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Hannes Luiro

Publications and source records attributed to Hannes Luiro.

13 recordsLinked to original sources
arXiv2020

Gradient and Lipschitz estimates for tug-of-war type games

We define a random step size tug-of-war game, and show that the gradient of a value function exists almost everywhere. We also prove that the gradients of value functions are uniformly bounded and converge weakly to the gradient of the corresponding $p$-harmonic function. Moreover, we establish an improved Lipschitz estimate when boundary values are close to a plane. Such estimates are known to play a key role in higher regularity theory of partial differential equations. The proofs are based on cancellation and coupling methods as well as improved version of the cylinder walk argument.

math.AP↗
arXiv2017

The Variation of the Fractional Maximal Function of a Radial Function

In this paper we study the regularity of the non-centered fractional maximal operator $M_β$. As the main result, we prove that there exists $C(n,β)$ such that if $q=n/(n-β)$ and $f$ is a radial function, then $\|DM_βf\|_{L^{q}(\mathbb{R}^n)}\leq C(n,β)\|Df\|_{L^{1}(\mathbb{R}^n)}$. The corresponding result was previously known only if $n=1$ or $β=0$. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all $f\in W^{1,1}(\mathbb{R}^n)$.

math.CA↗
arXiv2017

The variation of the maximal function of a radial function

We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variation of the non-centered Hardy-Littlewood maximal function of a radial function is comparable to the variation of the function itself.

math.CA↗
arXiv2016

Regularity for nonlinear stochastic games

We establish regularity for functions satisfying a dynamic programming equation, which may arise for example from stochastic games or discretization schemes. Our results can also be utilized in obtaining regularity and existence results for the corresponding partial differential equations.

math.AP↗
arXiv2016

Gradient walk and $p$-harmonic functions

We consider a class of stochastic processes and establish its connection to $p$-harmonic functions. In particular, we obtain stochastic approximations that converge uniformly to a $p$-harmonic function, with an explicit convergence rate, and also obtain a precise diffusion representation in continuous time. The main difficulty is how to deal with the zero set of the gradient of the underlying function.

math.AP↗
arXiv2015

Beyond local maximal operators

We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces are derived as consequences of our main result. We also apply our boundedness results by studying both generalized neighbourhood capacities and the Lebesgue differentiation of fractional weighted Sobolev functions.

math.CA↗
arXiv2015

On the uniqueness of quasihyperbolic geodesics

In this work we solve a couple of well known open problems related to the quasihyperbolic metric. In the case of planar domains, our first main result states that quasihyperbolic geodesics are unique in simply connected domains. As the second main result, we prove that for an arbitrary plane domain the geodesics are unique for all pairs of points with quasihyperbolic distance less than $π$. This bound is sharp and improves the best known bound. Concerning the $n$-dimensional case, we prove the existence of a universal constant $c$ (independent of dimension) such that any quasihyperbolic ball with radius less than $c$ is convex.

math.MG↗
arXiv2014

Local maximal operators on fractional Sobolev spaces

In this note we establish the boundedness properties of local maximal operators $M_G$ on the fractional Sobolev spaces $W^{s,p}(G)$ whenever $G$ is an open set in $\mathbb{R}^n$, $0<s<1$ and $1<p<\infty$. As an application, we characterize the fractional $(s,p)$-Hardy inequality on a bounded open set $G$ by a Maz'ya-type testing condition localized to Whitney cubes.

math.CA↗
arXiv2013

On the differentiability of directionally differentiable functions and applications

In the first part of this paper we establish, in terms of so called k-tangential sets, a kind of optimal estimate for the size and structure of the set of non-differentiability of Lipshitz functions with one-sided directional derivatives. These results can be applied to many important special functions in analysis, like distance functions or different maximal functions. In the second part, having the results from the first part in our use, we focus more carefully on the differentiability properties of the classical Hardy-Littlewood maximal function. For example, we will show that if f is continuous and differentiable outside a countable union of tangential sets, then the same holds to the maximal function Mf as well (if Mf is not identically infinity). As an another example, our results also imply that if f is differentiable almost everywhere, then Mf is differentiable a.e.

math.CA↗
arXiv2012

On the existence and uniqueness of p-harmonious functions

We give a self-contained and short proof for the existence, uniqueness and measurability of so called $p$-harmonious functions. The proofs only use elementary analytic tools. As a consequence, we obtain existence, uniqueness and measurability of value functions for the tug-of-war game with noise.

math.AP↗
arXiv2012

On the Hamilton-Jacobi Equation and Infimal Convolution in the Framework of Sobolev-functions

We study the regularity properties of the Hamilton-Jacobi flow equation and infimal convolution in the case where initial datum function is continuous and lies in given Sobolev-space $W^{1,p}(\rn)$. We prove that under suitable assumptions it holds for solutions $w(x,t)$ that $D_xw(\cdot,t)\to Du(\cdot)$ in $L^p(\rn)$. Moreover, we construct examples showing that our results are essentially optimal.

math.AP↗
arXiv2012

Harnack's inequality for p-harmonic functions via stochastic games

We give a proof of Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Harnack's inequality for p-harmonic functions in the case $p>2$ that avoids classical techniques like Moser iteration, but instead relies on suitable choices of strategies for the stochastic tug-of-war game.

math.AP↗