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تسجيل مستخدم جديدThe CMO-Dirichlet problem for the Schrodinger equation in the upper half-space and characterizations of CMO
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Let $mathcal{L}$ be a Schrodinger operator of the form $mathcal{L}=-Delta+V$ acting on $L^2(mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse Holder class ${RH}_q$ for some $qgeq (n+1)/2$. Let ${CMO}_{mathcal{L}}(mathbb{R}^n)$ denote the function space of vanishing mean oscillation associated to $mathcal{L}$. In this article we will show that a function $f$ of ${ CMO}_{mathcal{L}}(mathbb{R}^n) $ is the trace of the solution to $mathbb{L}u=-u_{tt}+mathcal{L} u=0$, $u(x,0)=f(x)$, if and only if, $u$ satisfies a Carleson condition $$ sup_{B: { balls}}mathcal{C}_{u,B} :=sup_{B(x_B,r_B): { balls}} r_B^{-n}int_0^{r_B}int_{B(x_B, r_B)} big|t abla u(x,t)big|^2, frac{ dx, dt } {t} <infty, $$ and $$ lim _{a rightarrow 0}sup _{B: r_{B} leq a} ,mathcal{C}_{u,B} = lim _{a rightarrow infty}sup _{B: r_{B} geq a} ,mathcal{C}_{u,B} = lim _{a rightarrow infty}sup _{B: B subseteq left(B(0, a)right)^c} ,mathcal{C}_{u,B}=0. $$ This continues the lines of the previous characterizations by Duong, Yan and Zhang cite{DYZ} and Jiang and Li cite{JL} for the ${ BMO}_{mathcal{L}}$ spaces, which were founded by Fabes, Johnson and Neri cite{FJN} for the classical BMO space. For this purpose, we will prove two new characterizations of the ${ CMO}_{mathcal{L}}(mathbb{R}^n)$ space, in terms of mean oscillation and the theory of tent spaces, respectively.
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