Vol. 2025
|
Engelstein, Max; Guillén-Mola, Ignasi Unique continuation for locally uniformly distributed measures In: The Journal of Geometric Analysis, vol. 35, no. 141, 2025. Abstract | Links @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}
}
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. |
Vol. 2024
|
Bortz, Simon; Poggi, Bruno; Tapiola, Olli; Tolsa, Xavier The A∞ condition, ε-approximators, and Varopoulos extensions in uniform domains To appear in The Journal of Geometric Analysis, 2024. Abstract | Links @unpublished{nokey,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 = {2023-02-26},
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 = {unpublished}
}
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. |
Bortz, Simon; Hofmann, Steve; Luna, José Luis; Mayboroda, Svitlana; Poggi, Bruno Critical Perturbations for Second Order Elliptic Operators. Part II: Non-tangential maximal function estimates In: Archive for Rational Mechanics and Analysis, vol. 248, no. 31, 2024. Abstract | Links @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}
}
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. |
Poggi, Bruno Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields To appear in Advances in Mathematics, 2024. Abstract | Links @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}
}
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. |
Chamorro, Diego; Poggi, Bruno On an almost sharp Liouville type theorem for fractional Navier-Stokes equations To appear in Publicacions Matemàtiques, 2024. Abstract | Links @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}
}
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. |
