Searcharxiv⌕ Search

arXiv subjects

Olli Saari

Publications and source records attributed to Olli Saari.

At least 19 recordsLinked to original sources
arXiv2026

The parabolic Dini-$\beta$ condition and absolute continuity of surface and caloric measure

We show if $\partial \Omega$ is the graph of a parabolic Lipschitz function, then parabolic surface measure $\sigma$ of $\partial \Omega$ is absolutely continuous with respect to its caloric measure if and only if a (square) Dini-$\beta$ condition is satisfied. More specifically, the (square) Dini-$\beta$ condition is that \[\int_0^1 \hat{\beta}(X,t,r)^2 \frac{dr}{r} < \infty, \quad \text{$\sigma$-a.e. } (X,t) \in \partial \Omega.\] Here $\hat{\beta}$ is a parabolic version of the Jones ($L^2$) $\beta$-numbers. We show that these conditions are satisfied if and only if the graph is covered by a countable collection of {\it regular} Lipschitz graphs, that is, graphs with additional in-time regularity in the form of a half order time derivative in the parabolic BMO space. This supports the view that covering by {\it regular} Lipschitz graphs is the right notion for qualitative parabolic rectifiability in the context of parabolic PDEs. We also show that if \[\int_0^1 \hat{\beta}(X,t,r)^2 \frac{dr}{r} < \infty\] up to a set of caloric measure zero then the caloric measure is absolutely continuous with respect to surface measure.

math.AP↗
arXiv2025

A duality approach to gradient H\"older estimates for linear divergence form elliptic equations

We prove a sparse bound in the context of Schauder theory for divergence form elliptic partial differential equations. In addition, we show how an iteration argument inspired by sparse domination bounds can be used to deduce gradient reverse H\"older inequalities for equations with non-constant coefficients from the theory for constant coefficient equations. We deal with coefficient matrices whose entries are either H\"older continuous or just uniformly continuous, leading to different results. The purpose of the approach is to highlight the connection between Schauder theory and duality of local Hardy spaces and local H\"older spaces.

math.AP↗
arXiv2025

Domains with globally exponentially integrable parabolic forward-in-time BMO

We characterize those open sets of the space time in which parabolic forward-in-time BMO functions are in a certain forward-in-time exponential integrability class. The characterization holds under qualitative connectivity assumptions on the domain and is formulated in terms of a quantitative growth bound on a forward-in-time version of the classical quasihyperbolic distance.

math.CA↗
arXiv2024

A stability result for parabolic measures of operators with singular drifts

We study the operator \[ \partial_t - \text{div} A \nabla + B \cdot \nabla \] in parabolic upper-half-space, where $A$ is an elliptic matrix satisfying an oscillation condition and $B$ is a singular drift with a Carleson control. Our main result establishes quantitative $A_{\infty}$-estimates for the parabolic measure in terms of oscillation of $A$ and smallness of $B$. The proof relies on new estimates for parabolic Green functions that quantify their deviations from linear functions of the normal variable and on a novel, quantitative Carleson measure criterion for anisotropic $A_{\infty}$-weights.

math.AP↗
arXiv2024

Sparse gradient bounds for divergence form elliptic equations

We provide sparse estimates for gradients of solutions to divergence form elliptic partial differential equations in terms of the source data. We give a general result of Meyers (or Gehring) type, a result for linear equations with VMO coefficients and a result for linear equations with Dini continuous coefficients. In addition, we provide an abstract theorem conditional on PDE estimates available. The linear results have the full range of weighted estimates with Muckenhoupt weights as a consequence.

math.AP↗
arXiv2024

Uniform bounds for bilinear symbols with linear K-quasiconformally embedded singularity

We prove bounds in the strict local $L^{2}(\mathbb{R}^{d})$ range for trilinear Fourier multiplier forms with a $d$-dimensional singular subspace. Given a fixed parameter $K \ge 1$, we treat multipliers with non-degenerate singularity that are push-forwards by $K$-quasiconformal matrices of suitable symbols. As particular applications, our result recovers the uniform bounds for the one-dimensional bilinear Hilbert transforms in the strict local $L^{2}$ range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range.

math.CA↗
arXiv2023

Carleson Conditions for Weights: The quantitative small constant case

We investigate the small constant case of a characterization of $A_\infty$ weights due to Fefferman, Kenig and Pipher. In their work, Fefferman, Kenig and Pipher bound the logarithm of the $A_\infty$ constant by the Carleson norm of a measure built out of the heat extension, up to a multiplicative and additive constant (as well as the converse). We prove, qualitatively, that when one of these quantities is small so is the other. In fact, we show that these quantities are bounded by a constant times the square root of the other, provided at least one of them is sufficiently small. We also give an application of our result to the study of elliptic measures associated to elliptic operators with coefficients satisfying the ``Dahlberg-Kenig-Pipher" condition. We suspect that the square root dependence in the bound used in this application is sharp and give some justification for this in the last section.

math.CA↗
arXiv2022

Phase space localizing operators

We construct phase space localizing operators in all dimensions. These are frequency localized variants of the conditional expectation operator related to a dyadic stopping time. Our construction is an improvement over the so-called phase plane projections of Muscalu, Tao and the third author in one dimension. The motivation for such operators comes from time-frequency analysis. They are used in particular to prove uniform estimates for multilinear modulation invariant operators.

math.CA↗
arXiv2022

$L^p-L^q$ local smoothing estimates for the wave equation via $k$-broad Fourier restriction

We explore the connection between $k$-broad Fourier restriction estimates and sharp regularity $L^p-L^q$ local smoothing estimates for the solutions of the wave equation in $\mathbb{R}^{n}\times \mathbb{R}$ for all $n \geq 3$ via a Bourgain--Guth broad-narrow analysis. An interesting feature is that local smoothing estimates for $e^{i t \sqrt{-Δ}}$ are not invariant under Lorentz rescaling.

math.AP↗
arXiv2022

A reverse Hölder inequality for the gradient of solutions to Trudinger's equation

We provide a higher integrability result for the gradient of positive solutions to Trudinger's equation (also known as the doubly non-linear equation) for the range $p\in [2,\infty)$. The estimate is achieved by refining a construction of intrinsic cylinders from the vectorial setting by incorporating estimates only available in the scalar case.

math.AP↗
arXiv2021

Note on time-regularity for weak solutions to parabolic systems of p-Laplace type

We show that local weak solutions to parabolic systems of p-Laplace type are H{ö}lder continuous in time with values in a spatial Lebesgue space and H{ö}lder continuous on almost every time line. We provide an elementary and self-contained proof building on the local higher integrability result of Kinnunen and Lewis.

math.AP↗
arXiv2021

Construction of a right inverse for the divergence in non-cylindrical time dependent domains

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be H\"older regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed H\"older regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier--Stokes equations in time dependent domains.

math.AP↗
arXiv2020

Weak differentiability for fractional maximal functions of general $L^{p}$ functions on domains

Let $Ω\subset \mathbb{R}^{n}$ be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to $Ω$ maps $L^{p}(Ω) \to W^{1,p}(Ω)$ for all $p > 1$, when the smoothness index $α\geq 1$. In particular, the results are valid in the range $p \in (1, n/(n-1)]$ that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.

math.CA↗
arXiv2019

Sobolev contractivity of gradient flow maximal functions

We prove that the energy dissipation property of gradient flows extends to the semigroup maximal operators in various settings. In particular, we show that the vertical maximal function relative to the $p$-parabolic extension does not increase the $\dot{W}^{1,p}$ norm of $\dot{W}^{1,p}(\mathbb{R}^n) \cap L^{2}(\mathbb{R}^n)$ functions when $p > 2$. We also obtain analogous results in the setting of uniformly parabolic and elliptic equations with bounded, measurable, real and symmetric coefficients, where the solutions do not have a representation formula via a convolution.

math.CA↗
arXiv2018

Regularity of fractional maximal functions through Fourier multipliers

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function maps $L^{p}$ into a first order Sobolev space in dimensions $n \geq 5$.

math.CA↗
arXiv2018

The Brunn-Minkowski inequality and a Minkowski problem for $\mathcal{A}$-harmonic Green's function

In this article we study two classical problems in convex geometry associated to $\mathcal{A}$-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation. Let $p$ be fixed with $2\leq n\leq p<\infty$. For a convex compact set $E$ in $\mathbb{R}^{n}$, we define and then prove the existence and uniqueness of the so called $\mathcal{A}$-harmonic Green's function for the complement of $E$ with pole at infinity. We then define a quantity $\mbox{C}_{\mathcal{A}}(E)$ which can be seen as the behavior of this function near infinity. In the first part of this article, we prove that $\mbox{C}_{\mathcal{A}}(\cdot)$ satisfies the following Brunn-Minkowski type inequality \[ \left[\mbox{C}_\mathcal{A} ( λE_1 + (1-λ) E_2 )\right]^{\frac{1}{p-n}} \geq λ\, \left[\mbox{C}_\mathcal{A} ( E_1 )\right]^{\frac{1}{p-n}} + (1-λ) \left[\mbox{C}_\mathcal{A} (E_2 )\right]^{\frac{1}{p-n}} \] when $n<p<\infty$, $0 \leq λ\leq 1$, and $E_1, E_2$ are nonempty convex compact sets in $\mathbb{R}^{n}$. We also show that $\mbox{C}_\mathcal{A}(\cdot)$ satisfies a similar inequality when $p=n$. Moreover, if equality holds in the either of these inequalities for some $E_1$ and $E_2$ then under certain regularity and structural assumptions on $\mathcal{A}$ we show that these two sets are homothetic. In the second part of this article we study a Minkowski type problem for a measure associated to the $\mathcal{A}$-harmonic Green's function for the complement of a convex compact set $E$ when $n\leq p<\infty$. If $μ_E$ denotes this measure, then we show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem. We also show that this problem has a unique solution up to translation.

math.AP↗