Maximal $${\gamma}$$ γ -regularity

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In this paper, we prove maximal regularity estimates in “square function spaces” which are commonly used in harmonic analysis, spectral theory, and stochastic analysis. In particular, they lead to a new class of maximal regularity results for both deterministic and stochastic equations in L p -spaces with \({1 < p < {\infty} }\). For stochastic equations, the case 1 < p < 2 was not covered in the literature so far. Moreover, the “square function spaces” allow initial values with the same roughness as in the L 2-setting.