Description of the spaces $L_p^r(R^n)$ in terms of singular difference integrals

This article gives a necessary and sufficient condition for a $p$-integrable function to have partial derivatives of specified orders which are $p$th power integrable over $R^n$. This condition is expressed using integrals of differences which in general converge conditionally in the $L_p$-norm. We also prove a Fubini theorem for these function spaces.
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