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Let (X,B, μ) be a probability space and let T1, . . . , Tl be l commuting invertible measure preserving transformations of X. We show that if T c1 1 . . . T cl l is ergodic for each (c1, . . . , cl) 6= (0, . . . , 0), then the averages 1 |ΦN | ∑ u∈ΦN ∏r i=1 T pi1(u) 1 . . . T pil(u) l fi converge in L (μ) for all polynomials pij : Z → Z, all fi ∈ L∞(μ), and all Folner sequences {ΦN}N=1 in Z.

Convergence of polynomial ergodic averages of several variables for some commuting transformations