| Popis: |
Abstract Let φ : R n × [ 0 , ∞ ) → [ 0 , ∞ ) $\varphi:\mathbb{R}^{n}\times[0, \infty) \to[0, \infty)$ satisfy that φ ( x , ⋅ ) $\varphi(x, \cdot)$ , for any given x ∈ R n $x\in\mathbb{R}^{n}$ , is an Orlicz function and φ ( ⋅ , t ) $\varphi(\cdot, t)$ is a Muckenhoupt A ∞ $A_{\infty}$ weight uniformly in t ∈ ( 0 , ∞ ) $t\in(0, \infty)$ . The Musielak-Orlicz Hardy space H φ ( R n ) $H^{\varphi}(\mathbb{R}^{n})$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the Musielak-Orlicz space L φ ( R n ) $L^{\varphi}(\mathbb{R}^{n})$ . In this paper, the authors establish the boundedness of Marcinkiewicz integral μ Ω $\mu _{\Omega}$ from H φ ( R n ) $H^{\varphi}(\mathbb{R}^{n})$ to L φ ( R n ) $L^{\varphi}(\mathbb{R}^{n})$ under weaker smoothness conditions assumed on Ω. This result is also new even when φ ( x , t ) : = ϕ ( t ) $\varphi(x, t):=\phi(t)$ for all ( x , t ) ∈ R n × [ 0 , ∞ ) $(x, t)\in\mathbb{R}^{n}\times[0, \infty)$ , where ϕ is an Orlicz function. |
