We introduce the fractional Littlewood–Paley g-function of order s,s>0\documentclass[12pt]{minimal}
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\begin{document}$$s, ~~s>0$$\end{document}, noted gsα,β\documentclass[12pt]{minimal}
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\begin{document}$$g_{s}^{\alpha ,\beta }$$\end{document}, associated with the Jacobi operator Δα,β\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{\alpha ,\beta }$$\end{document} on (0,∞\documentclass[12pt]{minimal}
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\begin{document}$$(0, \infty $$\end{document}) as the operator gsα,β(f)(x)=∫0∞t2s+1|∂uα,β(f)∂t(x,t)|2dt12,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}g_{s}^{\alpha , \beta }(f)(x)=\left( \int _{0}^{\infty }t^{2s+1}|\frac{\partial u_{\alpha , \beta }(f)}{\partial t}(x,t)|^{2}dt\right) ^{\frac{1}{2}}, \end{aligned}$$\end{document}where uα,β(f)\documentclass[12pt]{minimal}
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\begin{document}$$ u_{\alpha , \beta }(f)$$\end{document} is the Poisson integral defined by uα,β(f)=Ptα,β∗α,βf\documentclass[12pt]{minimal}
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\begin{document}$$u_{\alpha , \beta }(f)= P_{t}^{\alpha , \beta }*_{\alpha , \beta }f$$\end{document} (∗α,β\documentclass[12pt]{minimal}
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\begin{document}$$ *_{\alpha , \beta }$$\end{document} being the convolution in the Jacobi setting). We establish the following Hardy–Littlewood–Sobolev-type inequality: For 0<s<2(α+1),1<p<2(α+1)s\documentclass[12pt]{minimal}
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\begin{document}$$0<s<2(\alpha +1),~~1<p<\frac{2(\alpha +1)}{s}$$\end{document} and 1q=1p-s2(α+1)\documentclass[12pt]{minimal}
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\begin{document}$$\frac{1}{q}=\frac{1}{p}-\frac{s}{2(\alpha +1)}$$\end{document}, there exists a constant Cα,s,p\documentclass[12pt]{minimal}
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\begin{document}$$C_{\alpha , s, p}$$\end{document} such that for all f∈Lp([0,+∞[,dμα,β)\documentclass[12pt]{minimal}
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\begin{document}$$f \in L^{p}([0,+\infty [,d\mu _{\alpha ,\beta })$$\end{document}, ‖gsα,βf‖q,μ≤Cα,s,p‖f‖p,μ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert g_{s}^{\alpha ,\beta }f\Vert _{q,\mu }\le C_{\alpha , s, p} \Vert f\Vert _{p,\mu }. \end{aligned}$$\end{document}Next, if p=1\documentclass[12pt]{minimal}
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\begin{document}$$p=1$$\end{document}, 0<s<2(α+1)\documentclass[12pt]{minimal}
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\begin{document}$$0<s<2(\alpha +1)$$\end{document} and q=α+12(α+1)-s\documentclass[12pt]{minimal}
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\begin{document}$$q=\frac{\alpha +1}{2(\alpha +1)-s}$$\end{document}, we prove that the operator gsα,β\documentclass[12pt]{minimal}
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\begin{document}$$g_{s}^{\alpha ,\beta }$$\end{document} is of weak type (1, q).
