Interpolation of Sobolev spaces , Littlewood-Paley inequalities and Riesz transforms on graphs

  title={Interpolation of Sobolev spaces , Littlewood-Paley inequalities and Riesz transforms on graphs},
  author={N. Badr and Emmanuel Russ},
Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖p and ∥(I − P )f ∥∥ p uniformly in f for 1 < p < +∞. These conditions are different for p < 2 and p > 2. The proofs rely on recent techniques developed to handle operators beyond the class of Calderón-Zygmund operators. For our purpose, we also prove Littlewood… CONTINUE READING

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