I recently tackled a problem and I arrived at something of the following form,
$$ \frac{1}{n^2} \sum_{i=0}^{n-1}\sum_{j=0}^{n-1} f(T^ix, T^jy), $$
where $T$ is a measure preserving transformation and I am interested in the limit as $n$ tends to infinity. In my case it is the shift operator in a probability space.
In the uni-variate case Birkhoff's ergodic theorem states that for a measure preserving transformation $T$,in a measurable space $(X, \mathscr{B})$, with a $\sigma$-finite measure $\mu$
$$\frac{1}{n} \sum_{i=0}^{n-1} f(T^ix)$$
converges a.e. to a function $f*\in L^1$, with $f^* = \frac{1}{\mu(X)}\int fd\mu$. Is there an equivalent result for the multivariate case? Will the first equation converge to it's average?
Best Answer
Let me expand here the point made in the the comments. First, I misread slightly your question. I had in mind two-parametric ergodic means given by $$ A_{n,m}(f) = \sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} f(T^i x, T^j y). $$ Not the means $A_{n}(f) = A_{n,n}(f)$. For the first you need to be in $L \log L(\Omega)$. For the second I do not have a definitive answer. It may be enough to be in $L^1$, see the edit bellow.
First, the following result is known.
The proof uses the following steps
Then, copying the argument in [JMZ], you have that the map $f \mapsto f^\ast$ given by $$ f^\ast(x,y) = \limsup_{n, m \to \infty} A_{n,m}(|f|) $$ maps $L \log L(\Omega)$ into $L_1(\Omega)$. Just by composition, we have that $$ \limsup_{n, m \to \infty} A_{n,m}(|f|) \leq \limsup_{n \to \infty} A^1_{n} \bigg( \underbrace{\sup_{m \geq 0} A_m^2|f|}_{g} \bigg) $$ But the function $g$ is in $L^1(\Omega)$ as the function $f$ is in $L \log L$. Now, using that for every function in $L^1(\Omega)$ we have almost everywhere convergence (and therefore the limsup is exchangeable by the lim) we can conclude. The fact that $f^\ast$ is in $L^1$ gives almost everywhere convergence by the same argument that is used with maximal functions.
I will go further and conjecture that the following is true (probably known to the experts):
It is known, see [JMZ, Theorem 8]. That this holds in the case of differentiability of integrals. I will try to adapt the argument to the ergodic case. It is also likely to be true, since something similar holds for martingales [G].
[G] Gundy, R. F., On the class L log L, martingales, and singular integrals, Stud. Math. 33, 109-118 (1969). ZBL0181.44202.
[JMZ] Jessen, B.; Marcinkiewicz, J.; Zygmund, A., Note on the differentiability of multiple integrals., Fundamenta Math. 25, 217-234 (1935). ZBL61.0255.01.
[Y] Yano, Shigeki, Notes on Fourier analysis. XXIX. An extrapolation theorem, J. Math. Soc. Japan 3, 296-305 (1951). ZBL0045.17901.
P.D.: For your purposes, perhaps it will be enough if you take the usual ergodic averages with respect to $$ S = p (\mathrm{id} \otimes T) + q (T \otimes \mathrm{id}) $$ for $p + q = 1$. Its ergodic averages will be weighted summations concentrated as Gaussian around the diagonal $i = j$.