The Gagliardo-Nirenberg inequality on metric measure spaces

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In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent n (n≥ 2), then it has exactly the n-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if a complete n-dimensional Finsler manifold of nonnegative n-Ricci curvature satisfies the Gagliardo-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. The other one is that we give an alternative proof to Mao's main result in [23] for smooth metric measure spaces with nonnegative weighted Ricci curvature.