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Consider a RHP on the real line a jump is piece-wise H\"older continuous(or $L^2$), say for example the jump is $$g(x)=g_1(x)\chi_1+g_2(x)\chi_2,$$ where $g_j(x)$ are Holder continuous functions and $supp\chi_1\cap supp\chi_2=\emptyset$.

Do we have decomposition $m(z)=m_1(z)\chi_1(\Re(z))+m_2\chi_2(\Re(z))$, where $m_{j+}=m_{j-}g_j$ on the support of $\chi_j,j=1,2$.

My question: is my question even well-posed? Any existing theories in literature?

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    $\begingroup$ Maybe I misunderstood your question completely, but it makes no sense to me: $m(z)$ is supposed to be holomorphic in the upper and lower complex half-planes, while $m_1(z) \chi_1(\Re z) + m_2(z) \chi_2(\Re z)$, if non-zero, will necessarily be discontinuous almost everywhere on $x + i \mathbb{R}$ for every $x$ on the boundary of the support of $\chi_1$ or $\chi_2$. $\endgroup$
    – Mateusz Kwaśnicki
    Commented May 9, 2020 at 8:07
  • $\begingroup$ That's right. And there are two ways. One is put new jumps one those boundaries (which may be designed my our own will, hopefully). Second way is to find another possible decomposition. The decomposition i provided is just a native guess though. @MateuszKwaśnicki $\endgroup$
    – DuFong
    Commented May 9, 2020 at 20:34

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