Consider the multidimensional Bessel operator Bf(x)=-∑j=1N∂j2f(x)+αjxj∂jf(x),x∈(0,∞)N.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} B f(x) = -\sum _{j=1}^N \left( \partial _j^2 f(x) +\frac{\alpha _j}{x_j} \partial _j f(x) \right) , \quad x\in (0,\infty )^N. \end{aligned}$$\end{document}Let d=∑j=1Nmax(1,αj+1)\documentclass[12pt]{minimal}
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\begin{document}$$d = \sum _{j=1}^N \max (1,\alpha _j+1)$$\end{document} be the dimension of the space (0,∞)N\documentclass[12pt]{minimal}
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\begin{document}$$(0,\infty )^N$$\end{document} equipped with the measure x1α1…xNαNdx1…dxN\documentclass[12pt]{minimal}
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\begin{document}$$x_1^{\alpha _1}\ldots x_N^{\alpha _N} dx_1\ldots dx_N$$\end{document}. In the general case α1,…,αN>-1\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _1,\ldots ,\alpha _N >-1$$\end{document} we prove multiplier theorems for spectral multipliers m(B) on L1,∞\documentclass[12pt]{minimal}
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\begin{document}$$L^{1,\infty }$$\end{document} and the Hardy space H1\documentclass[12pt]{minimal}
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\begin{document}$$H^1$$\end{document}. We assume that m satisfies the classical Hörmander condition supt>0η(·)m(t·)W2,β(R)<∞\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sup _{t>0} \left\| \eta (\cdot ) m(t\cdot ) \right\| _{W^{2,\beta }(\mathbb {R})}<\infty \end{aligned}$$\end{document}with β>d/2\documentclass[12pt]{minimal}
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\begin{document}$$\beta > d/2$$\end{document}. Furthermore, we investigate imaginary powers Bib\documentclass[12pt]{minimal}
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\begin{document}$$B^{ib}$$\end{document}, b∈R\documentclass[12pt]{minimal}
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\begin{document}$$b\in \mathbb {R}$$\end{document}, and prove some lower estimates on L1,∞\documentclass[12pt]{minimal}
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\begin{document}$$L^{1,\infty }$$\end{document} and Lp\documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document}, 1<p<2\documentclass[12pt]{minimal}
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\begin{document}$$1<p<2$$\end{document}. As a consequence, we deduce that our multiplier theorem is sharp.
