Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators

Abstract

Let \phi be a growth function, and let A:=-(\nabla -\mathrm{i}a)\cdot (\nabla -\mathrm{i}a)+V be a magnetic Schrödinger operator on L^{\mathrm{2}}({ℝ}^n),n\ge \mathrm{2}, where \alpha :=({\alpha }_{\mathrm{1}},{\alpha }_{\mathrm{2}},\cdots ,{\alpha }_n)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{2}}({ℝ}^n,{ℝ}^n) and \mathrm{0}\le V\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{1}}({ℝ}^n). We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space H_{A,\phi }({ℝ}^n), defined by the Lusin area function associated with \{{\mathrm{e}}^{-t^{\mathrm{2}}A}\}t>\mathrm{0}, in terms of the Lusin area function associated with \{{\mathrm{e}}^{-t\sqrt{A}}\}t>\mathrm{0}, the radial maximal functions and the nontangential maximal functions associated with \{{\mathrm{e}}^{-t^{\mathrm{2}}A}\}t>\mathrm{0} and \{{\mathrm{e}}^{-t\sqrt{A}}\}t>\mathrm{0}, respectively. The boundedness of the Riesz transforms L_k{A}^{-\mathrm{1}/\mathrm{2}},k\in \{\mathrm{1},\mathrm{2},\cdots ,n\}, from H_{A,\phi }({ℝ}^n) to L^{\phi }({ℝ}^n) is also presented, where L_k is the closure of {\displaystyle \frac{\partial }{{\partial }_{x_k}}}-\mathrm{i}{\alpha }_k in L^{\mathrm{2}}({ℝ}^n). These results are new even when \phi (x,t):=\omega (x)t^p for all x\in {ℝ}^nand t ∈(0,+∞) with p ∈(0, 1] and \omega \in A_{\infty }({ℝ}^n) (the class of Muckenhoupt weights on {ℝ}^n).

Key words： Magnetic Schrödinger operator; Musielak-Orlicz-Hardy space; Lusin area function; growth function; maximal function; Riesz transform

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