In this paper, we characterize the weighted Hardy space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{\mathcal {L}}^{1}(\omega )$\end{document} related to the Schrödinger operator \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {L}=-\Delta +V$\end{document}, with V a non‐negative potential satisfying a reverse Hölder inequality, by atomic decomposition and Riesz transforms. We also get a characterization of its dual space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$B\hspace*{-1.0pt}M\hspace*{-1.0pt}O_{\mathcal {L}}(\omega )$\end{document} through a weighted Carleson measure. Then we prove the boundedness of some classical operators associated to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {L}$\end{document} on the weighted BMO space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$B\hspace*{-1.0pt}M\hspace*{-1.0pt}O_{\mathcal {L}}(\omega )$\end{document}.
In this paper, we characterize the weighted local Hardy spaces h p ρ (ω) related to the critical radius function ρ and weights ω ∈ A ρ,∞ ∞ (R n ) which locally behave as Muckenhoupt's weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. Then, we establish the equivalence of the weighted local Hardy space h 1 ρ (ω) and the weighted Hardy space H 1 L (ω) associated to Schrödinger operators L with ω ∈ A ρ,∞ 1 (R n ). By the atomic characterization, we also prove the existence of finite atomic decompositions associated with h p ρ (ω). Furthermore, we establish boundedness in h p ρ (ω) of quasi-Banach-valued sublinear operators.
We study the boundedness of weighted multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We also investigate weighted estimates for bilinear operators related to Schrödinger operator.
In this paper, we characterize the weighted local Hardy spaces h p ρ (ω) related to the critical radius function ρ and weights ω ∈ A ρ, ∞ ∞ (R n ) which locally behave as Muckenhoupt's weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. By the atomic characterization, we also prove the existence of finite atomic decompositions associated with h p ρ (ω). Furthermore, we establish boundedness in h p ρ (ω) of quasi-Banach-valued sublinear operators. As their applications, we establish the equivalence of the weighted local Hardy space h 1 ρ (ω) and the weighted Hardy space H 1 L (ω) associated to Schrödinger operators L with ω ∈ A ρ,∞ 1 (R n ).
Let p be an odd prime. In this paper, we consider the equationand we describe all its solutions. Moreover, we prove that this equation has no solution (x, y, m, n) when n > 3 is an odd prime and y is not the sum of two consecutive squares. This extends the work of Tengely [On the diophantine equation x 2 + q 2m = 2y p , Acta Arith. 127(1) (2007), 71-86].
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