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We present the ultralow-temperature specific heat and thermal conductivity measurements on single crystals of YbMgGaO4, which was recently argued to be a promising candidate for a quantum spin liquid (QSL). In a zero magnetic field, a large magnetic contribution of specific heat is observed, and exhibits a power-law temperature dependence (Cm ~ T0.74). On the contrary, we do not observe any significant contribution of thermal conductivity from magnetic excitations. In magnetic fields H ≥ 6 T, the exponential T dependence of Cm and the enhanced thermal conductivity indicate a magnon gap of the fully polarized state. The absence of magnetic thermal conductivity at the zero field in this QSL candidate puts a strong constraint on the theories of its ground state. [ABSTRACT FROM AUTHOR]