Abstract:Let T be a finite tree graph, T N be the Cartesian power graph of T , and d N be the graph distance metric on T N . Also let= r} be the sphere of radius r centered at x and M be the spherical maximal averaging operator on T N given byf (y) .We will show that for any fixed 1 ≤ p ≤ ∞, the L p operator norm of M , i.e.M p := supgrows exponentially in the dimension N . In particular, if r is the probability that a random vertex of T is a leaf, then M p ≥ r −N/p , although this is not a sharp bound. This exponentia… Show more
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