Articles

@article{nokey,

title = { Carleson's ε^2 conjecture in higher dimensions },

author = {Ian Fleschler and Xavier Tolsa and Michele Villa },

url = {https://arxiv.org/abs/2310.12316},

year  = {2025},

date = {2025-05-01},

urldate = {2025-05-01},

journal = { Inventiones Mathematicae},

volume = {241},

number = {1},

pages = {207-307},

abstract = { We prove a higher dimensional analogue of Carleson's ε^2 conjecture about the characterization of tangent points.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {Unique continuation for locally uniformly distributed measures},

author = {Max Engelstein and Ignasi Guillén-Mola},

url = {https://arxiv.org/abs/2501.13869},

doi = {https://doi.org/10.1007/s12220-025-01978-6},

year  = {2025},

date = {2025-03-27},

urldate = {2025-03-27},

journal = {The Journal of Geometric Analysis},

volume = {35},

number = {141},

abstract = {In this note we show that the support of a locally $k$-uniform measure in $mathbb{R}^{n+1}$ satisfies a kind of unique continuation property. As a consequence, we show that locally uniformly distributed measures satisfy a weaker unique continuation property. This continues work of Kirchheim and Preiss (Math. Scand. 2002) and David, Kenig and Toro (Comm. Pure Appl. Math. 2001) and lends additional evidence to the conjecture proposed by Kowalski and Preiss (J. Reine Angew. Math. 1987) that each connected component of the support of a locally $n$-uniform measure in $mathbb{R}^{n+1}$ is contained in the zero set of a quadratic polynomial.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@unpublished{nokey,

title = { Quantitative Carleson's conjecture for Ahlfors regular domains},

author = {Emily Casey and Xavier Tolsa and Michele Villa},

url = {https://arxiv.org/abs/2505.10666},

year  = {2025},

date = {2025-01-01},

abstract = { In this article, we prove a quantitative version of Carleson's conjecture in higher dimension: we characterise those Ahlfors-David regular domains for which the Carleson's coefficients satisfy the so-called strong geometric lemma. },

howpublished = {Preprint arXiv:2505.10666 },

keywords = {},

pubstate = {published},

tppubtype = {unpublished}

}

@unpublished{nokey,

title = {Quantitative unique continuation for Neumann problem in planar $C^{1,alpha}$ domains},

author = {Yingying Cai and Jiuyi Zhu and Jinping Zhuge},

year  = {2025},

date = {2025-01-01},

urldate = {2025-01-01},

abstract = {In this paper, we study the quantitative unique continuation property of the second-order elliptic operators under the vanishing Neumann boundary condition over $C^{1,alpha}$ or convex domains in two dimensions. We establish the optimal estimates of the number of critical points, doubling index and the total length of level curves. The key idea is to reduce the Neumann problem to the Dirichlet problem, which has been understood better, by a classical duality between an A-harmonic function and its stream function.},

keywords = {},

pubstate = {published},

tppubtype = {unpublished}

}

@article{nokey,

title = {The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domainss},

author = {Ignasi Guillén-Mola and Martí Prats and Xavier Tolsa},

url = {https://arxiv.org/abs/2406.11604},

year  = {2025},

date = {2025-01-01},

urldate = {2025-01-01},

journal = {Anal. Math. Phys.},

volume = {15},

issue = {4},

pages = { Paper No. 106, 79 pp.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {A counterexample regarding a two-phase problem for harmonic measure in VMO},

author = {Xavier Tolsa},

url = {https://arxiv.org/abs/2401.14568},

year  = {2025},

date = {2025-01-01},

urldate = {2025-01-01},

journal = {Potential Anal.},

volume = {63},

issue = {3},

pages = {1499-1515},

abstract = {Let U be a vanishing Reifenberg flat domain such that U and U^c have joint big pieces of chord-arc subdomains and the outer unit normal to the boundary belongs to VMO with respect to harmonic measure. In this paper we find a fractal type domain with the property that the outer unit normal is constant a.e. with respectto harmonic measure.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = { Extrapolation of solvability of the regularity problem in rough domains},

author = {Josep M. Gallegos and Mihalis Mourgoglou and Xavier Tolsa},

url = {https://arxiv.org/abs/2310.12316},

year  = {2025},

date = {2025-01-01},

urldate = {2023-06-09},

journal = {J. Funct. Anal. 288},

volume = {288},

number = {1},

pages = { Paper No. 110672},

abstract = { Let Ω⊂Rn+1, n≥2, be an open set satisfying the corkscrew condition with compact and uniformly n-rectifiable boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space M1,1(∂Ω) and the weak-A∞ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in M1,p(∂Ω) for p>1 implies solvability in M1,1(∂Ω) and, in the particular case of the Laplacian, solvability in M1,1(∂Ω) implies solvability in M1,p(∂Ω) for some p>1. Moreover, under the hypothesis that ∂Ω supports a 1-weak Poincaré inequality, we prove that the solvability of the regularity problem in the Hajłasz-Sobolev space M1,1(∂Ω) is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives. },

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {The dimension of harmonic measure on some AD-regular flat sets of fractional dimension},

author = {Xavier Tolsa},

url = {https://arxiv.org/abs/2301.04084},

year  = {2024},

date = {2024-08-01},

urldate = {2023-01-20},

journal = { Int. Math. Res. Not. IMRN},

volume = {2024},

number = {8},

pages = { 6579–6605},

abstract = { In this paper it is shown that if E⊂Rn+1 is an s-AD regular compact set, with s∈[n−12,n), and E is contained in a hyperplane or, more generally, in an n-dimensional C1 manifold, then the Hausdorff dimension of the harmonic measure for the domain R^{n+1}∖E is strictly smaller than s, i.e., than the Hausdorff dimension of E. },

howpublished = {To appear in Int. Math. Res. Not. IMRN },

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {The A∞ condition, ε-approximators, and Varopoulos extensions in uniform domains },

author = { Simon Bortz and Bruno Poggi and Olli Tapiola and Xavier Tolsa},

url = {https://arxiv.org/abs/2302.13294},

year  = {2024},

date = {2024-04-05},

urldate = {2024-04-05},

journal = {J. Geom. Anal. },

volume = {34},

number = {7},

pages = {Paper No. 218, 53 pp.},

abstract = {Suppose that Ω⊂Rn+1, n≥1, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Ω. We show that the corresponding elliptic measure ωL is quantitatively absolutely continuous with respect to surface measure of ∂Ω in the sense that ωL∈A∞(σ) if and only if any bounded solution u to Lu=0 in Ω is ε-approximable for any ε∈(0,1). By ε-approximability of u we mean that there exists a function Φ=Φε such that ∥u−Φ∥L∞(Ω)≤ε∥u∥L∞(Ω) and the measure μ˜Φ with dμ˜=|∇Φ(Y)|dY is a Carleson measure with L∞ control over the Carleson norm. As a consequence of this approximability result, we show that boundary BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy L1-type Carleson measure estimates with BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis. },

howpublished = {To appear in The Journal of Geometric Analysis},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {Critical Perturbations for Second Order Elliptic Operators. Part II: Non-tangential maximal function estimates},

author = {Simon Bortz and Steve Hofmann and José Luis Luna and Svitlana Mayboroda and Bruno Poggi},

url = {https://arxiv.org/abs/2302.02746},

doi = {10.1007/s00205-024-01977-x},

year  = {2024},

date = {2024-04-04},

urldate = {2024-02-19},

journal = {Archive for Rational Mechanics and Analysis},

volume = {248},

number = {31},

abstract = {This is the final part of a series of papers where we  study perturbations of divergence form second order elliptic operators $-operatorname{div} A nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi-Nash-Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-$L^p$ ``$N<S$'' estimate, which we eventually couple with square function bounds,  weighted extrapolation theory, and a bootstrapping argument to recover the full $L^2$ bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class. 

	

As a corollary, we claim the first results in   an unbounded domain concerning the $L^p$-solvability of boundary value problems for the magnetic Schr"odinger operator $-(nabla-i{bf a})^2+V$ when the magnetic potential ${bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space.},

howpublished = {To appear in Archive for Rational Mechanics and Analysis},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@unpublished{nokey,

title = {On an almost sharp Liouville type theorem for fractional Navier-Stokes equations},

author = {Diego Chamorro and Bruno Poggi},

url = {https://arxiv.org/abs/2211.13077},

year  = {2024},

date = {2024-04-03},

abstract = {We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power $(-Delta)^{frac{alpha}{2}}$ with $0<alpha<2$. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $dot{H}^{frac{alpha}{2}}(R)$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $alpha$ that the zero function is the unique smooth solution. The additional integrability condition is almost sharp for $3/5<alpha<5/3$. Moreover, in the case $1<alpha<2$ a gain of regularity is established under some conditions, however the study of regularity in the regime $0<alphaleq 1$ seems for the moment to be an open problem.},

howpublished = {To appear in Publicacions Matemàtiques},

keywords = {},

pubstate = {published},

tppubtype = {unpublished}

}

@unpublished{nokey,

title = {Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields},

author = {Bruno Poggi},

url = {https://arxiv.org/abs/2107.14103},

year  = {2024},

date = {2024-04-03},

abstract = {We resolve both a conjecture and  a problem of Z. Shen from the  90's regarding non-asymptotic   bounds on the eigenvalue counting function  of the magnetic Schr"odinger operator  $L_$, under a mild decay   condition on the Green's function. For  $L_V$,  we prove a priori exponential decay of Green's function,  eigenfunctions, and Lax-Milgram solutions in an Agmon distance with weight $1/u$, which may degenerate. Similar a priori results hold for $L_{{bf a},V}$. Furthermore, when $ngeq3$ and $V$ satisfies a scale-invariant Kato condition and a weak doubling property, we show that $1/sqrt u$ is pointwise equivalent to the   Fefferman-Phong-Shen maximal function $m(cdot,V)$ (also known as Shen's critical radius function); in particular this gives a setting where the Agmon distance with weight $1/u$ is not too degenerate.  Finally, we  extend results from the literature for $L_{{bf a},V}$ regarding exponential decay of the fundamental solution and eigenfunctions,   to the  situation of   irregular magnetic fields with directionality assumptions.},

howpublished = {To appear in Advances in Mathematics},

keywords = {},

pubstate = {published},

tppubtype = {unpublished}

}

@unpublished{nokey,

title = {Solvability of the Neumann problem for elliptic equations in chord-arc domains with very pieces of good superdomains},

author = {Mihalis Mourgoglou and Xavier Tolsa},

url = {https://arxiv.org/abs/2407.20385},

year  = {2024},

date = {2024-01-01},

urldate = {2024-01-01},

abstract = {Let Omega subset mathbb{R}^{n+1} be a bounded chord-arc domain, let mathcal L=-{rm div} Anabla be an elliptic operator in Omega associated with a matrix A having Dini mean oscillation coefficients, and let 1<pleq 2. In this paper we show that if the regularity problem for mathcal L is solvable in L^q for some q>p in Omega, partial Omega supports a weak p-Poincaré inequality, and Omega has very big pieces of superdomains for which the Neumann problem for mathcal L is solvable uniformly in L^q, then the Neumann problem for mathcal L is solvable in L^p in Omega. },

keywords = {},

pubstate = {published},

tppubtype = {unpublished}

}

@article{nokey,

title = {The two-phase problem for harmonic measure in VMO and the chord-arc condition},

author = {Xavier Tolsa and Tatiana Toro},

url = {https://arxiv.org/abs/2209.14346},

year  = {2024},

date = {2024-01-01},

urldate = {2022-09-28},

journal = {Trans. Amer. Math. Soc. Ser. B},

volume = {11},

pages = {1294–1315},

abstract = { Let Ω+⊂Rn+1 be a bounded δ-Reifenberg flat domain, with δ>0 small enough, possibly with locally infinite surface measure. Assume also that Ω−=Rn+1∖Ω+¯ is an NTA domain as well and denote by ω+ and ω− the respective harmonic measures of Ω+ and Ω− with poles p±∈Ω±. In this paper we show that the condition that logdω−dω+∈VMO(ω+) is equivalent to Ω+ being a chord-arc domain with inner normal belonging to VMO(Hn|∂Ω+). },

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {The regularity problem for the Laplace equation in rough domains},

author = {Mihalis Mourgoglou and Xavier Tolsa},

url = {https://arxiv.org/abs/2110.02205},

year  = {2024},

date = {2024-01-01},

urldate = {2021-10-05},

journal = {Duke Math. J. },

volume = {173},

issue = {9},

pages = {1731–1837},

abstract = {Let Ω⊂Rn+1, n≥2, be a bounded open and connected set satisfying the corkscrew condition with uniformly n-rectifiable boundary. In this paper we study the connection between the solvability of (Dp′), the Dirichlet problem for the Laplacian with boundary data in Lp′(∂Ω), and (Rp) (resp. (R~p)), the regularity problem for the Laplacian with boundary data in the Hajłasz Sobolev space W1,p(∂Ω) (resp. W~1,p(∂Ω), the usual Sobolev space in terms of the tangential derivative), where p∈(1,2+ε)  and 1/p+1/p′=1. Our main result shows that (Dp′) is solvable if and only if so is (Rp). Under additional geometric assumptions (two-sided local John condition or weak Poincaré inequality on the boundary), we prove that (Dp′)⇒(R~p). In particular, we deduce that in bounded chord-arc domains (resp. two-sided chord-arc domains) there exists p0∈(1,2+ε) so that (Rp0) (resp. (R~p0)) is solvable. We also extend the results to unbounded domains with compact boundary and show that in two-sided corkscrew domains with n-Ahlfors-David regular boundaries the single layer potential operator is invertible from Lp(∂Ω) to the inhomogeneous Sobolev space W1,p(∂Ω). Finally, we provide a counterexample of a chord-arc domain Ω0⊂Rn+1, n≥3, so that (R~p) is not solvable for any p∈[1,∞).},

howpublished = {To appear in Duke Math. J.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {Connectivity conditions and boundary Poincaré inequalities},

author = {Olli Tapiola and Xavier Tolsa},

url = {https://arxiv.org/abs/2205.11667},

year  = {2024},

date = {2024-01-01},

urldate = {2022-05-23},

journal = {Analysis & PDE },

volume = {17},

number = {5},

pages = {1831–1870},

abstract = {Inspired by recent work of Mourgoglou and the second named author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets Ω⊂Rn+1, with codimension 1 Ahlfors--David regular boundaries. First, we prove that if Ω satisfies both the local John condition and the exterior corkscrew condition, then Ω also satisfies the Harnack chain condition (and hence, is a chord-arc domain). Second, we show that if Ω is a 2-sided chord-arc domain, then the boundary ∂Ω supports a Heinonen--Koskela type weak 1-Poincaré inequality. We also construct an example of a set Ω⊂Rn+1 such that the boundary ∂Ω is Ahlfors--David regular and supports a weak boundary 1-Poincaré inequality but Ω is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories. },

howpublished = {To appear in Anal. & PDE},

keywords = {},

pubstate = {published},

tppubtype = {article}

}