Multipliers and imaginary powers of the schrödinger operators characterizing UMD Banach spaces

In this paper we establish $L^p$-boundedness properties for Laplace type transform spectral multipliers associated with the Schrödinger operator $\mathcal{L}=-Δ+V$. We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the $L^p$-boundedness of the imaginary powers $\mathcal{L}^{iγ}$, $γ\in \mathbb{R}$, of $\mathcal{L}$.

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