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We prove sharp maximal inequalities for    L  q   ‐valued stochastic integrals with respect to any Hilbert space‐valued local martingale. Our proof relies on new Burkholder–Rosenthal type inequalities for martingales taking values in an    L  q   ‐space.

L q ‐valued Burkholder–Rosenthal inequalities and sharp estimates for stochastic integrals