Abstract. We study two problems. The first one is the similarity problem for indefinite Sturm--Liouville spectral problems. The second object is the so-called HELP inequality, a version of the classical Hardy-Littlewood inequality proposed by W.N. Everitt in 1971. Both problems are well understood when the corresponding Sturm-Liouville differential expression is regular. Our main main objective is to give criteria for both the validity of the HELP inequality and the similarity to a self adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl-Titchmarsh $m$-functions at $0$ and at $\infty$. As a biproduct of this result we show that both problems are closely connected. Next we characterize the behavior of $m$-functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients. Finally, we apply these results for the study of the two-way diffusion equation, also known as the time-independent Fokker-Plank equation.

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