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• Uniform rectifiability and ε-approximability of harmonic functions in Lp 

  Hofmann, Steve; Tapiola, Olli (Centre Mersenne; l'Institut Fourier,, 2020)

  Suppose that E⊂Rn+1 is a uniformly rectifiable set of codimension 1. We show that every harmonic function is ε-approximable in Lp(Ω) for every p∈(1,∞), where Ω:=Rn+1∖E. Together with results of many authors this shows that ...

• Uniform rectifiability implies Varopoulos extensions 

  Hofmann, Steve; Tapiola, Olli (Elsevier, 2021)

  We construct extensions of Varopolous type for functions f∈BMO(E), for any uniformly rectifiable set E of codimension one. More precisely, let Ω⊂Rn+1 be an open set satisfying the corkscrew condition, with an n-dimensional ...

• A proof of Carleson's 𝜀2-conjecture 

  Jaye, Benjamin; Tolsa, Xavier; Villa, Michele (Mathematics Department, Princeton University, 2021)

  In this paper we provide a proof of the Carleson 𝜀2-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the ...

• Gradient estimates for heat kernels and harmonic functions 

  Coulhon, Thierry; Jiang, Renjin; Koskela, Pekka; Sikora, Adam (Elsevier, 2020)

  Let (X,d,μ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a “carré du champ”. Assume that (X,d,μ,E) supports a scale-invariant L2-Poincaré inequality. In this article, we study the following ...

• On a Continuous Sárközy-Type Problem 

  Kuca, Borys; Orponen, Tuomas; Sahlsten, Tuomas (Oxford University Press (OUP), 2023)

  We prove that there exists a constant ϵ>0ϵ>0 with the following property: if K⊂R2K⊂R2 is a compact set that contains no pair of the form {x,x+(z,z2)}{x,x+(z,z2)} for z≠0z≠0⁠, then dimHK≤2−ϵdimH⁡K≤2−ϵ⁠.