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Ouhabaz proved a $p$-specific $L^p$-spectral multiplier theorem for the Grushin operator acting on $\\mathbb{R}^{d_1}\\times\\mathbb{R}^{d_2}$ which is given by \\[ L =-\\sum_{j=1}^{d_1} \\partial_{x_j}^2 - \\bigg( \\sum_{j=1}^{d_1} |x_j|^2\\bigg) \\sum_{k=1}^{d_2}\\partial_{y_k}^2. \\] Their approach yields an $L^p$-spectral multiplier theorem within the range $1< p\\le \\min\\{ \\frac{2d_1}{d_1+2},\\frac{2(d_2+1)}{d_2+3} \\}$ under a regularity condition on the multiplier which is sharp only when $d_1\\ge d_2$. 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Chen and E. M. Ouhabaz proved a $p$-specific $L^p$-spectral multiplier theorem for the Grushin operator acting on $\\mathbb{R}^{d_1}\\times\\mathbb{R}^{d_2}$ which is given by \\[ L =-\\sum_{j=1}^{d_1} \\partial_{x_j}^2 - \\bigg( \\sum_{j=1}^{d_1} |x_j|^2\\bigg) \\sum_{k=1}^{d_2}\\partial_{y_k}^2. \\] Their approach yields an $L^p$-spectral multiplier theorem within the range $1< p\\le \\min\\{ \\frac{2d_1}{d_1+2},\\frac{2(d_2+1)}{d_2+3} \\}$ under a regularity condition on the multiplier which is sharp only when $d_1\\ge d_2$. 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