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\begin{document}

New Integral: 
  \begin{equation*}
    \left\|\left( \Psi_m A^a\right)_{2^j}\right\|_{H^{s}}\lesssim
    \left\|\left( (\psi_m A^a)_{2^m}\right)_{2^{j-m}}\right\|_{H^{s}}\leq C(2^{j-m})
    \left\| (\psi_m A^a)_{2^m}\right\|_{H^{s}}.
  \end{equation*} 
\begin{equation}
\label{eq:new:2}
\int_{a+1} A^{a} dx = \sum\limits_a^b \lambda  \varrho 
\end{equation}
\begin{equation}
\label{eq:klaus-t15sep:6}
\int\limits_a^b \quad \qquad \qquad \qquad 
\end{equation}
\begin{equation}
\int_a
\end{equation}
\begin{equation}
  \label{eq:new:1}
  \int_1 y_{n+1} \int_{n} \quad 
\end{equation}
\end{document}


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