We give a self-contained and short proof for the existence, uniqueness and measurability of so ca... more 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.
We study the size of the set of non-differentiability points for classical Hardy-Littlewood maxim... more We study the size of the set of non-differentiability points for classical Hardy-Littlewood maximal function Mf . Our first main result states that Mf is differentiable a.e. if function f is differentiable a.e. Another main theorem is that if f is differentiable (and Mf 6≡ ∞), then for every 0 < δ < 1 2 the set of non-differentiability points of Mf is included in a countable union of δ-porous sets. This also implies that the Hausdorff-dimension of the non-differentiability points is at most n − 1. The results can be also applied to other maximal operators as well as to other important special functions, like convex functions and distance functions.
Abstract In the first part of this paper we study the regularity properties of a wide class of ma... more Abstract In the first part of this paper we study the regularity properties of a wide class of maximal operators. These results are used to show that the spherical maximal operator is continuous W 1 , p ( R n ) ↦ W 1 , p ( R n ) , when p > n n − 1 . Other given applications include fractional maximal operators and maximal singular integrals. On the other hand, we show that the restricted Hardy–Littlewood maximal operator M λ , where the supremum is taken over the cubes with radii greater than λ > 0 , is bounded from L p ( R n ) to W 1 , p ( R n ) but discontinuous.
In this article, we study the regularity of the non-centered fractional maximal operator $M_{\bet... more In this article, we study the regularity of the non-centered fractional maximal operator $M_{\beta}$. As the main result, we prove that there exists $C(n,\beta)$ such that if $q=n/(n-\beta)$ and $f$ is radial function, then $\|DM_{\beta}f\|_{L^{q}({\mathbb{R}^n})}\leq C(n,\beta)\|Df\|_{L^{1}({\mathbb{R}^n})}$. The corresponding result was previously known only if $n=1$ or $\beta=0$. 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 $f\in W^{1,1}({\mathbb{R}^n})$.
Proceedings of the Edinburgh Mathematical Society, 2010
We establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is ... more We establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is an arbitrary subdomain and 1 < p < ∞. Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces Fps,q (Ω) for 1 < p,q < ∞ and 0 < s < 1.
We give a self-contained and short proof for the existence, uniqueness and measurability of so ca... more 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.
We study the size of the set of non-differentiability points for classical Hardy-Littlewood maxim... more We study the size of the set of non-differentiability points for classical Hardy-Littlewood maximal function Mf . Our first main result states that Mf is differentiable a.e. if function f is differentiable a.e. Another main theorem is that if f is differentiable (and Mf 6≡ ∞), then for every 0 < δ < 1 2 the set of non-differentiability points of Mf is included in a countable union of δ-porous sets. This also implies that the Hausdorff-dimension of the non-differentiability points is at most n − 1. The results can be also applied to other maximal operators as well as to other important special functions, like convex functions and distance functions.
Abstract In the first part of this paper we study the regularity properties of a wide class of ma... more Abstract In the first part of this paper we study the regularity properties of a wide class of maximal operators. These results are used to show that the spherical maximal operator is continuous W 1 , p ( R n ) ↦ W 1 , p ( R n ) , when p > n n − 1 . Other given applications include fractional maximal operators and maximal singular integrals. On the other hand, we show that the restricted Hardy–Littlewood maximal operator M λ , where the supremum is taken over the cubes with radii greater than λ > 0 , is bounded from L p ( R n ) to W 1 , p ( R n ) but discontinuous.
In this article, we study the regularity of the non-centered fractional maximal operator $M_{\bet... more In this article, we study the regularity of the non-centered fractional maximal operator $M_{\beta}$. As the main result, we prove that there exists $C(n,\beta)$ such that if $q=n/(n-\beta)$ and $f$ is radial function, then $\|DM_{\beta}f\|_{L^{q}({\mathbb{R}^n})}\leq C(n,\beta)\|Df\|_{L^{1}({\mathbb{R}^n})}$. The corresponding result was previously known only if $n=1$ or $\beta=0$. 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 $f\in W^{1,1}({\mathbb{R}^n})$.
Proceedings of the Edinburgh Mathematical Society, 2010
We establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is ... more We establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is an arbitrary subdomain and 1 < p < ∞. Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces Fps,q (Ω) for 1 < p,q < ∞ and 0 < s < 1.
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Papers by Hannes Luiro