Variable anisotropic fractional integral operators

Abstract

In 2011, Dekel et al. introduced a highly geometric Hardy spaces \(H^p(\Theta)\), for the full range \(0<p\le 1\), which are constructed over a continuous multilevel ellipsoid cover \(\Theta\) of \(\mathbb{R}^n\) with high anisotropy in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. We introduce a new class of fractional integral operators \(T_{\alpha}\) adapted to ellipsoid cover \(\Theta\) and obtained their boundedness from \(H^p(\Theta)\) to \(H^q(\Theta)\) and from \(H^p(\Theta)\) to \(L^q(\mathbb{R}^n)\), where \(\frac{1}{q}=\frac{1}{p}+\alpha\) and \(0<\alpha<1\).