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We prove that in every bipartite Cayley graph of every non-amenable group, there is a perfect matching that is obtained as a factor of independent uniform random variables. We also discuss expansion properties of factors and improve the Hoffman spectral bound on independence number of finite graphs. A perfect matching in a graph is a set of its edges that(More)
Persistence and permanence are properties of dynamical systems that describe the long-term behavior of the solutions, and in particular specify whether positive solutions approach the boundary of the positive orthant. Mass-action systems (or more generally power-law systems) are very common in chemistry, biology, and engineering, and are often used to(More)
1. Introduction 7 2. Dyadic cubes and lattices 8 3. The Three Lattice Theorem 13 4. The forest structure on a subset of a dyadic lattice 17 5. Stopping times and augmentation 20 6. Sparse and Carleson families 21 7. From the theory to applications 29 8. The multilinear Calderón-Zygmund operators 30 9. Controlling values of T [f 1 ,. .. , f m ] on a cube 32(More)
We prove that if µ is a d-dimensional Ahlfors-David regular measure in R d+1 , then the boundedness of the d-dimensional Riesz transform in L 2 (µ) implies that the non-BAUP David-Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of µ.
We consider functions f (A, B) of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class B 1 ∞,1 (R 2), then we have the following Lipschitz type estimate in the trace norm: f (A 1 , B 1) − f (A 2 , B 2) S 1 ≤ const(A 1 − A 2 S 1 + B 1 − B 2 S 1). However, the(More)
We generalize earlier results of [2], [3], [6], [13], [14] to the case of functions of n-tuples of commuting self-adjoint operators. In particular, we prove that if a function f belongs to the Besov space B 1 ∞,1 (R n), then f is operator Lipschitz and we show that if f satisfies a Hölder condition of order α, then f (A 1 · · · , An) − f (B 1 , · · · , Bn)(More)
Fix d ≥ 2, and s ∈ (d−1, d). We characterize the non-negative locally finite non-atomic Borel measures µ in R d for which the associated s-Riesz transform is bounded in L 2 (µ) in terms of the Wolff energy. This extends the range of s in which the Mateu-Prat-Verdera characterization of measures with bounded s-Riesz transform is known. As an application, we(More)
Background Joint work with Fedor Nazarov and Igor Verbitsky (preprint) Start with general functional analysis theorem: Background Joint work with Fedor Nazarov and Igor Verbitsky (preprint) Start with general functional analysis theorem: Given an integral operator T on a σ-finite measure space (Ω, ω) with kernel K : Background Joint work with Fedor Nazarov(More)
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