Variable anisotropic Hardy spaces with variable exponents (Q1983825)

Let \(\Theta\) be a continuous ellipsoid consisting of ellipsoids \(\theta_{x,t}\) with center \(x\in \mathbb{R}^n\) and scale \(t\in \mathbb{R}\) of the form \(\theta_{x,t}={M}_{x,t}(\mathbb{B}_n)+x\), where \({M}_{x,t}\) is a nonsingular matrix and \(\mathbb{B}_n\) is the unit ball in \(\mathbb{R}^n\). Let \(p(\cdot): \mathbb{R}^n \to (0,\infty]\) be a variable exponent function satisfying the globally log-Hölder continuous condition. The main aim of the paper is to develop the theory of highly geometric Hardy spaces \(H^{p(\cdot)}(\Theta)\). Similarly to classical results, the authors characterize the Hardy space \(H^{p(\cdot)}(\Theta)\) via the radial grand maximal function and the atomic decomposition. As an application, they establish a number of criteria for a singular integral to be bounded from \(H^{p(\cdot)}(\Theta)\) to \(L^{p(\cdot)}(\Theta)\) and bounded on \(H^{p(\cdot)}(\Theta)\).