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On the exponential integrability of fractional integrals on spaces of homogeneous type. (English) Zbl 1002.42501

In their earlier work [Lect. Notes Pure Appl. Math. 122, 171-216 (1990; Zbl 0695.43006)], the authors introduced a notion of a normal space of homogeneous type \((X,\delta,\mu)\) and a notion of a fractional integral \(I_\alpha\). Here \(0<\alpha<1\). For a nonexpert reader let us mention that in the classical case when \(X=\mathbb{R}\), \(\delta\) is the usual Euclidean metric on \(\mathbb{R}\) and \(\mu\) is the Lebesgue measure, the fractional integral \(I_\alpha f(x)\) of a compact supported function \(f\) at \(x\in\mathbb{R}\) is given by the formula \[ I_\alpha f(x)= \int_{\mathbb{R}} f(y)/|x- y|^{(1-\alpha)} dy. \] The first theorem of this work asserts that there are constants \(C_1\) and \(c\) such that for any ball \(B\subseteq X\) and any function \(f\in L^{1/\alpha}(B)\) \[ \int_B \exp\Biggl(\Biggl({|I_\alpha f(x)|\over C_1\|f\|_{1/\alpha}}\Biggr)^{1/(1- \alpha)}\Biggr) d\mu(x)\leq c\mu(B). \] The second theorem is a generalization of the first one to the context of functions with unbounded support, in terms of a modified fractional integral \(\widetilde I_\alpha\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
43A85 Harmonic analysis on homogeneous spaces

Citations:

Zbl 0695.43006
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