Geodesic
Convergence of the imaginary parts of simplest fractions in $L_p(\mathbb R)$ for $p1$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 41, Tome 416 (2013), pp. 108-116

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For $p\in(1/2,1)$, the $L_p(\mathbb R)$-convergence of the series $\sum_{k=1}^\infty|\operatorname{Im}(t-z_k)^{-1}|$ is studied, where the $z_k$ are some points on the complex plane. The problem is solved completely in the case where the sequence $\{\operatorname{Re}z_k\}$ has no limit points. Also, the case where this sequence has finitely many limit points is studied. 
@article{ZNSL_2013_416_a5,
     author = {I. R. Kayumov and A. V. Kayumova},
     title = {Convergence of the imaginary parts of simplest fractions in $L_p(\mathbb R)$ for $p<1$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {108--116},
     publisher = {mathdoc},
     volume = {416},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_416_a5/}
}
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I. R. Kayumov; A. V. Kayumova. Convergence of the imaginary parts of simplest fractions in $L_p(\mathbb R)$ for $p<1$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 41, Tome 416 (2013), pp. 108-116. http://geodesic.mathdoc.fr/item/ZNSL_2013_416_a5/