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We study Fourier multipliers on free group $\mathbb{F}_\infty$ associatedwith the first segment of the reduced words, and prove that they are completelybounded on the noncommutative $L^p$ spaces $L^p(\hat{\mathbb{F}}_\infty)$ ifftheir restriction on $L^p(\hat{\mathbb{F}}_1)=L^p(\mathbb{T})$ are completelybounded. As a consequence, every classical Mikhlin multiplier extends to a$L^p$ Fourier multiplier on free groups for all $1<p<\infty$.

Free Fourier Multipliers associated with the firstSegment