Let Ω \Omega be either R n \mathbb {R}^n or a strongly Lipschitz domain of R n \mathbb {R}^n , and ω ∈ A ∞ ( R n ) \omega \in A_{\infty }(\mathbb {R}^n) (the class of Muckenhoupt weights). Let L L be a second-order divergence form elliptic operator on L 2 ( Ω ) L^2 (\Omega ) with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by L L has the Gaussian property ( G 1 ) (G_1) with the regularity of their kernels measured by μ ∈ ( 0 , 1 ] \mu \in (0,1] . Let Φ \Phi be a continuous, strictly increasing, subadditive, positive and concave function on ( 0 , ∞ ) (0,\infty ) of critical lower type index p Φ − ∈ ( 0 , 1 ] p_{\Phi }^-\in (0,1] . In this paper, the authors first introduce the “geometrical” weighted local Orlicz-Hardy spaces h ω , r Φ ( Ω ) h^{\Phi }_{\omega ,\,r}(\Omega ) and h ω , z Φ ( Ω ) h^{\Phi }_{\omega ,\,z}(\Omega ) via the weighted local Orlicz-Hardy spaces h ω Φ ( R n ) h^{\Phi }_{\omega }(\mathbb {R}^n) , and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by L L when p Φ − ∈ ( n / ( n + μ ) , 1 ] p_{\Phi }^-\in (n/(n+\mu ),1] . Second, the authors furthermore establish three equivalent characterizations of h ω , r Φ ( Ω ) h^{\Phi }_{\omega ,\,r}(\Omega ) in terms of the grand maximal function, the radial maximal function and the atomic decomposition when the complement of Ω \Omega is unbounded and p Φ − ∈ ( 0 , 1 ] p_{\Phi }^-\in (0,1] . Third, as applications, the authors prove that the operators ∇ 2 G D \nabla ^2{\mathbb G}_D are bounded from h ω , r Φ ( Ω ) h^{\Phi }_{\omega ,\,r}(\Omega ) to the weighted Orlicz space L ω Φ ( Ω ) L^{\Phi }_{\omega }(\Omega ) , and from h ω , r Φ ( Ω ) h^{\Phi }_{\omega ,\,r}(\Omega ) to itself when Ω \Omega is a bounded semiconvex domain in R n \mathbb {R}^n and p Φ − ∈ ( n n + 1 , 1 ] p_{\Phi }^-\in (\frac {n}{n+1},1] , and the operators ∇ 2 G N \nabla ^2{\mathbb G}_N are bounded from h ω , z Φ ( Ω ) h^{\Phi }_{\omega ,\,z}(\Omega ) to L ω Φ ( Ω ) L^{\Phi }_{\omega }(\Omega ) , and from h ω , z Φ ( Ω ) h^{\Phi }_{\omega ,\,z}(\Omega ) to h ω , r Φ ( Ω ) h^{\Phi }_{\omega ,\,r}(\Omega ) when Ω \Omega is a bounded convex domain in R n \mathbb {R}^n and p Φ − ∈ ( n n + 1 , 1 ] p_{\Phi }^-\in (\frac {n}{n+1},1] , where G D {\mathbb G}_D and G N {\mathbb G}_N denote, respectively, the Dirichlet Green operator and the Neumann Green operator.
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Aug 21, 2013
A. F. M. ter Elst + 1
Open Access
