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We discuss in a stochastic framework the interplay between Riemann-Liouvilletype operators applied to stochastic processes, real interpolation, boundedmean oscillation, and an approximation problem for stochastic integrals. Weprovide upper and lower bounds for gradient processes on the L\'evy-It\^ospace, which arise in the special case of the Wiener space from the Feynman-Kactheory for parabolic PDEs. The upper bounds are formulated in terms ofBMO-conditions on the fractional integrated gradient, the lower bounds in termsof oscillatory quantities. On the general L\'evy-It\^o space we are concernedwith gradient processes with values in a Hilbert space, where the regularitydepends on the direction in this Hilbert space. We discuss two applications ofour techniques: on the Wiener space an approximation problem for H\"olderfunctionals and on the L\'evy-It\^o space an orthogonal decomposition ofH\"older functionals into a sum of stochastic integrals with a control of thecorresponding integrands.

On Riemann-Liouville type operators, BMO, gradient estimates in the L\'evy-It\^o space, and approximation