# On the convergence sets of operator sequences
on spaces of homogeneous type

Grigori A. Karagulyan  Institute of Mathematics of NAS of RA, Marshal
Baghramian ave., 24/5, Yerevan, 0019, Armenia  Faculty of Mathematics and
Mechanics, Yerevan State University, Alex Manoogian, 1, 0025, Yerevan, Armenia
<EMAIL_ADDRESS>
###### Abstract.

We consider sequences of operators $U_{n}:L^{1}(X)\to M(X)$, where $X$ is a
space of homogeneous type. Under certain conditions on the operators $U_{n}$
we give a complete characterization of convergence (divergence) sets of
functional sequences $U_{n}(f)$, where $f\in L^{p}(X)$, $1\leq p\leq\infty$.
The results are applied to characterize convergence sets of some specific
operator sequences in classical analysis.

###### Key words and phrases:

Calderón-Zygmund operator, weighted inequalities, sparse operators, maximal
function, martingale transform, homogeneous spaces

###### 2010 Mathematics Subject Classification:

42C05, 42C10, 42C25, 42A55

The work was supported by the Science Committee of RA, in the frames of the
research project 21AG‐1A045

##  1\. Introduction

###  1.1. A historical overview

Let

$f=\\{f_{k}(x),\quad k=1,2,\ldots\\}$ (1.1)

be an infinite sequence of real functions. Denote by $C(f)\subset\mathbb{R}$
the convergence set of sequence (1.1), i.e. the set of points $x\in\mathbb{R}$
such that $\lim f_{n}(x)$ exists. A classical theorem of Hahn-Sierpinski
[Hahn, Sie] asserts that if functions (1.1) are continuous, then $C(f)$ is a
$F_{\sigma\delta}$-set, and conversely, every $F_{\sigma\delta}$-set is a
convergence set for a sequence of continuous functions. The first part of this
statement is immediate, since the convergence set of (1.1) may be written in
the form

$C(f)=\cap_{m=1}^{\infty}\cup_{n=1}^{\infty}\cap_{k=1}^{\infty}\\{x:\,|f_{n+k}(x)-f_{n}(x)\leq
1/m\\},$

and the latter is a $F_{\sigma\delta}$-set if the functions are continuous
(see Definition 1.1). The second part of the Hahn-Sierpinski theorem requires
certain construction of a sequence of continuous functions for which a given
$F_{\sigma\delta}$-set is a convergence set (see also [Hau], page 307). Note
that the complement of the convergence set $C(f)$ is the divergence set of
(1.1), which we denote by $D(f)$. So the $G_{\delta\sigma}$ sets completely
characterize the divergence sets of sequences of continuous functions. One can
also consider unbounded divergence set $UD(f)\subset D(f)$ of (1.1), which
consists of the points $x$, satisfying $\limsup|f_{n}(x)|=\infty$. It is also
known that for a set to be the $UB$-set for a sequence of continuous functions
it is necessary and sufficient to be a $G_{\delta}$-set (see [Hahn, Sie,
Hau]). Hence, we have certain topological characterization of the $C$, $D$ and
$UD$ sets in the class of sequences consisting of continuous functions.

Such characterization problems were commonly considered in many fields of
analysis (Fourier series, analytic functions on the unit ball, power series,
differentiation theory) and there are many published papers and open problems,
some of which will be considered in the last section. Earlier reviews of
problems, concerning the convergence or divergence sets of Fourier series in
trigonometric, Walsh and Haar orthogonal systems the reader can find in papers
[Wade, Uly1, Uly2]. We also refer the monograph [CoLo] related to boundary
exceptional sets of analytic functions.

Let us state few remarkable examples of characterization theorems below. The
first one provides a complete characterization of non-differentiability points
sets of continuous functions.

######  Theorem A (Zahorski, [Zah]).

A set $E\subset\mathbb{R}$ can be a set of non-differentiability points of a
real continuous function if and only if it is a union of a $G_{\delta}$-set
and a $G_{\delta\sigma}$-null-set.

In fact, the main ingredient of this theorem is the construction of a
continuous function, whose non-differentiability points set coincides with a
given set, which is a union of a $G_{\delta}$-set and a $G_{\delta\sigma}$-set
of measure zero. For the realization of this construction in [Zah] a subtle
technique was used (see also [Pir] and [Fow] for alternative simplified proofs
of Theorem A). The next well-known result is an extension of Kolmogorov’s
celebrated theorem of [Kol] on an example of an integrable function, whose
Fourier series is everywhere divergent.

######  Theorem B (Zeller, [Zel]).

Let $E$ be a $F_{\sigma}$-set in the circle $\mathbb{T}=\mathbb{R}/2\pi$. Then
there exists a function $f\in L^{1}(\mathbb{T})$ whose Fourier series
convergence at any point $x\in E$ and unboundedly diverges whenever
$x\in\mathbb{T}\setminus E$.

This theorem, combining with above mentioned general results, provides a
complete characterization of $UD$-sets in the class of Fourier series of
functions from $L^{1}(\mathbb{T})$. Later, an analogous result for the Walsh
series was proved in [Luk1, Luk2]. Note that it is an open problem to
characterize $D$ or $UD$ sets of Fourier series of $L^{p}(\mathbb{T})$,
$1<p\leq\infty$, or continuous functions (see [Uly1], page 107). Note that
according to Carleson-Hunt [Car, Hunt] results, Fourier series of functions
from $L^{p}(\mathbb{T})$, $p>1$ converge a.e., so the devergence sets of those
series are null-sets. Some partial results concerning the characterization of
convergence or divergence sets of Fourier series one can find in [Bug, Buz,
Buz1, Tai, Gog, Khe, KaKa, Kar3, Kar4, KarKar, Lun, KaGa]. Mostly, those
papers provide construction of different examples of Fourier series, diverging
on a given null-set and in this context the most general is Kahane-
Katznelson’s [KaKa] theorem on existence of a continuous function, whose
trigonometric Fourier series diverges on a given null-set (see also [Kat],
chap. II.3, for a detailed discussion of divergence sets). We also refer
papers [Kar3, Lun], which provide complete characterization of $C$, $D$ and
$UD$ sets of Fourier-Haar series. The author in [Kar2] obtained a general
characterization theorems for certain operator sequences, which cover many
results of above mentioned papers, providing complete characterization of
convergence sets of Fourier series in Haar, Franklin systems, as well as
positive order Cezaro means of trigonometric and Walsh Fourier series. We
remark here that Körner [Kor] constructed a $G_{\delta}$-set, which is not a
convergence set for any kind of trigonometric series. This example says that
for the ordinary partial sums of trigonometric or Walsh Fourier series a pure
topological characterization of convergence sets in some open problems may
fail (see discussion in [Uly1, Uly2]).

The next result provides a complete characterization of radial convergence
sets of bounded analytic functions on the unit disc. It was a solution of a
longstanding problem posed by Collingwood and Lohwater in [CoLo]. Namely,

######  Theorem C (Kolesnikov, [Kol]).

Let $D$ be the unit disc with the boundary circle $\Gamma$. For a set
$E\subset\Gamma$ to be the radial convergence set for a bounded analytic
function on $D$ it is necessary and sufficient to be a $F_{\sigma\delta}$ set
of full measure.

###  1.2. Main results of the paper

In the present paper we obtain complete characterization theorems for certain
operator sequences living in general spaces of homogeneous type. Those
generalize the results of [Kar1, Kar2], where the operators were considered on
the interval $[0,1]$. By a quasi-distance on a set $X$ we mean a non-negative
function $d(x,y)$ defined on $X\times X$ such that

  1. (1)

$d(x,y)=0$ if and only if $x=y$,

  2. (2)

$d(x,y)=d(y,x)$ for any $x,y\in X$,

  3. (3)

$d(x,y)\leq K(d(x,z)+d(z,y))$,

where $K>0$ is a constant. Such a distance defines a topology on $X$, for
which the balls $B(x,r)=\\{y\in X:\,d(y,x)<r\\}$ form a base. Namely, a set
$G\subset X$ in this topology is open if and only if for any point $x\in G$
there exists a ball $B(x,r)\subset G$. However, the balls themselves need not
to be open when $K>1$. Let $\mu$ be measure on a $\sigma$-algebra in $X$,
containing all Borel sets and balls, such that

$\mu(B(x,2r))\leq C\mu(B(x,r))<\infty.$ (1.2)

Such a combination $(X,d,\mu)$ is called a space of homogeneous type, which
definition is adapted from [CoWe] (see also [Christ]).

######  Definition 1.1.

Recall that a set in a topological space is of $G_{\delta}$ type provided it
is a countable intersection of open sets, and a set is $G_{\delta\sigma}$ if
it is a countable union of $G_{\delta}$-sets. The $F_{\sigma}$ sets are the
countable unions of closed sets, and $F_{\sigma\delta}$ are the countable
intersections of $F_{\sigma}$ sets.

######  Definition 1.2.

The distance of two sets $A$ and $B$ in a space of homogeneous type $X$ will
be denoted by

${\rm dist}(A,B)=\inf_{x\in A,\,y\in B}d(x,y)$ (1.3)

The notation $A\Subset B$ will stand for the relation ${\rm dist}(A,B^{c})>0$.
Note that $A\Subset B$ implies $B^{c}\Subset A^{c}$.

Let $L^{1}(X)$ denote the space of Lebesgue integrable functions on a space of
homogeneous type $(X,d,\mu)$ and suppose that $M(X)$ is the normed space of
bounded measurable functions on $X$ with the norm $\|f\|_{M}=\sup_{x\in
X}|f(x)|$. We consider sequences of linear operators

$U_{n}:L^{1}(X)\to M(X),\quad n=1,2,\ldots,$ (1.4)

which may have some of the following properties

######  U 1.

$\rho_{n}=\|U_{n}\|_{L^{1}\to M}<\infty$, $n=1,2,\ldots$,

######  U 2.

$\varrho=\sup_{n}\|U_{n}\|_{L^{\infty}\to M}<\infty$,

######  U 3.

if $f\in L^{1}(X)$ is constant on an open set $G\subset X$, then $U_{n}(x,f)$
uniformly converges over any set $E\Subset G$.

######  U 4.

if $G\subset X$ is an open set with $\mu(G)<\infty$, then
$U_{n}(x,\textbf{1}_{G})\to\textbf{1}_{G}(x)$ almost everywhere as
$n\to\infty$.

We will always suppose that $X$ is a space of homogeneous type unless other
assumptions are suggested. The main results of the paper read as follows.

######  Theorem 1.1.

If an operator sequence (1.4) satisfies U1-U4, then for any $G_{\delta\sigma}$
null-set $E\subset X$ and any $\varepsilon>0$, there exists a function $f\in
L^{\infty}(X)$ such that

1) $\mu({\rm supp\,}f)<\varepsilon$,

2) $U_{n}(x,f)$ diverges at any $x\in E$,

3) $U_{n}(x,f)\to f(x)$ if $x\in X\setminus E$.

######  Theorem 1.2.

Let an operator sequence (1.4) satisfy U1-U4, and for a function
$\psi:[0,\infty)\to[0,\infty)$ we have

$\lim_{t\to\infty}\psi(t)=\infty.$ (1.5)

Then for every $G_{\delta}$ null-set $E\subset X$ and $\varepsilon>0$, there
exists a measurable function $f$ such that

1) $\mu({\rm supp\,}f)<\varepsilon,\quad\int_{X}\psi(|f|)<\infty$,

2) $\limsup_{n\to\infty}|U_{n}(x,f)|=\infty$ for all $x\in E$,

3) $U_{n}(x,f)\to f(x)$ if $x\in X\setminus E$.

Observe that if besides of properties U1-U4 we also require the continuity of
$U_{n}(x,f)$ as functions on $x$, then Theorem 1.1 gives the complete
characterization of $C$ and $D$ sets of sequences $U_{n}(x,f)$, where $f\in
L^{p}(X)$ and $1\leq p\leq\infty$. Similarly, Theorem 1.2 provides complete
characterization of $UD$-sets of those sequences when $1\leq p<\infty$. One
can not claim for the function $f$ in Theorem 1.2 to be bounded like in
Theorem 1.1, since in that case condition U2 would imply the boundedness of
$U_{n}(x,f)$ for any $x\in X$.

In the case when $X$ coincides with $[0,1]$, Theorems 1.1 and 1.2 were proved
in [Kar2], and we essentially use some arguments of [Kar1, Kar2] in the
present work. Note that in a recent paper [KaGa] authors consider kernel
operator sequences

$U_{n}(x,f)=\int_{[0,1]^{d}}K_{n}(x,t)f(t)dt$

on the $d$-dimensional cube $[0,1]^{d}$ (or $\mathbb{R}^{d}$), where the
kernels $K_{n}(x,t)$ satisfy certain conditions in the spirit of an
approximation of identity. Those conditions provide properties U1-U3 for the
operators $U_{n}$, but instead of U4 the paper requires the convergence of
$U_{n}(x,f)$ at any continuity point $x$ of $f$. Under these requirements on
$U_{n}$ the main result of [KaGa] proves that any countable set is a
divergence set for a sequence $U_{n}(x,f)$ with a function $f\in M[0,1]^{d}$.

The approach provided in the proofs of Theorems 1.1 and 1.2 enables to obtain
also the following pure divergence result, where the operator sequences
satisfy only conditions U1 and U3. This result is a generalization of an
analogous theorem of [Kar1] for operators living on $[0,1]$.

######  Theorem 1.3.

If an operator sequence (1.4) satisfies U1 and U3, then for every null-set
$E\subset X$, there exists a set $G$ such that for the indicator function
$f=\textbf{1}_{G}$ we have a divergence of $U_{n}(x,f)$ at any point $x\in E$.
Moreover,

$\limsup_{n\to\infty}U_{n}(x,f)\geq 1,\quad\liminf_{n\to\infty}U_{n}(x,f)\leq
0\text{ for all }x\in E.$ (1.6)

######  Remark 1.1.

With a misery change in the proofs, Theorems 1.1-1.3 can be stated and proved
for operators $U_{r}$, where the parameter $r$ varies in an infinite partially
ordered set $R$. Namely, we can consider a partially ordered set $R$ with a
relation $<$, satisfying

1) there is a unique maximal element $r_{\infty}\in R$,

2) for any $r<r_{\infty}$ there exists an infinitely many elements $s$,
satisfying $s>r$.

Then given a process (or a sequence) $x_{r}$, $r\in R$, in $X$, the
convergence $\lim_{r\to r_{\infty}}x_{r}=a$ requires the following: for any
open set $G\supset a$ there exists a $r\in R$ such that for any $s$ with
$r<s<r_{\infty}$ we have $x_{r}\in G$.

We note that Theorem 1.3 as well as the result of paper [Kar1] generalize the
results of papers [Khe, Bug, Gog, Luk1, Luk2, Pro], where authors consider
different Fourier partial sums (trigonometric, Walsh, Haar) instead of
operators $U_{n}$. Examples of operators satisfying U1-U4 and corollaries of
Theorems 1.1 and 1.2 will be considered in the next section.

##  2\. Examples and applications

###  2.1. Approximation of the identity on metric measure spaces

Let $X$ be a space of homogeneous type. A sequence of kernel functions
$K_{n}(x,y)\in L^{\infty}(X\times X)\cap L^{1}(X\times X)$ is said to be an
approximation of the identity if it satisfies the conditions

$\displaystyle\int_{X}K_{n}(x,y)dy\rightrightarrows 1\text{ as }n\to\infty,$
(2.1)
$\displaystyle\int_{\\{y:\,d(x,y)>\delta\\}}K_{n}^{*}(x,y)dy\rightrightarrows
0\text{ as }n\to\infty,\text{ for every }\delta>0,$ (2.2)
$\displaystyle\sup_{n,x}\int_{X}|K_{n}^{*}(x,y)|dy<\infty.$ (2.3)

where the convergence in (2.1) and (2.2) holds uniformly with respect to $x$
over $X$, and

$K_{n}^{*}(x,y)=\sup_{t:\,d(x,t)\geq d(x,y)}|K_{n}(x,t)|.$

Consider the operator sequence

$U_{n}(x,f)=\int_{X}K_{n}(x,y)f(y)d\mu(y),$ (2.4)

where the kernels $K_{n}$ satisfy above conditions. It is known that if a
space of homogeneous type satisfies the following property that the space of
functions

$C_{K}(X)=\left\\{f\in C(X):\,{\rm supp\,}(f)\text{ is bounded and }\sup_{x\in
X}|f(x)|<\infty\right\\}$ (2.5)

is dense in $L^{1}(X)$, then operators (2.4) enjoy the following properties
(see for example [CoWe, Christ]):

1) If a function $f\in L^{1}(X)$ is uniformly continuous (in particular is
constant) on an open set $G\subset X$, then $U_{n}(x,f)\rightrightarrows f(x)$
uniformly on any subset $E\Subset G$.

2) If $f\in L^{1}(X)$, then $U_{n}(x,f)\to f(x)$ almost everywhere.

Obviously, such operators satisfy conditions U1-U4. Properties U1 and U2
immediately follow from the definition of approximation of the identity, while
properties U3 and U4 are weakened cases of above properties 1) and 2)
respectively. So we can state the following

######  Proposition 2.1.

If a space of homogeneous type $X$ is such that $C_{K}(X)$ is dense in
$L^{1}(X)$, then operators (2.4) obey the conditions of Theorems 1.1-1.3.

###  2.2. Walsh functions

We consider Walsh orthonormal system, defined on the set of all sequences

$(x_{0},x_{1},x_{2},\ldots),\quad x_{j}=0\text{ or }1,\quad j=0,1,2,\ldots.$
(2.6)

(see [GES], chap. 1). Every such a sequence generates a series

$\sum_{j=0}^{\infty}x_{j}\cdot 2^{-j-1},$ (2.7)

which is the dyadic decomposition of some number from $[0,1]$. We note that
this correspondence is surjective, but it is not injective, since any dyadic
rationals of $[0,1]$ have two decompositions (2.7), infinite and finite. For a
geometric understanding of the set of sequences (2.6) introduce the extended
interval $[0,1]^{*}$, where each dyadic rational number $x$ is doubled,
generating the left point $x-$ corresponding to the finite dyadic
decomposition and the right point $x+$ that corresponds to the infinite
decomposition of $x$. Writing $x+$ or $x-$ for a dyadic irrational number $x$,
we mean just the point $x\in[0,1]^{*}$. Define a generalized dyadic interval
by

$[a,b]=\\{x+,x-\in[0,1]^{*}:\,a<x<b\\}\cup\\{a+,b-\\},$ (2.8)

where

$a=\frac{j-1}{2^{k}},\quad b=\frac{j}{2^{k}},\quad 1\leq j\leq 2^{k},\quad
k=0,1,\ldots.$ (2.9)

One can easily check that the dyadic distance

$d(x,y)=\inf\\{b-a:\,[a,b]\text{ is a generalized dyadic
interval},\,x,y\in[a,b]\\},$ (2.10)

between two points $x,y\in[0,1]^{*}$ defines a quasi-distance on $[0,1]^{*}$.
Then any set $E^{*}\subset[0,1]^{*}$ has its counterpart $E\subset[0,1]$,
obtained by an identifying all the pairs of points $x+,x-\in[0,1]^{*}$ in one.
So we define the measure of $E^{*}$ to be the Lebesgue measure of the set $E$.
Hence $[0,1]^{*}$ equipped with such a measure becomes a space of homogeneous
type. Moreover, one can easily check that $[0,1]^{*}$ is a compact space (we
will not need it).

To define the Walsh functions, recall the group operation on $[0,1]^{*}$,
defining the sum of two sequences of type (2.6) $\\{x_{j}\\}_{j=0}^{\infty}$
and $\\{y_{j}\\}_{j=0}^{\infty}$ to be the sequence
$\\{z_{j}\\}_{j=0}^{\infty}$, where

$z_{j}=\left\\{\begin{array}[]{rcl}1&\hbox{ if }&x_{j}+y_{j}=1,\\\ 0&\hbox{ if
}&x_{j}+y_{j}=0\hbox{ or }2.\end{array}\right.$

Now we can define the Walsh system of functions
$\\{w_{n}(x)\\}_{n=0}^{\infty}$ on $[0,1]^{*}$. We set $w_{0}(x)\equiv 1$. For
$n\geq 1$ we write its dyadic representation
$n=\sum_{j=0}^{k}\varepsilon_{j}2^{j},$ then define

$w_{n}(x)=(-1)^{\sum_{j=1}^{k}\varepsilon_{j}x_{j}}\text{ for
}x=(x_{0},x_{1},\ldots)\in[0,1]^{*}.$ (2.11)

One can see that Walsh functions are continuous in the topology of
$[0,1]^{*}$. Besides, it is well-known that $(C,\alpha)$-means
$\sigma_{n}^{\alpha}(x,f)$ $\alpha>0$ of the partial sums of Walsh-Fourier
series satisfy properties U1-U4 (see [GES], chap. 4). So we can state the
following results.

######  Corollary 2.1.

For a set $E\subset[0,1]^{*}$ to be the divergence (unbounded divergence) set
of $\sigma_{n}^{\alpha}(x,f)$ for a function $L^{p}([0,1]^{*})$, $1\leq
p\leq\infty$ it is necessary and sufficient to be a $G_{\delta\sigma}$-set
($G_{\delta}$-set) in the topology of $[0,1]^{*}$.

######  Remark 2.1.

The $(C,\alpha)$-means $\sigma_{n}^{\alpha}$ of Walsh-Fourier series can be
considered as kernel operators. Nevertheless, it is known (see [GES]) that the
kernels of $\sigma_{n}^{\alpha}$ do not form an approximation of the identity.

###  2.3. Splines

Recall the definition of splines on an interval $[a,b]$. For a knot-collection
$\Delta=\\{t_{j}\\}_{j=1}^{n+k}\subset[a,b]$ such that

$\displaystyle t_{i}\leq t_{i+1},\quad t_{i}\leq t_{i+k},$ (2.12)
$\displaystyle t_{1}=\ldots=t_{k}=a,\quad t_{n+1}=\ldots=t_{n+k}=b$ (2.13)

let denote by $\mathcal{S}_{k}(\Delta)$ the space of $k$-order splines with
the notes $\Delta$. Those are the functions, which are polynomials of degree
$\leq k-1$ on each interval $[t_{j},t_{j+1}]$ and has a $k-1-m_{j}$ continuous
derivatives at any node $t_{j}\in\Delta$ of a multiplicity $m_{i}$. Let
$P_{\Delta}$ be the orthoprojection operator onto $\mathcal{S}_{k}(\Delta)$.
Denote $|\Delta|=\max_{j}(t_{j+1}-t_{j})$. Given sequence of fixed $k$-order
knot-collections $\Delta_{n}$ with $|\Delta_{n}|\to 0$ consider the operator
sequence $P_{\Delta_{n}}$. It is known such an operator sequence satisfy
properties U1-U4. Moreover, Shadrin [Sha] solving de Boor’s conjecture proved
the uniformly boundedness of these operators on $C[a,b]$ that implies
$P_{\Delta_{n}}(f)\rightrightarrows f$ whenever $f\in C[a,b]$. Then
Passenbrunner and Shadrin in [PaSh] proved $P_{\Delta_{n}}(f)\to f$ a.e. for
any $f\in L^{1}[a,b]$, where it was also proved that the kernels of operators
$P_{\Delta_{n}}$ form an approximation of the identity. Thus we can state the
following.

######  Corollary 2.2.

Let $\Delta_{n}$ be a sequence of knot-collections in $[a,b]$ such that
$|\Delta_{n}|\to 0$. Then for a set $E\subset[a,b]$ to be the divergence
(unbounded divergence) set of $P_{\Delta_{n}}(f)$ for a function $L^{p}[a,b]$,
$1\leq p\leq\infty$ it is necessary and sufficient to be a
$G_{\delta\sigma}$-set ($G_{\delta}$-set) in the standard topology of $[a,b]$.

###  2.4. More specific examples

Let us also discuss some more specific examples of operator sequences that
obey the conditions of Theorem 1.1 and 1.2, and those can be deduced from one
of above considered general examples.

  1. (1)

Partial sums of Fourier series in Haar and Franklin systems with general nodes
are spline operator sequences, so those obey the conditions of Corollary 2.2.
For the Haar series the characterization of $C$, $D$ and $UD$ sets first were
given in [Lun] (for $UD$ sets) and [Kar3] (for $D$ sets)

  2. (2)

The characterization of convergence and divergence sets of $(C,\alpha)$-means
of trigonometric Fourier series with a parameter $\alpha>0$ was given by
Zahorski in [Zah1], where the same problem was considered also for Poisson
integrals on the unit disc. We just note that both operators are
approximations of the identity on the unit circle.

  3. (3)

Let $(X,d,\mu)$ be a space of homogeneous type such that $C_{K}(X)$ is dense
in $L^{1}(X)$. Consider a sequence of measurable sets $G_{n}\subset X$, which
are equivalent to balls $B(0,r_{n})$, i.e.

$G_{n}\subset B(0,r_{n}),\quad\mu(G_{n})>c\mu(B(0,r_{n}))$ (2.14)

where $c>0$ is a constant. Clearly the operator sequence

$U_{n}(x,f)=\frac{1}{\mu(G_{n})}\int_{G_{n}}f(t)d\mu(t)$ (2.15)

is a particular case of operators (2.4). This sequence in particular is
interesting when $X$ coincides with $\mathbb{R}^{n}$.

  4. (4)

Dyachkov in [Dya] considered the $C$-$D$ sets characterization problem for
operators

$U_{\varepsilon}(x,f)=\int_{\mathbb{R}^{d}}f(x-t)\phi_{\varepsilon}(t)dt\hbox{
as }\varepsilon\searrow 0,$

for an approximation of the identity kernels
$\phi_{\varepsilon}(t)=\varepsilon^{-d}\phi(t/\varepsilon)$, where $\phi\in
L^{1}(\mathbb{R}^{d})$.

  5. (5)

The main part of Zahorski’s theorem of [Zah] (Theorem A) is a construction of
a continuous function whose non-differentiability set is a given set
$G=G_{1}\cup G_{2}$, where $G_{1}$ is $G_{\delta}$ set and $G_{2}$ is a
$G_{\delta\sigma}$ null set. Without loss of generality one can suppose that
$G_{1}\cap G_{2}=\varnothing$. So the construction can be split into
contractions of two different functions $f_{1}$ and $f_{2}$, which non-
differentiability sets are $G_{1}$ and $G_{2}$ respectively. Then the desired
functions is $f=f_{1}+f_{2}$. The function $f_{2}$ can be seeked in the form

$f_{2}(x)=\int_{-\infty}^{x}g(t)dt$ (2.16)

where $g\in L^{\infty}(\mathbb{R})$. Then the differentiability points set of
the function $f_{2}$ coincides with the set of points $x\in\mathbb{R}$ such
that the limit

$\lim_{|I|\to 0,\,I\ni x}\frac{1}{|I|}\int_{I}g(t)dt$ (2.17)

exists, where $I$ are closed intervals containing the point $x$. The existence
of a function $g\in L^{\infty}(\mathbb{R})$ for which the limit (2.17) exists
only at the points of a given $G_{\delta\sigma}$ null set, is a part of
general Proposition 2.1.

##  3\. Auxiliary lemmas

It is known from [MaSe] that for any quasi-distance $d$ there exists an
alternative quasi-distance $d^{\prime}$, which is equivalent to $d$, i.e.
$C_{1}d(x,y)\leq d^{\prime}(x,y)\leq C_{2}d(x,y)$, and satisfies a Lipschits
type condition

$|d^{\prime}(x,z)-d^{\prime}(y,z)|\leq
L(d^{\prime}(x,y))^{\alpha}(\max\\{d^{\prime}(x,z),d^{\prime}(y,z)\\})^{1-\alpha},$
(3.1)

where $L>0$ and $0<\alpha<1$ are some constants. One can see that $d^{\prime}$
induces the same topology as $d$ does and in view of (3.1) $d^{\prime}$-balls
become open sets there. Using this we can prove the following.

######  Lemma 3.1.

Any open set $G$ in a space of homogeneous type permits a representation
$G=\cup_{k=1}^{\infty}F_{k}$, where each $F_{k}$ is a closed set and
$F_{k}\Subset G$.

###### Proof.

Let ${\rm dist}^{\prime}$ denote the distance corresponding to the quasi-
distance $d^{\prime}$ from (3.1). Define

$F_{k}=\\{x\in G:\,{\rm dist}^{\prime}(x,G^{c})\geq 1/k\\},\quad
k=1,2,\ldots.$

It is clear that $F_{k}\Subset G$. It remains to show that each $F_{k}$ is
closed or equivalently $F_{k}^{c}$ is open. For any $x\in F_{k}^{c}$ we have
${\rm dist}^{\prime}(x,G^{c})<1/k$. Then for small enough $r>0$ the condition
$d^{\prime}(y,x)<r$ implies ${\rm dist}^{\prime}(y,G^{c})<1/k$, and it follows
from (3.1). Thus we obtain $B^{\prime}(x,r)\subset F_{k}^{c}$, i.e.
$F_{k}^{c}$ is open.

∎

The following lemma is a standard property of a topology induced by a quasi-
distance.

######  Lemma 3.2.

If sets $A$ and $B$ in a quasi-metric space $(X,d)$ satisfy ${\rm
dist}(A,B)>0$, then there are disjoint open sets $U$ and $V$ such that
$A\Subset U$ and $B\Subset V$.

###### Proof.

Let $d^{\prime}$ be a Lipschits distance satisfying (3.1). Equivalency of
distances $d$ and $d^{\prime}$ implies ${\rm dist}^{\prime}(A,B)>0$, besides
we know that each $d^{\prime}$-ball is an open set. Define an open set
$U\supset A$ to be the union of all the balls $B^{\prime}(x,r)$ with $x\in A$
and $r={\rm dist}^{\prime}(x,B)/(2K)>0$, where $K$ is the triangle inequality
constant of the distance $d^{\prime}$. Similarly defining the open
neighborhood $V\supset B$, we claim that $U\cap V=\varnothing$. Suppose to the
contrary that there is a point $z\in U\cap V$. Then we find $x\in A$ and $y\in
B$ such that

$d^{\prime}(x,z)<{\rm dist}^{\prime}(x,B)/(2K),\quad d^{\prime}(y,z)<{\rm
dist}^{\prime}(y,A)/(2K).$

This implies

$d^{\prime}(x,y)\leq K(d^{\prime}(x,z)+d^{\prime}(y,z))<\max\\{{\rm
dist}^{\prime}(x,B),{\rm dist}^{\prime}(y,A)\\},$

which is a contradiction and so $U\cap V=\varnothing$. It remains just observe
that $A\Subset U$ and $A\Subset V$, which hold with respect to both distances
$d$ and $d^{\prime}$. ∎

It is known in the geometric measure theory (see for example [Sim], Theorem
1.11) that if in a topological measurable space $(X,\mu)$ every open set is of
$F_{\sigma}$ and the measure $\mu$ is open $\sigma$-finite (i.e. $X$ is a
countable union of finite measure open sets), then

$\mu(E)=\inf_{G\text{ open},\,G\supset E}\mu(G),$ (3.2)

for any Borel set $E$. Any space of homogeneous type $X$ satisfies such
properties, since every open set is of $F_{\sigma}$ by Lemma 3.1, and $X$ can
be written as a countable union $d^{\prime}$-balls, which are open and have
finite measure. Hence we can state the following.

######  Lemma 3.3.

Any space of homogeneous type $X$ enjoys the property (3.2).

######  Definition 3.1.

A sequence $\Omega=\\{\omega_{k}:\,k=1,2,\ldots\\}$ of Borel sets in a
topological space $X$ is said to be

1) a partition of an open set $B\subset X$ if

$B=\cup_{k}\omega_{k},\quad\omega_{k}\cap\omega_{k^{\prime}}=\varnothing\text{
if }k\neq k^{\prime},$ (3.3)

2) locally-finite if there are open sets $V_{k}\supset\omega_{k}$ such that
each of them has only finite number of intersection with others, i.e.

$\\#\\{j\in\mathbb{N}:\,V_{j}\cap V_{k}\neq\varnothing\\}<\infty\text{ for any
}k=1,2,\ldots,$ (3.4)

3) a regular partition for an open set $B$ if it is a partition of $B$ and
there are open sets $V_{j}$, satisfying (3.4) (so it is locally-finite) as
well as,

$\omega_{k}\Subset V_{k}\subset B,\quad k\geq 1.$ (3.5)

######  Lemma 3.4.

Let $\omega_{k}$, $k=1,2,\ldots$, be a locally-finite family of measurable
sets in a topological measurable space $X$. Then for any measurable set
$E\subset\cup_{k}\omega_{k}$ and numbers $\varepsilon_{k}>0$ there exists an
open set $G\supset E$ such that

$\mu((G\setminus E)\cap\omega_{k})<\varepsilon_{k},\quad k=1,2,\ldots.$

###### Proof.

Let $V_{k}\supset\omega_{k}$, $k=1,2,\ldots$, be open sets such that each
$V_{k}$ has intersection only with finite number of elements $V_{j}$ and we
denote this number by $l_{k}$. One can check that

$\varepsilon^{\prime}_{k}=\min_{j:\,V_{j}\cap
V_{k}\neq\varnothing}\varepsilon_{j}/l_{j}>0\text{ for all }k=1,2,\ldots,$
(3.6)

because only finite number of $j$’s in this minimum satisfy the relation
$V_{j}\cap V_{k}\neq\varnothing$. Thus, applying Lemma 3.3, we may define open
sets $G_{k}$, $k=1,2,\ldots$, such that

$\displaystyle E\cap\omega_{k}\subset G_{k}\subset
V_{k},\quad\mu(G_{k}\setminus E)<\varepsilon^{\prime}_{k}.$ (3.7)

Thus we have $G=\cup_{k}G_{k}\supset E$ and

$\displaystyle\mu((G\setminus E)\cap\omega_{k})$
$\displaystyle\leq\sum_{j:\,V_{j}\cap\omega_{k}\neq\varnothing}\mu((G_{j}\setminus
E)\cap\omega_{k})$ (3.8) $\displaystyle\leq\sum_{j:\,V_{j}\cap
V_{k}\neq\varnothing}\mu(G_{j}\setminus E)<\sum_{j:\,V_{j}\cap
V_{k}\neq\varnothing}\varepsilon^{\prime}_{j}$ (3.9)
$\displaystyle<\sum_{j:\,V_{j}\cap
V_{k}\neq\varnothing}\varepsilon_{k}/l_{k}=\varepsilon_{k}.$ (3.10)

This completes the proof. ∎

######  Lemma 3.5.

Any open set $B\subset X$ in a space of homogeneous type $X$ enjoys a regular
partition.

###### Proof.

For any open set $B\subset X$ we can find closed sets $F_{k}$, $k=1,2,\ldots$,
such that

$B=\cup_{k}F_{k},\quad F_{k}\Subset{\rm Int}(F_{k+1}),$ (3.11)

where ${\rm Int}(E)$ denotes the interior of a set $E\subset X$. Indeed, first
we write $B=\cup_{k}A_{k}$, where $A_{k}\Subset B$ are closed sets (see Lemma
3.1). Since ${\rm dist}(A_{1},B^{c})>0$, by Lemma 3.2 the sets $A_{1}$ and
$B^{c}$ have disjoint open neighborhoods $D_{1}\Supset A_{1}$ and
$G_{1}\Supset B^{c}$. So we will have $A_{1}\Subset D_{1}$ and
$\overline{D_{1}}\subset(G_{1})^{c}\Subset B$. Then we do the same with the
closed sets $A_{2}\cup\overline{D_{1}}\Subset B$ and $B^{c}$ and we get an
open set $D_{2}\Supset A_{2}\cup\overline{D_{1}}$ such that
$\overline{D_{2}}\Subset B$. Continuing this procedure to infinity, finally we
get a sequence of open sets $D_{k}$ such that $D_{k}\Supset
A_{k}\cup\overline{D_{k-1}}$ and $\overline{V_{k}}\subset B$. Let us see that
$F_{1}=A_{1}$, $F_{k}=A_{k}\cup\overline{D_{k-1}}$, $k\geq 2$, are closed
sets, satisfying (3.11). Clearly we have $B=\cup_{k}F_{k}$, besides
$F_{k}\Subset D_{k}\subset F_{k+1}$. Since $D_{k}$ is open, we get
$F_{k}\Subset{\rm Int}(F_{k+1})$.

Having (3.11), we define Borel sets $\omega_{1}=F_{1}$,
$\omega_{k}=F_{k}\setminus F_{k-1}$, $k\geq 2$, which clearly satisfy the
partition condition (3.3). One can also see that the open sets $V_{1}={\rm
Int}(F_{2}),V_{2}={\rm Int}(F_{3})$, $V_{k}={\rm Int}(F_{k+1})\setminus
F_{k-2}$, $k>2$, satisfy (3.4). Besides by (3.11) we have

$\omega_{k}\subset F_{k}\setminus\Subset{\rm Int}(F_{k+1})\subset{\rm
Int}(F_{k-1})\subset V_{k},$

which implies (3.5).

∎

The notation $f_{n}(x)\rightrightarrows f(x)$ will stand for the uniformly
convergence of sequences of functions on a set.

######  Lemma 3.6.

Let $(X,d,\mu)$ be a space of homogeneous type, and let an operator sequence
(1.4) satisfy conditions U1 and U3. If $G\subset X$ is an open set of finite
measure and $F\Subset G$ is measurable, then

$d_{G}(F)=\sup_{n\in\mathbb{N}}\sup_{x\in G^{c}}\,\,\sup_{\|f\|_{1}\leq
1}|U_{n}(x,f\cdot\textbf{1}_{F})|<\infty.$ (3.12)

###### Proof.

First we note that by property U3 we have

$U_{n}(x,f\cdot\textbf{1}_{F})\rightrightarrows 0\hbox{ on }G^{c},\hbox{ as
}n\to\infty,$ (3.13)

for any $f\in L^{1}(X)$, since $f\cdot\textbf{1}_{F}\in L^{1}(X)$ is
identically zero on the on the open set $F^{c}$ and $G^{c}\Subset F^{c}$.
Suppose to the contrary that $d_{G}(F)=\infty$. The boundedness property U1 of
operators (1.4) implies

$\displaystyle d_{G,m}(F)=\max_{1\leq n\leq m}\sup_{x\in G^{c}}$
$\displaystyle\sup_{\|f\|_{1}\leq 1}|U_{n}(x,f\cdot\textbf{1}_{F})|$ (3.14)
$\displaystyle\leq\max_{1\leq n\leq m}\rho_{n}<\infty,\text{ for any
}m\in\mathbb{N},$ (3.15)

and by the contrary assumption we have

$d_{G,m}(F)\to\infty,\hbox{ as }m\to\infty.$ (3.16)

Applying induction, one can find integers $n_{k}\in\mathbb{N}$, points
$x_{k}\in G^{c}$ and functions $f_{k}\in L^{1}(X)$, $k=1,2,\ldots$, such that

$\displaystyle\|f_{k}\|_{1}\leq 1,\quad{\rm supp\,}f_{k}\subset F,\,k\geq 1,$
(3.17) $\displaystyle|U_{n_{k}}(x_{k},f_{k})|>k^{3}\cdot(1+\max_{1\leq
i<k}\rho_{n_{i}}),\,k\geq 1$ (3.18) $\displaystyle\sup_{1\leq
i<k}|U_{n_{k}}(x,f_{i})|<1,\quad x\in G^{c},\,k>1.$ (3.19)

Indeed, using (3.16), first we define $f_{1}(x)$, $n_{1}\in\mathbb{N}$ and
$x_{1}\in G^{c}$ satisfying

$\displaystyle\|f_{1}\|_{1}\leq 1,\quad{\rm supp\,}f_{1}\subset F,$
$\displaystyle|U_{n_{1}}(x_{1},f_{1})|>1,$

which is the base of induction. Then suppose that we have already defined
$f_{k}\in L^{1}(X)$, $n_{k}\in\mathbb{N}$ and $x_{k}\in G^{c}$ satisfying
conditions (3.17)-(3.19) for $k=1,2,\ldots,p$. Once again using (3.16), we
find $f_{p+1}(x)$, $n_{p+1}\in\mathbb{N}$ and $x_{p+1}\in G^{c}$, satisfying
(3.17) and (3.18) for $k=p+1$. Moreover, by (3.15) the number $n_{p+1}$ can be
chosen enough bigger and so, using (3.13), we can also ensure (3.19) for
$k=p+1$. This completes the induction process. Let us also add the bound

$|U_{n_{k}}(x,f_{i})|\leq\rho_{n_{k}},\quad x\in X,\quad i=1,2,\ldots,$ (3.20)

which immediately follows from U1 and the bound $\|f_{i}\|_{1}\leq 1$. Now
consider the function

$f(x)=\sum_{k=1}^{\infty}\alpha_{k}f_{k}(x),\quad\alpha_{k}=\frac{1}{k^{2}\cdot(1+\max_{1\leq
i<k}\rho_{n_{i}})}.$ (3.21)

Clearly $f\in L^{1}(X)$ and moreover, ${\rm supp\,}f\subset F$. Using
(3.17),(3.18),(3.19),(3.20) and (3.21), we obtain

$\begin{split}|U_{n_{k}}(x_{k},f)|\geq\\\
&\geq\alpha_{k}|U_{n_{k}}(x_{k},f_{k})|-\sum_{i=1}^{k-1}\alpha_{i}|U_{n_{k}}(x_{k},f_{i})|-\sum_{i=k+1}^{\infty}\alpha_{i}|U_{n_{k}}(x_{k},f_{i})|\\\
&\geq
k-\sum_{i=1}^{k-1}\frac{1}{i^{2}}-\sum_{i=k+1}^{\infty}\frac{1}{i^{2}}\geq
k-2.\end{split}$

This gives a contradiction, since ${\rm supp\,}f\subset F$ and
$U_{n}(x,f)\rightrightarrows 0$ on $G^{c}$ according to (3.13). ∎

For a sequence of pairwise disjoint measurable sets $\Omega=\\{\omega_{k}\\}$
in a space of homogeneous type $X$ and a mapping $\nu:\Omega\to\mathbb{N}$ we
denote

$U_{(\Omega,\nu)}(x,f)=\sum_{k\geq
1}U_{\nu(\omega_{k})}(x,f)\cdot\textbf{1}_{\omega_{k}}(x).$ (3.22)

######  Lemma 3.7.

Let an operator sequence $U_{n}$ satisfy U3 and functions $f_{k}\in L^{1}(X)$,
$k=1,2,\ldots,n$ be constant on an open set $G\subset X$, which has a
partition $\Omega=\\{\omega_{k}\\}$ such that $\omega_{k}\Subset G$. Then for
any $\varepsilon>0$ and $R>0$ there exists a mapping $\nu:\Omega\to\mathbb{N}$
such that

$\displaystyle\min_{\omega\in\Omega}\nu(\omega)>R,$ (3.23)
$\displaystyle|U_{(\Omega,\nu)}(x,f_{i})-f_{i}(x)|<\varepsilon,\quad x\in
G,\quad i=1,2,\ldots,n.$ (3.24)

###### Proof.

From condition U3 it follows that $U_{n}(x,f_{i})\rightrightarrows f_{i}(x)$
on each set $\omega_{k}$, for any $i=1,2,\ldots n$. Thus for every
$\omega_{k}$ we may find an integer $\nu(\omega_{k})\in\mathbb{N}$ such that
$\nu(\omega_{k})>R$ and

$|U_{\nu(\omega_{k})}(x,f_{i})-f_{i}(x)|<\varepsilon,\quad
x\in\omega_{k},\quad i=1,2,\ldots,n.$

Thus we get (3.23) and (3.24). ∎

######  Lemma 3.8.

Let an operator sequence $U_{n}$ satisfy U1 and U3, $G\subset X$ be an open
set with a regular partition $\Omega=\\{\omega_{k}\\}$ and let
$\nu:\Omega\to\mathbb{N}$ be a mapping. Then if $C\subset G$ is a measurable
set and a function $f\in L^{\infty}(X)$ satisfies

$\|f\|_{\infty}\leq 1,\quad{\rm supp\,}(f)\subset C,$ (3.25)

then it holds the inequality

$|U_{(\Omega,\nu)}(x,f)|\leq\sum_{j}c(\omega_{j})|C\cap\omega_{j}|,\quad x\in
X,$ (3.26)

where $c(\omega_{j})>0$ are constants independent of $C$ and the function $f$.
In fact, those depend only on the operators $U_{n}$, the open set $G$ and its
partition $\omega_{k}$ (see (3.34)).

###### Proof.

For $x\not\in G$ (3.26) trivially holds, since we have
$U_{(\Omega,\nu)}(x,f)=0$ (see (3.22)). By the definition of regular partition
we may fix open sets $V_{k}\supset\omega_{k}$, $k=1,2,\ldots$, satisfying
(3.3). For $x\in G$ we can write

$x\in\omega_{k}\text{ for a unique }k\geq 1,$ (3.27)

and then

$\displaystyle|U_{(\Omega,\nu)}(x,f)|\leq\sum_{j:\,V_{j}\cap
V_{k}=\varnothing}$
$\displaystyle|U_{\nu(\omega_{k})}(x,f\cdot\textbf{1}_{\omega_{j}})|$ (3.28)
$\displaystyle+\sum_{j:\,V_{j}\cap
V_{k}\neq\varnothing}|U_{\nu(\omega_{k})}(x,f\cdot\textbf{1}_{\omega_{j}})|=S_{1}(x)+S_{2}(x).$
(3.29)

The condition $V_{j}\cap V_{k}=\varnothing$ implies $x\in\omega_{k}\Subset
V_{k}\subset(V_{j})^{c}$. So applying Lemma 3.6 and relations (3.25), for the
first sum we obtain

$\displaystyle S_{1}(x)\leq\sum_{j:\,V_{j}\cap V_{k}=\varnothing}$
$\displaystyle\|f\cdot\textbf{1}_{\omega_{j}}\|_{1}d_{V_{j}}(\omega_{j})$
(3.30) $\displaystyle\leq\sum_{j:\,V_{j}\cap
V_{k}=\varnothing}d_{V_{j}}(\omega_{j})|C\cap\omega_{j}|.$ (3.31)

Applying U1 and (3.25), we have

$\displaystyle S_{2}(x)$ $\displaystyle\leq\sum_{j:,V_{j}\cap
V_{k}\neq\varnothing}\rho_{\nu(\omega_{k})}|C\cap\omega_{j}|$ (3.32)
$\displaystyle\leq\sum_{j:,V_{j}\cap
V_{k}\neq\varnothing}\max\\{\rho_{\nu(\omega_{i})}:\,V_{i}\cap
V_{j}\neq\varnothing\\}|C\cap\omega_{j}|.$ (3.33)

Combining (3.29), (3.31) and (3.33) we get (3.26) with the constants

$c(\omega_{j})=\max\left\\{\max\\{\rho_{\nu(\omega_{i})}:\,V_{i}\cap
V_{j}\neq\varnothing\\},d_{V_{j}}(\omega_{j})\right\\}.$ (3.34)

It remains just note that by U1 and Lemma 3.6 we have $c(\omega_{j})<\infty$
for any $j=1,2,\ldots$. ∎

For an open set $A\subset X$ of bounded measure we denote

$\lambda(A)=A\cup\left\\{x\in
X:\limsup_{n\to\infty}|U_{n}(x,\textbf{1}_{A})|>0\right\\}.$

If an operator sequence $\\{U_{n}\\}$ satisfies U4, then

$\lim_{n\to\infty}U_{n}(x,\textbf{1}_{A})=0,\hbox{ a.e. on }X\setminus A,$

so we obtain

$\mu(\lambda(A)\setminus A)=0.$ (3.35)

######  Lemma 3.9.

Let an operator sequence (1.4) satisfy U1, U3 and U4. If open sets $A$ and $B$
in $X$ have finite measures and $\lambda(A)\subset B$, then for any
$\varepsilon>0$ there exists an open set $G\subset B$ such that

$\displaystyle\lambda(A)\subset G,\quad\lambda(G)\subset B,$ (3.36)
$\displaystyle\mu(G\setminus A)<\varepsilon.$ (3.37)

###### Proof.

Applying Lemma 3.5, we find for the open set $B$ a regular partition
$\Omega=\\{\omega_{k}:\,k=1,2,\ldots\\}$. By Lemma 3.6 and relation (3.5) we
conclude $d_{B}(\omega_{k})<\infty$ for any $k=1,2,\ldots$. Applying Lemma
3.4, we find an open set $G\subset B$ such that

$\displaystyle\lambda(A)\subset G,$ (3.38) $\displaystyle\mu((G\setminus
A)\cap\omega_{k})<\varepsilon_{k}=\frac{\varepsilon}{2^{k}(1+d_{B}(\omega_{k}))}.$
(3.39)

The first relation in (3.36) immediately follows from (3.38). Then we have

$\mu(G\setminus A)\leq\sum_{k=1}^{\infty}\mu((G\setminus
A)\cap\omega_{k})<\sum_{k=1}^{\infty}\frac{\varepsilon}{2^{k}}=\varepsilon.$
(3.40)

It remains to prove that $\lambda(G)\subset B$. Chose an arbitrary point
$x\not\in B$. The relation $\lambda(A)\subset B$ implies

$\lim_{n\to\infty}U_{n}(x,\textbf{1}_{A})=0.$ (3.41)

Thus, by the linearity of operator $U_{n}$ and the pairwise disjointness of
the sets $\omega_{k}$ we can write

$\displaystyle\limsup_{n\to\infty}|U_{n}(x,\textbf{1}_{G})|$
$\displaystyle=\limsup_{n\to\infty}|U_{n}(x,\textbf{1}_{G\setminus A})|$
(3.42)
$\displaystyle\leq\limsup_{n\to\infty}\sum_{k=0}^{\infty}|U_{n}(x,\textbf{1}_{(G\setminus
A)\cap\omega_{k}})|$ (3.43)

By property U3 we have

$\lim_{n\to\infty}U_{n}(x,\textbf{1}_{(G\setminus A)\cap\omega_{k}})=0\text{
for every }k\geq 1.$ (3.44)

Besides by Lemma 3.6 and (3.39) we can write

$\displaystyle|U_{n}(x,\textbf{1}_{(G\setminus A)\cap\omega_{k}})|\leq
d_{B}(\omega_{k})\|\textbf{1}_{(G\setminus
A)\cap\omega_{k}}\|_{1}<\varepsilon_{k}d_{B}(\omega_{k})\leq\varepsilon/2^{k}.$
(3.45)

Combining (3.43), (3.44) and (3.45), we obtain

$\limsup_{n\to\infty}U_{n}(x,\textbf{1}_{G})=0,$

i.e. $x\not\in\lambda(G)$ and therefore $\lambda(G)\subset B$. ∎

######  Lemma 3.10.

Let an operator sequence (1.4) satisfy U1, U3 and U4. If $A$ and $B$ are open
sets of finite measure in $(X,\mu)$ with $\lambda(A)\subset B$, then there
exists a family of open sets

$G_{r},\quad r\in\mathcal{D}=\left\\{i/2^{k},\,0\leq i\leq
2^{k},\,k=0,1,\ldots\right\\}$

such that

$\displaystyle G_{1}=A,\,G_{0}=B,$ (3.46) $\displaystyle\lambda(G_{r})\subset
G_{r^{\prime}}\text{ whenever }r>r^{\prime}.$ (3.47)

###### Proof.

Define $G_{1}=A$, $G_{0}=B$ and apply Lemma 3.9 to the pair of open sets
$G_{1},\,G_{0}$. With this we find an open set $G=G_{1/2}$, satisfying (3.36).
Then we apply induction. Denote

$\mathcal{D}_{k}=\left\\{\frac{i}{2^{k}}:\,0\leq i\leq 2^{k}\right\\},$

and suppose that we have already defined $G_{r}$ for each
$r\in\mathcal{D}_{k}$ such that (3.47) holds whenever
$r,r^{\prime}\in\mathcal{D}_{k}$. Applying Lemma 3.9 to each pair of sets
$G_{i/2^{k}},\,G_{(i+1)/2^{k}}$, we obtain intermediate sets
$G_{(2i+1)/2^{k+1}}$, $0\leq i\leq 2^{k}-1$. Obviously a new family of sets
$\\{G_{r},\,r\in\mathcal{D}_{k+1}\\}$ will also satisfy (3.47). Continuing the
induction procedure we finally get sets $G_{r}$ defined for all
$r\in\mathcal{D}$, and satisfying (3.47) for full range of dyadic indexes
$r,r^{\prime}$. ∎

######  Lemma 3.11.

Let operators (1.4) satisfy U1-U4. If $\varepsilon>0$, $G\subset X$ is a
finite measure open set and $E\subset G$ is a null-set, then there exists an
open set $A$, with $E\subset A\subset G$, and a function $h(x)$, $x\in X$,
such that

$\displaystyle\mu(A)<\varepsilon,$ (3.48) $\displaystyle{\rm supp\,}(h)\subset
G,\quad h(x)=1,\,x\in A,$ (3.49) $\displaystyle 0\leq h(x)\leq 1,\quad x\in
X,$ (3.50) $\displaystyle|U_{n}(x,h)|\leq\varepsilon,\quad x\in G^{c},\quad
n=1,2,\ldots,$ (3.51) $\displaystyle U_{n}(x,h)\to h(x)\hbox{ for every }x\in
X.$ (3.52)

###### Proof.

Applying Lemma 3.5, we find a regular partition
$\Omega=\\{\omega_{k}:\,k\in\mathbb{N}\\}$ for $G$. Then, using Lemma 3.4 we
define an open set $B$, satisfying

$\displaystyle E\subset B\subset G,$ (3.53)
$\displaystyle\mu(B\cap\omega_{k})<\varepsilon_{k}=\frac{\varepsilon}{2^{k}(1+d_{G}(\omega_{k}))}.$
(3.54)

Applying Lemma 3.9, we get an open set $A$, satisfying $E\subset A$,
$\lambda(A)\subset B$. Bound (3.54) implies $\mu(B)<\varepsilon$ and so
(3.48). According to Lemma 3.10, there exists a family of open sets
$\\{G_{r}:\,r\in\mathcal{D}\\}$, satisfying (3.46) and (3.47). Define a
function

$h(x)=\left\\{\begin{array}[]{ccl}\sup\\{r:\,x\in G_{r}\\}&\hbox{ if }&x\in
B,\\\ 0&\hbox{ if }&x\in X\setminus B.\end{array}\right.$ (3.55)

Obviously, $h(x)$ satisfies conditions (3.49), (3.50) and moreover, ${\rm
supp\,}(h)\subset B$. Using Lemma 3.6 and (3.54), for any $x\in G^{c}$ we
obtain

$\displaystyle|U_{n}(x,h)|$
$\displaystyle\leq\sum_{k=1}^{\infty}|U_{n}(x,h\cdot\textbf{1}_{\omega_{k}})|$
(3.56)
$\displaystyle<\sum_{k=1}^{\infty}d_{G}(\omega_{k})|B\cap\omega_{k}|\leq\sum_{k=1}^{\infty}\frac{\varepsilon}{2^{k}}=\varepsilon,$
(3.57)

that gives (3.51). It remain to check condition (3.52). To this end we
consider the function

$p(x)=\textbf{1}_{G_{r_{0}}}(x)+\sum_{k=0}^{m-1}r_{k}\textbf{1}_{G_{r_{k+1}}\setminus
G_{r_{k}}}(x),$ (3.58)

where the dyadic rational numbers $r_{k}\in\mathcal{D}$ satisfy

$1=r_{0}>r_{1}>\ldots>r_{m}=0,\quad r_{k}-r_{k+1}<\varepsilon.$ (3.59)

Observe that

$|h(x)-p(x)|<\varepsilon,\quad x\in X.$ (3.60)

Indeed, we have

$\displaystyle h(x)=p(x)=1,\quad x\in G_{1}=A,$ (3.61) $\displaystyle
h(x)=p(x)=0,\quad x\in X\setminus G_{0}=X\setminus B.$ (3.62)

If $x\in G_{0}\setminus G_{1}$, then we have $x\in G_{r_{k+1}}\setminus
G_{r_{k}}$ for some $k=0,1,\ldots,m-1$. Then from the definition of $h(x)$ it
follows that

$r_{k}\geq h(x)\geq r_{k+1},\quad p(x)=r_{k}.$ (3.63)

Thus we get (3.60). From (3.60) and the bound $\|U_{n}\|_{L^{\infty}\to
M}\leq\varrho$ (see property U2) it follows that

$\limsup_{n\to\infty}|U_{n}(x,h)-h(x)|\leq\limsup_{n\to\infty}|U_{n}(x,h-p)-h(x)+p(x)|\\\
+\limsup_{n\to\infty}|U_{n}(x,p)-p(x)|\leq(\varrho+1)\varepsilon+\limsup_{n\to\infty}|U_{n}(x,p)-p(x)|.$
(3.64)

Consider the following cases.

Case 1: $x\in G_{0}=G_{r_{0}}$. We have $h(t)=1$ as $t\in G_{0}$. By Lemma 3.1
there exists a closed set $F$ such that $x\in F\Subset G_{0}$. Thus, from
property U3 we obtain

$\lim_{n\to\infty}U_{n}(x,h)=1=h(x).$

Case 2: $x\in B\setminus G_{0}$. In this case we have

$x\in G_{r_{k+1}}\setminus G_{r_{k}}$ (3.65)

for some $k=0,1,\ldots,m-1$. Again by U3 it follows that

$\lim_{n\to\infty}U_{n}(x,\textbf{1}_{G_{r_{i}}})=1,\quad i\geq k+1.$ (3.66)

On the other hand, using the property $\lambda(G_{r_{i}})\subset G_{r_{i+1}}$,
we can write

$\lim_{n\to\infty}U_{n}(x,\textbf{1}_{G_{r_{i}}})=0,\quad i\leq k-1,$ (3.67)

whenever $k\geq 1$. Thus we obtain

$\lim_{n\to\infty}U_{n}(x,\textbf{1}_{G_{r_{i+1}}\setminus
G_{r_{i}}})=\lim_{n\to\infty}U_{n}(x,\textbf{1}_{G_{r_{i+1}}})\\\
-\lim_{n\to\infty}U_{n}(x,\textbf{1}_{G_{r_{i}}})=0,\,i\in\mathbb{N},\,i\neq
k,k-1.$ (3.68)

Thus, first using (3.63), (3.68) then (3.66), (3.67) and the boundedness
property U2, in the case $k>0$ we get

$\displaystyle\limsup_{n\to\infty}$ $\displaystyle|U_{n}(x,p)-p(x)|$ (3.69)
$\displaystyle=\limsup_{n\to\infty}|r_{k-1}U_{n}(x,\textbf{1}_{G_{r_{k}}\setminus
G_{r_{k-1}}})+r_{k}U_{n}(x,\textbf{1}_{G_{r_{k+1}}\setminus
G_{r_{k}}})-r_{k}|$ (3.70)
$\displaystyle\leq\varrho\varepsilon+\limsup_{n\to\infty}|r_{k}U_{n}(x,\textbf{1}_{G_{r_{k}}\setminus
G_{r_{k-1}}})+r_{k}U_{n}(x,\textbf{1}_{G_{r_{k+1}}\setminus
G_{r_{k}}})-r_{k}|$ (3.71)
$\displaystyle=\varrho\varepsilon+r_{k}\limsup_{n\to\infty}|U_{n}(x,\textbf{1}_{G_{r_{k+1}}})-U_{n}(x,\textbf{1}_{G_{r_{k-1}}})-1|$
(3.72) $\displaystyle=\varrho\varepsilon.$ (3.73)

If $k=0$, then similarly we have

$\limsup_{n\to\infty}|U_{n}(x,p)-p(x)|\leq\varrho\varepsilon+r_{0}\limsup_{n\to\infty}|U_{n}(x,\textbf{1}_{G_{r_{1}}})-1|=\varrho\varepsilon.$
(3.74)

Combining the last two estimates with (3.64), we conclude $U_{n}(x,h)\to
h(x)$.

Case 3: $x\in X\setminus B$. From the relations $\lambda(G_{r_{i}})\subset
G_{r_{m}}=B$, $i=1,2,\ldots,m-1$, we have

$\lim_{n\to\infty}U_{n}(x,\textbf{1}_{G_{r_{i+1}}\setminus G_{r_{i}}})=0,\quad
i=1,2,\ldots,m-2,$

and therefore using also (3.58) and the equation $r_{m-1}<\varepsilon$ (see
(3.59)), we obtain

$\limsup_{n\to\infty}|U_{n}(x,p)|=\limsup_{n\to\infty}r_{m-1}|U_{n}(x,\textbf{1}_{G_{r_{m}}\setminus
G_{r_{m-1}}})|<\varepsilon\varrho,$

which completes the proof of (3.52).

∎

######  Lemma 3.12.

If operators (1.4) satisfy U1-U4, then for every $G_{\delta}$ null-set
$E\subset X$ and $\varepsilon>0$, there exists a function $g\in M(X)$,
satisfying

$\displaystyle\mu({\rm supp\,}(g))<\varepsilon,$ (3.75) $\displaystyle 0\leq
g(x)\leq 1,$ (3.76) $\displaystyle U_{n}(x,g)\to g(x),\quad x\in X\setminus
E,$ (3.77) $\displaystyle\limsup_{n\to\infty}U_{n}(x,g)\geq
1,\quad\liminf_{n\to\infty}U_{n}(x,g)\leq 0,\quad x\in E.$ (3.78)

###### Proof.

We have $E=\cap_{k=1}^{\infty}E_{k}$, where $E_{k}\subset X$ are open sets and
we can also suppose that each

$\mu(E_{k})<\varepsilon/2^{k}.$ (3.79)

We claim there are functions $h_{k}(x)$, $x\in X$, and open sets $A_{k}\subset
X$, $k=1,2,\ldots,$ satisfying the following conditions h1)-h8):

h1) $E\subset A_{k}\subset E_{k}$, $A_{k}\subset A_{k-1}$, $k\geq 1$
($A_{0}=X$),

h2) ${\rm supp\,}h_{k}(x)\subset A_{k-1}$, $h_{k}(x)=1$, $x\in A_{k}$, $k\geq
1$,

h3) $0\leq h_{k}(x)\leq 1$, $x\in X$, $k\geq 1$,

h4) $|U_{n}(x,h_{k})|<2^{-k}$, $x\in A_{k-1}^{c}$, $k\geq 1$,

h5) $U_{n}(x,h_{k})\to h_{k}(x)$ as $n\to\infty$, $x\in X$.

For any open set $A_{k}$ there is a regular partition $\Omega_{k}$ and a
function $\nu_{k}:\Omega_{k}\to\mathbb{N}$, satisfying

h6) $\min_{\omega\in\Omega_{k}}\nu_{k}(\omega)>k$ for any $k=1,2,\ldots$.

h7) $|U_{(\Omega_{k},\nu_{k})}(x,h_{i})-1|<\frac{1}{k^{2}}$, $\quad x\in
A_{k}$, $i\leq k$,

h8) $|U_{(\Omega_{i},\nu_{i})}(x,h_{k})|<\frac{1}{k^{2}}$, $\quad x\in
X$,$\quad i<k$.

We realize this construction by induction. Applying Lemma 3.11, we find an
open set $A_{1}$, with $E\subset A_{1}\subset E_{1}$, and a function
$h_{1}(x)$, $x\in X$ such that

$\displaystyle h_{1}(x)=1,\,x\in A_{1},$ $\displaystyle 0\leq h_{1}(x)\leq
1,\quad x\in X,$ $\displaystyle U_{n}(x,h_{1})\to h_{1}(x)\hbox{ for all }x\in
X.$

By Lemma 3.7 there exists a regular partition $\Omega_{1}$ of the open set
$A_{1}$ and a function $\nu_{1}:\Omega_{1}\to\mathbb{N}$ such that

$|U_{(\Omega_{1},\nu_{1})}(x,h_{1})-1|<1,\quad x\in A_{1}.$

This gives the base of induction. Now suppose that we have already chosen sets
$A_{k}$ and functions $h_{k}(x)$, sarisfying h1)-h8) for $k=1,2,\ldots,p$. Let
$c_{k}(\omega)$, $\omega\in\Omega_{k}$, be the constants from (3.26),
corresponding to the regular partition $\Omega_{k}$ of the set $A_{k}$,
$k=1,2,\ldots,p$. For any $l=1,2,\ldots,p$ we fix a collection of positive
numbers $\\{\varepsilon_{l}(\omega)>0:\,\omega\in\Omega_{l}\\}$, satisfying

$\sum_{\omega\in\Omega_{l}}\varepsilon_{l}(\omega)<\frac{1}{(p+1)^{2}}.$
(3.80)

Then, applying Lemma 3.4, for any $l\leq p$ we chose an intermediate open set
$C_{l}$, satisfying

$\displaystyle E\subset C_{l}\subset A_{p}\cap E_{p+1},$ (3.81)
$\displaystyle|C_{l}\cap\omega|<\frac{\varepsilon_{l}(\omega)}{c_{l}(\omega)},\quad\omega\in\Omega_{l},\,l=1,2,\ldots,p.$
(3.82)

Denote $C=\cap_{l=1}^{p}C_{l}$. Thus, applying Lemma 3.8, for any function $f$
satisfying (3.25), we get

$\displaystyle|U_{(\Omega_{l},\nu_{l})}(x,f)|$
$\displaystyle\leq\sum_{\omega\in\Omega_{l}}c_{l}(\omega)|C\cap\omega|$ (3.83)
$\displaystyle\leq\sum_{\omega\in\Omega_{l}}c_{l}(\omega)|C_{l}\cap\omega|\leq\sum_{\omega\in\Omega_{l}}\varepsilon_{l}(\omega)<\frac{1}{(p+1)^{2}},$
(3.84)

which holds for every $x\in X$ and $l\leq p$. Then using Lemma 3.11, we find
an open set $A_{p+1}$ and a function $h_{p+1}(x)$, $x\in X$, such that

$\displaystyle E\subset A_{p+1}\subset C\subset A_{p}\cap E_{p+1},$
$\displaystyle{\rm supp\,}h_{p+1}\subset C,\quad h_{p+1}(x)=1,\,x\in A_{p+1},$
$\displaystyle 0\leq h_{p+1}(x)\leq 1,\quad x\in X,$
$\displaystyle|U_{n}(x,h_{p+1})|\leq 2^{-p-1},\quad x\in(A_{p})^{c},$
$\displaystyle U_{n}(x,h_{p+1})\to h_{p+1}(x)\hbox{ for all }x\in X.$

Thus we obtain conditions h1)-h5) for $k=p+1$. From (3.84) we can also get the
inequality

$|U_{(\Omega_{l},\nu_{l})}(x,h_{p+1})|\leq\frac{1}{(p+1)^{2}},\,x\in X,\,l\leq
p,$

which implies h8) in the case $k=p+1$. Then, applying Lemma 3.7, we find a
locally-finite partition $\Omega_{p+1}$ for $A_{p+1}$ and a mapping
$\nu_{p+1}:\Omega_{p+1}\to\mathbb{N}$ satisfying h6) and h7) for $k=p+1$. This
completes the induction. From h1) we obtain

$E=\cap_{i=1}^{\infty}A_{i}.$

Define

$g(x)=\sum_{i=1}^{\infty}(-1)^{i+1}h_{i}(x)\text{ if }x\in X\setminus E$
(3.85)

and $g(x)=0$ if $x\in E$. Observe that series (3.85) converges in $L^{1}(X)$,
since by conditions h1), h2), h3) and (3.79) it follows that
$\|h_{k}\|_{1}<\varepsilon 2^{-k+1}$. Thus by the boundedness of the operator
$U_{n}:L^{1}(X)\to M(X)$ we conclude

$U_{n}(g)=\sum_{i=1}^{\infty}(-1)^{i+1}U_{n}(h_{i}),$ (3.86)

where the series converges uniformly. Clearly, h1) and (3.79) imply (3.75).
Fix a point $x\in E^{c}$. We have

$x\in A_{k-1}\setminus A_{k},$ (3.87)

for some $k\geq 1$. Taking into account h2), the latter implies

$\displaystyle h_{i}(x)=0,\quad i>k,$ (3.88) $\displaystyle h_{i}(x)=1,\quad
i<k.$ (3.89)

Thus we obtain

$g(x)=\sum_{i=1}^{k}(-1)^{i+1}h_{i}(x)=\sum_{i=1}^{k-1}(-1)^{i+1}+(-1)^{k+1}h_{k}(x).$
(3.90)

If $k$ is even, then $g(x)=1-h_{k}(x)$, for an odd $k$ we have
$g(x)=h_{k}(x)$. From this and property h3) of function $h_{k}(x)$, we obtain
condition (3.76). Then condition h4) implies

$|U_{n}(x,h_{i})|\leq 2^{-i},\quad i>k.$ (3.91)

Thus, using (3.86), (3.90) as well as h5), for $m>k$ we get

$\limsup_{n\to\infty}|U_{n}(x,g)-g(x)|=\limsup_{n\to\infty}|\sum_{i=m}^{\infty}(-1)^{i+1}U_{n}(x,h_{i})|\\\
\leq\sum_{i=m}^{\infty}|U_{n}(x,h_{i})|\leq\sum_{i=m}^{\infty}2^{-i}=2^{-m+1}.$

Since $m$ can be arbitrarily bigger, this implies condition (3.77). To prove
(3.78), now suppose $x\in E$. Then we have $x\in A_{k}$ for all
$k=1,2,\ldots$. Using this, for any $k=1,2,\ldots$ we may fix a set
$\omega_{k}\in\Omega_{k}$ such that $\omega_{k}\ni x$. By condition h8) we
have

$|U_{\nu_{k}(\omega_{k})}(x,h_{i})|\leq\frac{1}{i^{2}},\quad i>k,$ (3.92)

and form h7) it follows that

$|U_{\nu_{k}(\omega_{k})}(x,h_{i})-1|\leq\frac{1}{k^{2}},\quad i\leq k.$
(3.93)

Thus, using (3.86), we obtain

$\begin{split}|U_{\nu_{k}(\omega_{k})}(x,g)-\sum_{i=1}^{k}(-1)^{i+1}|\\\
&\leq\sum_{i=1}^{k}|U_{\nu(\omega_{k})}(x,h_{i})-1|+\sum_{i=k+1}^{\infty}|U_{\nu(\omega_{k})}(x,h_{i})|\\\
&\leq
k\cdot\frac{1}{k^{2}}+\sum_{i=k+1}^{\infty}\frac{1}{i^{2}}<\frac{2}{k}.\end{split}$

Since by h6) we have $\nu_{k}(\omega_{k})\to\infty$ and the sum
$\sum_{i=1}^{k}(-1)^{k+1}$ takes values $0$ or $1$ alternatively, we get
(3.78) for any $x\in E$, finalizing the proof of lemma. ∎

##  4\. Proof of theorems

###### Proof of Theorem 1.1.

Suppose $E$ is a $G_{\delta\sigma}$ null-set and we have
$E=\bigcup_{k=1}^{\infty}E_{k}$, where $E_{k}$ are $G_{\delta}$ null-sets and
we may additionally suppose that $E_{k}\subset E_{k+1}$. Applying Lemma 3.12,
we attach to each $E_{k}$ a function $g_{k}(x)$ such that

g1) $\mu({\rm supp\,}(g_{k}))<\varepsilon/2^{k}$,

g2) $0\leq g_{k}(x)\leq 1$,

g3) $U_{n}(x,g_{k})\to g_{k}(x)$ at every point $x\not\in E_{k}$,

g4) for every $x\in E_{k}$ we have $\delta(x,g_{k})\geq 1/2$, where we use the
notation

$\delta(x,g)=\limsup_{n\to\infty}|U_{n}(x,g)-g(x)|.$ (4.1)

We claim that

$f(x)=\sum_{i=1}^{\infty}\frac{g_{i}(x)}{(4\varrho+5)^{i}}$

is our desired function, where $\varrho$ is the constant in U2. The
boundedness of $f$ follows from condition g2), and condition g1) implies
$\mu({\rm supp\,}(f))<\varepsilon$. Chose an arbitrary point $x\in E$. For
some $k$ we have

$x\in E_{k}\setminus E_{k-1}.$

This together with g3), g4) implies

$\displaystyle\lim_{n\to\infty}U_{n}\left(x,g_{i}\right)=g_{i}(x)\text{ if
}i\leq k-1,$ (4.2) $\displaystyle\delta(x,g_{k})\geq 1/2,$ (4.3)
$\displaystyle\delta(x,g_{i})\leq\sup_{n}|U_{n}(x,g_{i})|+|g_{i}(x)|\leq\varrho+1,\quad
i=1,2,\ldots.$ (4.4)

Thus, using also U2, we obtain

$\displaystyle\delta(x,f)$
$\displaystyle\geq\frac{\delta(x,g_{k})}{(4\varrho+53)^{k}}-\sum_{i=k+1}^{\infty}\frac{\delta(x,g_{i})}{(4\varrho+5)^{i}}$
$\displaystyle\geq\frac{1}{2\cdot(4\varrho+5)^{k}}-\sum_{i=k+1}^{\infty}\frac{\varrho+1}{(4\varrho+5)^{i}}$
$\displaystyle=\frac{1}{4\cdot(4\varrho+5)^{k}}>0,$

which means that the sequence $U_{n}(x,f)$ diverges for any $x\in E$. Letting
$x\not\in E$, we have $x\not\in E_{i}$ for any $i\in\mathbb{N}$ and so
$U_{n}(x,g_{i})\to g_{i}(x)$ as $n\to\infty$. This implies

$\displaystyle\delta(x,f)$
$\displaystyle=\delta\left(x,\sum_{i=k+1}^{\infty}(4\varrho+5)^{-i}g_{i}\right)$
$\displaystyle\leq\sup_{n}\left|U_{n}\left(x,\sum_{i=k+1}^{\infty}(4\varrho+5)^{-i}g_{i}\right)\right|+\left|\sum_{i=k+1}^{\infty}(4\varrho+5)^{-i}g_{i}(x)\right|$
$\displaystyle\leq(\varrho+1)\sum_{i=k+1}^{\infty}(4\varrho+5)^{-i}.$

Since the latter goes to zero, we obtain $\delta(x,f)=0$. Theorem is proved. ∎

###### Proof of Theorem 1.2.

We will make use the same functions $h_{k}$ constructed in the proof of Lemma
3.12 with an additional bound $\mu(A_{k})<\alpha_{k}$ which can be clearly
ensured. Then for small enough numbers $\alpha_{k}<2^{-k}$ we will have

$f(x)=\sum_{k=1}^{\infty}h_{k}(x)\in L_{\phi}(X),$ (4.5)

and $\mu({\rm supp\,}(f))<\varepsilon$. By the construction of the functions
$h_{k}$ the initial set $E$ can be written in the form $E=\cap_{k}A_{k}$.
Since $\|h_{k}\|_{1}<2^{-k+1}$, series (4.5) converges in $L^{1}(X)$ and by
the boundedness of the operator $U_{n}:L^{1}\to M$ we can write

$U_{n}(f)=\sum_{i=1}^{\infty}U_{n}(h_{i}),$ (4.6)

where the series converges uniformly on $X$. Letting $x\in E^{c}$, we have
$x\in A_{k-1}\setminus A_{k}$ for some $k\geq 1$ ($A_{0}=X$). Then we have
(3.88), (3.89) and therefore,

$f(x)=\sum_{j=1}^{k}h_{j}(x)=k-1+h_{k}(x).$ (4.7)

Besides for such a point $x\in A_{k-1}\setminus A_{k}$ we have (3.91). Thus
using h5) together with (3.91), (4.6) and (4.7), for $m>k$ we get

$\limsup_{n\to\infty}|U_{n}(x,f)-f(x)|=\limsup_{n\to\infty}|\sum_{i=m}^{\infty}U_{n}(x,h_{i})|\leq\sum_{i=m}^{\infty}2^{-i}.$
(4.8)

Since $m$ can be arbitrarily bigger, this implies convergence condition b) of
the theorem. To prove condition a), suppose $x\in E$ and so $x\in A_{k}$ for
all $k=1,2,\ldots$. Using this we may fix sets $\omega_{k}\in\Omega_{k}$ with
$\omega_{k}\ni x$. Then using (3.92) and (3.93), we obtain

$\displaystyle|U_{\nu_{k}(\omega_{k})}(x,f)-k|$
$\displaystyle\leq\sum_{i=1}^{k}|U_{\nu_{k}(\omega_{k})}(x,h_{i})-1|+\sum_{i=k+1}^{\infty}|U_{\nu_{k}(\omega_{k})}(x,h_{i})|$
(4.9) $\displaystyle\leq
k\cdot\frac{1}{k^{2}}+\sum_{i=k+1}^{\infty}\frac{1}{i^{2}}<\frac{2}{k},$
(4.10)

which implies the unboundedly divergence of $U_{n}(x,f)$ at a point $x\in E$.
∎

###### Proof of Theorem 1.3.

We construct open sets $G_{k}$, $k=1,2,\ldots,$ together with a regular
partition $\Omega_{k}$ and a mapping $\nu_{k}:\Omega_{k}\to\mathbb{N}$,
satisfying following conditions.

g1) $E\subset G_{k}\subset G_{k-1}$, $k\geq 1$ ($G_{0}=X$),

g2) $\min_{\omega\in\Omega_{k}}\nu_{k}(\omega)>k$, $k\geq 1$,

g3) $|U_{(\Omega_{k},\nu_{k})}(x,\textbf{1}_{G_{i}})-1|<\frac{1}{k^{2}}$,
$\quad x\in G_{k}$, $i\leq k$,

g4) $|U_{(\Omega_{i},\nu_{i})}(x,\textbf{1}_{G_{k}})|<\frac{1}{k^{2}}$, $\quad
x\in X$,$\quad i<k$.

We do it by induction. Define the open set $G_{1}$ arbitrarily. By Lemma 3.7
there exists a regular partition $\Omega_{1}$ of $G_{1}$ and a function
$\nu_{1}:\Omega_{1}\to\mathbb{N}$ such that

$|U_{(\Omega_{1},\nu_{1})}(x,\textbf{1}_{G_{1}})-1|<1,\quad x\in G_{1}.$

Then suppose that we have already chosen the sets $G_{k}$ and corresponding
partitions $\Omega_{k}$ for $k=1,2,\ldots,p$. Applying Lemma 3.4, we may
choose open sets $C_{l}$, $l=1,2,\ldots,p$ with $E\subset C_{l}\subset G_{p}$,
satisfying

$\sum_{\omega\in\Omega_{l}}c_{l}(\omega)|C_{l}\cap\omega|<\frac{1}{(p+1)^{2}},\quad
l=1,2,\ldots,p,$

where $c_{l}(\omega)$ are the constants from (3.26). Then, denoting
$G_{p+1}=\cap_{l=1}^{p}C_{l}$ and applying Lemma 3.8, we obtain the inequality

$|U_{(\Omega_{l},\nu_{l})}(x,\textbf{1}_{G_{p+1}})|\leq\sum_{\omega\in\Omega_{l}}c_{l}(\omega)|G_{p+1}\cap\omega|\leq\frac{1}{(p+1)^{2}},$
(4.11)

which holds for every $x\in X$ and $l\leq p$. This implies g4) for $k=p+1$.
Then, applying Lemma 3.7, we find a regular partition $\Omega_{p+1}$ for
$G_{p+1}$ and a mapping $\nu_{p+1}:\Omega_{p+1}\to\mathbb{N}$ satisfying g2)
and g3) for $k=p+1$. This completes the induction. Now we can define our
desired set by

$G=\bigcup_{i=1}^{\infty}(G_{2i-1}\setminus G_{2i}).$ (4.12)

One can check,

$U_{n}(x,\textbf{1}_{G})=\sum_{k=1}^{\infty}(-1)^{k+1}U_{n}(x,\textbf{1}_{G_{k}}).$
(4.13)

Suppose $x\in E$. Then we have $x\in G_{k}$, $k=1,2,\ldots$. Using this we may
fix $\omega_{k}\in\Omega_{k}$ such that $\omega_{k}\ni x$. By condition g4) we
have

$|U_{\nu_{k}(\omega_{k})}(x,\textbf{1}_{G_{i}})|\leq\frac{1}{i^{2}},\quad
i>k,$ (4.14)

and form g3) it follows that

$|U_{\nu_{k}(\omega_{k})}(x,\textbf{1}_{G_{i}})-1|\leq\frac{1}{k^{2}},\quad
i\leq k.$ (4.15)

Thus we obtain

$\displaystyle|U_{\nu_{k}(\omega_{k})}(x,f)-\sum_{i=1}^{k}(-1)^{i+1}|$ (4.16)
$\displaystyle\leq\sum_{i=1}^{k}|U_{\nu(\omega_{k})}(x,\textbf{1}_{G_{i}})-1|+\sum_{i=k+1}^{\infty}|U_{\nu(\omega_{k})}(x,\textbf{1}_{G_{i}})|$
(4.17) $\displaystyle\leq
k\cdot\frac{1}{k^{2}}+\sum_{i=k+1}^{\infty}\frac{1}{i^{2}}<\frac{2}{k}$ (4.18)

Thus we get divergence of $U_{\nu(\omega_{k})}(x,f)$ for any $x\in E$. ∎

##  5\. Open problems

Some open problems listed here one can find also in the papers [Uly1, Uly2,
CoLo, Wade].

  1. (1)

Find the complete characterization of $D$-sets ($UD$-sets) of ordinary
trigonometric series and Fourier series of $L^{p}$, $1\leq p\leq\infty$, or
$C(\mathbb{T})$ functions. Note that Körner [Kor] in 1961 constructed a
$G_{\delta}$-set, which is not a convergence set for any kind of trigonometric
series. This example shows that for the partial sums of Fourier or ordinary
trigonometric series a pure topological characterization of $C$-$D$ sets may
fail ([Uly1, Uly2]).

  2. (2)

The same problems are open also for the Walsh series.

  3. (3)

Kahane-Katznelson’s [KaKa] theorem analogue for the Walsh system is an open
problem([Wade]). It claims, given null-set $E$, construct a continuous
function which Walsh-Fourier series diverges at any $x\in E$ . Concerning to
this problem we note that Harris [Har] has proved that for any compact null
set $e\subset[0,1]$ there exists a continuous function, whose Walsh-Fourier
series diverges at any $x\in e$. As we noted the original proof of [KaKa]
essentially uses analytic functions technique, which hardly can be applied in
Walsh case. A real functions approach to Kahane-Katznelson’s theorem one can
find in [Kar4].

  4. (4)

Characterize the sets, which are radial $D$-sets of an univalent function on
the unit disc ([CoLo]).

  5. (5)

It was considered in [Kar4] the exceptional null set problem for the Hilbert
transform

$Hf(x)=\lim_{\varepsilon\to 0}\int_{|t-x|>\varepsilon}\frac{f(t)}{x-t}dt.$
(5.1)

It is well-known the almost everywhere existence of this limit whenever $f\in
L^{1}(\mathbb{R})$ (see for example [Zyg], ch. 4.3). It was proved in [Kar4]
that for any closed null set $e\subset\mathbb{R}$ there exists a continuous
function $f\in C(\mathbb{R})\cap L^{1}(\mathbb{R})$ such that the limit in
(5.1) doesn’t exists at any point of $e$. We do not know whether the set $e$
in this statement can be an arbitrary null set.

## References