We　investigate　the　joint　effects　of　a　magnetic　field　and　electron-electron　interaction　on　the　tunneling　density　of　states　(DOS)　of　a　quantum　dot　coupled　to　the　Luttinger　liquid　leads　in　the　Kondo　regime.　We　find　that　for　intralead　electron　interaction,　the　DOS　develops　two　peaks　deviated　from　the　origin　by　the　Zeeman　energy.　With　the　increase　of　the　intralead　interaction,　a　phase　transition　occurs.　For　moderately　strong　interaction,　the　Zeeman　splitting　peaks　develop　into　two　dips.　The　splitting　of　the　Kondo　peak　and　dip　is　not　symmetric　with　respect　to　up　and　down　spins.　In　the　limit　of　strong　interaction　the　Zeeman　splitting　behavior　disappears　and　there　appears　a　power-law　scaling　behavior.