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Mapping properties of fundamental harmonic analysis operators in the exotic Bessel framework

We prove sharp power-weighted $L^p$, weak type and restricted weak type inequalities for the heat semigroup maximal operator and Riesz transforms associated with the Bessel operator $B_ν$ in the exotic range of the parameter $-\infty < ν< 1$. Moreover, in the same framework, we characterize basic mapping properties for other fundamental harmonic analysis operators, including the heat semigroup based vertical $g$-function and fractional integrals (Riesz potential operators).