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We prove qualitatively sharp heat kernel bounds in the setting ofFourier-Bessel expansions when the associated type parameter $\nu$ ishalf-integer. Moreover, still for half-integer $\nu$, we also obtain sharpestimates of all kernels subordinated to the heat kernel. Analogous estimatesfor general $\nu > -1$ are conjectured. Some consequences concerning therelated heat semigroup maximal operator are discussed.

On sharp heat and subordinated kernel estimates in the Fourier-Bessel setting