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In this paper, we consider the integral functional $$ {\\mathcal C}^H_t(a):=\\lim_{\\varepsilon\\downarrow 0}\\int_0^t1_{\\{|B^H_s-a|>\\varepsilon\\}}\\frac1{B^H_s-a}ds^{2H}\\equiv \\frac1{\\pi}{\\mathscr H}{\\mathscr L}^H(\\cdot,t)(a) $$ in $L^2(\\Omega)$ with $ a\\in {\\mathbb R}, t\\geq 0$ and ${\\mathscr H}$ denoting the Hilbert transform. 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