arXiv subjects

Toni Heikkinen

Publications and source records attributed to Toni Heikkinen.

8 recordsLinked to original sources

arXiv2018

Generalized Lebesgue points for Hajł asz functions

Let $X$ be a quasi-Banach function space over a doubling metric measure space $\mathcal P$. Denote by $α_X$ the generalized upper Boyd index of $X$. We show that if $α_X<\infty$ and $X$ has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajł asz function $u\in\dot M^{s,X}$. Moreover, if $α_X<(Q+s)/Q$, then quasievery point is a Lebesgue point of $u$. As an application we obtain Lebesgue type theorems for Lorentz--Hajłasz, Orlicz--Hajłasz and variable exponent Hajłasz functions.

math.FA↗

arXiv2018

A median approach to differentiation bases

We study a version of the Lebesgue differentiation theorem in which the integral averages are replaced with medians over Busemann--Feller differentiation bases. Our main result gives several characterizations for the differentiation property in terms of the corresponding median maximal function. As an application, we study pointwise behaviour in Besov and Triebel--Lizorkin spaces, where functions are not necessarily locally integrable. Most of our results apply also for functions defined on metric measure spaces.

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arXiv2015

Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions

We show that, for $0<s<1$, $0<p<\infty$, $0<q<\infty$, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin functions can be approximated in the norm by discrete median convolutions. This allows us to show that, for these functions, the limit of medians, \[ \lim_{r\to 0}m_u^γ(B(x,r))=u^*(x), \] exists quasieverywhere and defines a quasicontinuous representative of $u$. The above limit exists quasieverywhere also for Hajłasz functions $u\in M^{s,p}$, $0<s\le 1$, $0<p<\infty$, but approximation of $u$ in $M^{s,p}$ by discrete (median) convolutions is not in general possible.

math.FA↗

arXiv2015

Approximation by Hölder functions in Besov and Triebel-Lizorkin spaces

In this paper, we show that Besov and Triebel-Lizorkin functions can be approximated by a Hölder continuous function both in the Lusin sense and in norm. The results are proven in metric measure spaces for Hajłasz-Besov and Hajłasz-Triebel-Lizorkin functions defined by a pointwise inequality. We also prove new inequalities for medians, including a Poincaré type inequality, which we use in the proof of the main result.

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arXiv2014

Measure density and extension of Besov and Triebel-Lizorkin functions

We show that a domain is an extension domain for a Hajłasz-Besov or for a Hajłasz-Triebel-Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case $0<p<1$. The necessity of the measure density condition is derived from embedding theorems; in the case of Hajłasz-Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Hajłasz-Besov spaces are intermediate spaces between $L^p$ and Hajłasz-Sobolev spaces. Our results are proved in the setting of a metric measure space, but most of them are new even in the Euclidean setting, for instance, we obtain a characterization of extension domains for classical Besov spaces $B^s_{p,q}$, $0<s<1$, $0<p<\infty$, $0<q\le\infty$, defined via the $L^p$-modulus of smoothness of a function.

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arXiv2013

Regularity of the local fractional maximal function

This paper studies smoothing properties of the local fractional maximal operator, which is defined in a proper subdomain of the Euclidean space. We prove new pointwise estimates for the weak gradient of the maximal function, which imply norm estimates in Sobolev spaces. An unexpected feature is that these estimates contain extra terms involving spherical and fractional maximal functions. Moreover, we construct several explicit examples which show that our results are essentially optimal. Extensions to metric measure spaces are also discussed.

math.FA↗

arXiv2013

Fractional maximal functions in metric measure spaces

We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

math.FA↗

arXiv2013

Smoothing properties of the discrete fractional maximal operator on Besov and Triebel--Lizorkin spaces

Motivated by the results of Korry and Kinnunen and Saksman, we study the behaviour of the discrete fractional maximal operator on fractional Hajlasz spaces, Hajlasz-Besov and Hajlasz-Triebel-Lizorkin spaces on metric measure spaces. We show that the discrete fractional maximal operator maps these spaces to the spaces of the same type with higher smoothness. Our results extend and unify aforementioned results. We present our results in general setting, but they are new already in the Euclidean case.

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