Moderate deviations for the capacity of the random walk range in dimension four

In this paper we find a natural four-dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen (Mem. Amer. Math. Soc. 198 (2009) viii+82) concerning the volume of the random walk range for d=2. We find that the deviation statistics of the capacity of the random walk can be related to the following constant of generalized Gagliardo–Nirenberg inequalities: inff:‖∇f‖L2<∞ ‖f‖L21/2‖∇f‖L21/2 [∫(R4)2f2(x)G(x−y)f2(y)dxdy]1/4.