) and \(1 < q_{2} < \infty\). Then, for any locally integrable functions \((f_{\nu})\) such that \(\|(M_{\alpha}f_{\nu})\|_{q_{0}, q_{1},q_{2}} < \infty\), \[ \|(I_{\alpha}f_{\nu})\|_{q_{0},q_{1},q_{2}} \leq C\|(M_{\alpha}f_{\nu})\|_{q_{0},q_{1},q_{2}}. \]

MSC:

42B35 Function spaces arising in harmonic analysis
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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[1] DOI: 10.1080/03605309508821161 · Zbl 0838.35017 · doi:10.1080/03605309508821161
[2] DOI: 10.2307/1996833 · Zbl 0289.26010 · doi:10.2307/1996833
[3] DOI: 10.4134/JKMS.2007.44.6.1233 · Zbl 1132.42308 · doi:10.4134/JKMS.2007.44.6.1233
[4] Gilbarg, Elliptic Partial Differential Equations of Second Order (1983) · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0
[5] Garcia-Cuerva, Weighted Norm Inequalities and Related Topics (1985)
[6] DOI: 10.1090/S0273-0979-1983-15154-6 · Zbl 0526.35080 · doi:10.1090/S0273-0979-1983-15154-6
[7] DOI: 10.1512/iumj.1994.43.43028 · Zbl 0809.42007 · doi:10.1512/iumj.1994.43.43028
[8] DOI: 10.1512/iumj.2005.54.2714 · Zbl 1078.42010 · doi:10.1512/iumj.2005.54.2714
[9] DOI: 10.1512/iumj.2004.53.2470 · Zbl 1100.31009 · doi:10.1512/iumj.2004.53.2470
[10] Stein, Singular Integrals and Differentiability Properties of Functions (1970) · Zbl 0207.13501
[11] DOI: 10.1007/s10114-005-0660-z · Zbl 1129.42403 · doi:10.1007/s10114-005-0660-z
[12] Pérez, Ann. Inst. Fourier (Grenoble) 45 pp 809– (1995) · Zbl 0820.42008 · doi:10.5802/aif.1475
[13] Chiarenza, Rend. Mat. 7 pp 273– (1987)
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