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In this paper we prove a randomized difference norm characterization forBessel potential spaces with values in UMD Banach spaces. The main ingredientsare $\mathcal{R}$-boundedness results for Fourier multiplier operators, whichare of independent interest. As an application we characterize the pointwisemultiplier property of the indicator function of the half-space on thesespaces. All results are proved in the setting of weighted spaces.

Difference Norms for Vector-Valued Bessel Potential Spaces with an Application to Pointwise Multipliers