On logarithmic Holder continuity of mappings on the boundary
E. A. Sevost'yanov
math.CV 30C65, 1A15, 30C62, 30C80


E. SEVOST'YANOV ON LOGARITHMIC
HÖLDER...




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ON LOGARITHMIC HÖLDER CONTINUITY OF MAPPINGS ON THE
BOUNDARY
    EVGENY SEVOST'YANOV

    December 30, 2023
=============================================================





We study mappings satisfying the so-called inverse Poletsky
inequality. Under integrability of the corresponding majorant, it is
proved that these mappings are logarithmic Hölder continuous in
the neighborhood of the boundary points. In particular, the
indicated properties hold for homeomorphisms whose inverse satisfy
the weighted Poletsky inequality.


2010 Mathematics Subject Classification: 30C65, 31A15, 30C62,
30C80



§ INTRODUCTION

As is known, quasiconformal mappings and mappings with bounded
distortion are Hölder continuous with some exponent (see, e.g.,
<cit.>, <cit.>,
and <cit.>). In this regard, there are
also classical results concerning mappings with bounded distortion,
or quasiregular mappings, which are rightly called quasiconformal
mappings with branch points (see, for example,
<cit.> and <cit.>). There are
many generalizations of these results to more general classes of
mappings with finite distortion. In this case, quite often, the
usual estimates of the Hölder type do not hold for mappings,
however, more general logarithmic estimates may be satisfied (see,
for example, <cit.>,
<cit.>, <cit.>,
<cit.> and <cit.>).

Distance distortion theorems and Hölder-type estimates have been
discussed and studied in our last two articles, <cit.> and
<cit.>. In particular, in <cit.>, we obtained estimates for
the distortion of mappings with the inverse Poletsky inequality at
the inner points of a given domain. The main purpose of this
manuscript is to obtain similar estimates not only at the inner, but
also at the boundary points of a given domain, which, for the sake
of simplicity, will be assumed to be the unit ball.

In what follows, for the sets A, B⊂ R^n we set, as
usual,


    diam A=sup_x, y∈ A|x-y| ,   dist (A, B)=inf_x∈ A,
    y∈ B|x-y| .


Let x_0∈D, x_0∞,


    B(x_0, r)={x∈ R^n: |x-x_0|<r} ,    B^n=B(0, 1) .


Sometimes, instead of dist (A, B), we also write d(A, B),
if a misunderstanding is impossible. A Borel function ρ:
R^n → [0,∞] is called an admissible for a
family Γ of paths γ in R^n, if the relation


    ∫_γρ (x)  |dx|⩾ 1


holds for any locally rectifiable path γ∈Γ.

A modulus of Γ is defined as follows:


    M(Γ)=inf_ρ∈  adm Γ∫_ R^nρ^n (x) dm(x) .


Let Q: R^n→ [0, ∞] be a Lebesgue measurable
function. We say that f satisfies the inverse Poletsky
inequality, if the relation


    M(Γ)⩽∫_f(D) Q(y)·ρ_*^n(y)  dm(y)


holds for any family of (locally rectifiable) paths Γ in D
and any ρ_*∈ adm f(Γ).

Note that estimates (<ref>) hold in many classes of mappings
(see, e.g., <cit.>, <cit.>
and <cit.>). A mapping f:D→
R^n is called a discrete if {f^-1(y)}
consists of isolated points for any y ∈  R^n, and an open, if the image of any open set U⊂ D is an open set
in R^n. A mapping f between domains D and
D^ ' is said to be a closed if f(E) is closed in
D^ ' for any closed set E⊂ D (see, e.g.,
<cit.>).


Given δ>0, a non-degenerate continuum A⊂ B^n
and a Lebesgue measurable function Q:  B^n→ [0,
∞] denote S_δ, A, Q a family of all open,
discrete and closed mappings f of the open unit ball onto itself
such that the relation (<ref>) holds and, in addition, dist (f^ -1(A),
∂ B^n)⩾ δ. The following statement holds.


 Let Q∈ L^1( B^n). Then any f∈ S_δ,
A, Q has a continuous extension f:
B^n→ B^n, and, in addition, for any
x_0∈∂ B^n there is C_n>0 and r_0=r_0(x_0)>0
such that


    |f(x)-f(x_0)|⩽C_n· (‖
    Q‖_1)^1/n/log^1/n(1+r_0/|x-x_0|)


for x∈ B(x_0, r_0)∩ B^n,

where ‖ Q‖_1 denotes the L^1-norm of the function Q in
L^1( B^n).
 






§ AUXILIARY LEMMA AND PROOF OF THEOREM <REF>
 Before
by proving the basic statement we prove the following important
lemma.


  Let E be a continuum in B^n. Now, there is
δ_1>0 such that S_δ, A, Q ⊂
S_δ_1, E, Q . In other words, if f is an open discrete
and closed mapping of the unit ball onto itself with condition
(<ref>), such that dist (f^ -1(A),∂
B^n)⩾ δ, then there is δ_1> 0, independent on
f such that dist (f^ -1(E),∂ B^n)⩾ δ_1.
 



Let us prove this statement by contradiction. Suppose that the
conclusion of the lemma is not correct. Then there are sequences
y_m∈ E, f_m⊂ S_δ, A, Q and x_m∈
B^n such that f_m(x_m)=y_m and dist (x_m, ∂
B^n)→ 0 as m→∞. Without loss of
generality, we may assume that x_m→ x_0 as
m→∞. By <cit.> f_m has a
continuous extension at x_0, moreover, {f_m}_m=1^∞ is
equicontinuous at x_0 (see, e.g., <cit.>). Now,
for any ε>0 there is m_0∈ N such that
|f_m(x_m)-f_m(x_0)|<ε for m⩾ m_0. On the other
hand, f_m(x_0)∈ S^n-1 because f_m is closed. Thus, by
the triangle inequality, |f_m(x_m)-f_m(x_0)|⩾
1-|f_m(x_m)|⩾ 1-δ_0, where δ_0=sup_x∈
E|x|. Finally, we have a contradiction, because
|f_m(x_m)-f_m(x_0)|⩾ 1-δ_0 and, simultaneously
|f_m(x_m)-f_m(x_0)|<ε as m⩾ m_0. The resulting
contradiction refutes the assumption made above. Lemma is
proved. 



Proof of Theorem <ref>. The possibility of a continuous
extension of f to the boundary of B^n is established
in <cit.>. In particular, the weakly flatness of
∂ B^n= S^n-1 follows by <cit.>.

Let us prove the logarithmic Hölder continuity (<ref>). It
suffices to prove relation (<ref>) for the case when x∈ B^n∩ B(x_0, r_0), since the general case x∈ B^n∩ B(x_0, r_0) is attained by passing to the
limit as x→ x_*, where x_*∈∂ B^n∩
B(x_0, r_0). Let x_0 ∈∂ B^n, let 0<δ<1 and let E=B(0,
δ/2)⊂ B^n. By Lemma <ref>, there
is δ_1>0 such that dist (f^ -1(E),
∂ B^n)⩾δ_1 for any f∈ S_δ, A, Q.
By <cit.>, the family S_δ, A, Q
is equicontinuous in B^n. Thus, for a number
0<δ_0<1/4 there is a neighborhood U⊂ B(x_0,
δ_1/2) of x_0 such that |f(x)-f(y)|<δ_0 for any x,
y∈ U∩ B^n and f∈ S_δ, A, Q. Let
f(x) f(y) and


    ε_0:=|f(x)-f(y)|<δ_0 .


Let us join the points f(x)
and 0 by segment I. The points  f(x), 0 and f(y) form the
plane P. Consider a circle


    S={z∈ P:
    |z-f(x)|=ε_0} .


The position of the point z=f(y) on the circle S is completely
determined by the angle φ, φ∈ [-π, π), between
the vector -f(x) and the radius-vector of the point z. The
points on the circle are denoted further in the polar coordinates by
the pairs z=(ε_0, φ). Our further goal is to
investigate the main three cases regarding the intervals of change
of this angle.

Case 1. ”Large angles”:  φ∈ [π/4, 3π/4], or
φ∈ [-π/4, -3π/4]. Let φ∈ [π/4, 3π/4].
Consider the ray


    r=r(t)=f(y)+te ,  
    e=-f(x)/|f(x)| ,   t>0.


By construction, the ray r is parallel to the segment I. For
t=|f(x)|, we have r(|f(x)|)=f(y)-f(x) and
|r(|f(x)|)|=ε_0<δ_0, i.e., the point r(|f(x)|)
belongs to E. Let J be a segment of the ray r, contained
between the points f(y) and r(|f(x)|). The distance between I
and J is calculated as follows:


    dist (I, J)=ε_0sinφ⩾√(2)/2ε_0 ,


see Figure <ref> for illustration.



Similarly, the situation φ∈ [-π/4, -3π/4] is
considered. We may show that in this case the formula (<ref>)
also holds.

Let α_1 and β_1 be (total) f-liftings of paths I
and J starting at points x and y, correspondingly. These
liftings are well defined and exist by <cit.>.
Now, by the definition, |α_1|∩ f^ -1(E)∅
|β_1|∩ f^ -1(E). Since dist (f^ -1(E),
∂ B^n)⩾δ_1 and x, y∈ B(x_0, δ_1/2),



    diam (α_1)⩾δ_1/2 ,  
    diam (β_1)⩾δ_1/2 .


Let


    Γ:=Γ(α_1, β_1,  B^n) .



Then on the one hand, by <cit.>


    M(Γ)⩾ (1/2)· M(Γ(α_1, β_1, 
    R^n)) ,


and on the other hand, by <cit.>


    M(Γ(α_1, β_1,  R^n))⩾
    c_n·log(1+1/m) ,


where c_n>0 is come constant depending only on n,


    m= dist(α_1, β_1)/min{ diam (α_1),
     diam (β_1)} .


Combining (<ref>) and (<ref>), and taking into account that
dist (α_1, β_1)⩽ |x-y|, we obtain that


    M(Γ)⩾c_n·log(1+δ_1/2 dist(α_1,
    β_1))⩾c_n·log(1+δ_1/2|x-y|) ,


where c_n>0 is some constant depending only on n.

We now obtain an upper bound for M(Γ). Set


    ρ(x)= {[ √(2)/ε_0,   x∈ B^n,;        0,  x∉ B^n . ].


By (<ref>) ρ satisfies the relation (<ref>) for the
family f(Γ). Then by the definition of the family S_δ, A, Q we obtain that



    M(Γ)⩽2^n/2/ε_0^n∫_
    B^n Q(y) dm(y)=2^n/2·‖
    Q‖_1/|f(x)-f(y)|^n ,


where ‖ Q‖_1 denotes the L^1-norm of the function Q in
B^n. By (<ref>) and (<ref>) we obtain that


    c_n·log(1+δ_1/2|x-y|)⩽
    2^n/2·‖ Q‖_1/|f(x)-f(y)|^n .


The desired inequality (<ref>) follows from the last relation
by passing to the limit as y→ x_0 , where
C_n:=2^n/2·c_n^-1/n and r_0=min{δ_1/2,
d(x_0, ∂ U)}.

Case 2. 

”Small angles”: φ∈ [-π/4, π/4), see
Figure <ref> for the illustration. In this case, we denote by
K the segment joining the point f(x) with the unit sphere in the
direction of the vector f(x)/|f(x)|. Then


    dist (K, J)=ε_0 .




Let α_2 be a maximal f-lifting of K^ ' starting
at x, where K^ ' is obtained from K by discarding its
endpoint lying on the unit sphere. Such a lift exists and tends with
its end to ∂ B^n= S^n-1 (see, e.g.,
<cit.>). Let also β_2 be a total
f-lifting of J starting at the point y (such a lifting exists
by <cit.>). Arguing similarly to a case 1, we
obtain that


    diam (β_2)⩾δ_1/2 .


Let


    Γ:=Γ(α_2∪ S^n-1, β_2,  B^n) .



According to the above, α_2∖α_2⊂ S^n-1. Thus α_2∪
S^n-1=α_2∪ S^n-1=α_2∪
S^n-1, consequently, α_2∪ S^n-1 is a closed
set. Observe that α_2 is connected (see, e.g.,
<cit.>). Moreover, the set α_2∪ S^n-1=α_2∪ S^n-1 is connected
as the union of two connected sets α_2 and S^n-1, that have at least one common point (see, e.g.,
<cit.>).

In this case, on the one hand, by <cit.>


    M(Γ)⩾ (1/2)· M(Γ(α_2∪ S^n-1,
    β_2,  R^n)) ,


and on the other hand,  by <cit.>


    M(Γ(α_2∪ S^n-1, β_2,  R^n))⩾
    c_n·log(1+1/m) ,


where c_n>0 is some constant depending only on n,


    m= dist(α_2∪ S^n-1, β_2)/min{ diam (α_2∪ S^n-1), 
    diam (β_2)} .


Then combining (<ref>) and (<ref>), and taking into
account that dist (α_2∪ S^n-1,
β_2)⩽ |x-y|, we obtain that


    M(Γ)⩾c_n·log(1+δ_1/2 dist(α_2∪ S^n-1,
    β_2))⩾c_n·log(1+δ_1/2|x-y|) ,


where c_n>0 is some constant depending only on n.

Let us now establish an upper bound for M(Γ). First of all,
note that

    Γ(α_2∪ S^n-1, β_2, 
    B^n)⊃Γ(α_2∪ S^n-1, β_2, 
    B^n)
 and 
    Γ(α_2∪ S^n-1, β_2,
     B^n)>Γ(α_2∪ S^n-1, β_2,
     B^n) .
 Therefore, by the principle of minority and in view
of the monotonicity of the module we obtain that


    M(Γ(α_2∪ S^n-1, β_2, 
    B^n))=M(Γ(α_2∪ S^n-1, β_2, 
    B^n))=M(Γ) .


Set


    ρ(x)= {[   1/ε_0,  x∈ B^n,;       0, x∉ B^n . ].


Observe that, by (<ref>), ρ satisfies the
relation (<ref>) for f(Γ(α_2∪ S^n-1,
β_2,  B^n)). In this case, by (<ref>) and the
definition of the family S_δ, A, Q we obtain that



    M(Γ)=M(Γ(α_2∪ S^n-1, β_2, 
    B^n))⩽1/ε_0^n∫_ B^n
    Q(y) dm(y)=‖ Q‖_1/|f(x)-f(y)|^n .


By (<ref>) and (<ref>) we obtain that


    c_n·log(1+δ_1/2|x-y|)⩽‖ Q‖_1/|f(x)-f(y)|^n .


The desired inequality (<ref>) follows from the last relation,
which is achieved by passing to the limit as y→ x_0 ,
where C_n:=c_n^-1/n and r_0=min{δ_1/2,
d(x_0, ∂ U)}.

Case 3.   <<Very large angles>>: either φ∈ (3π/4,
π], or φ∈ (-π, -3π/4), see Figure <ref> for
the illustration. Let, for example, φ∈ (3π/4, π].
Using the methods of elementary geometry, one can show that points
z=(ε_0, φ), having the largest absolute values of
the number φ, also have the largest Euclidean modulus of the
number z. Thus, if z=(ε_0, φ)∈ B^n,
then the whole arc of the circle


    z=z(φ)={(ε_0, θ): θ∈ [π/2,
    φ]}
 also belongs to B^n. Let us denote by L_1
the line segment


    l=l(t)=(ε_0, π/2)+te,  
    e=-f(x)/|f(x)| ,


corresponding to the values of the parameter t∈ [0, |f(x)|].
Now, we obtain that l(|f(x)|)=(ε_0, π/2)-f(x), while
|l(|f(x)|)|=ε_0 by the construction. Then L_1
intersects the ball E. Set L=z(φ)∪ L_1. We obtain that


    dist (L, I)=ε_0 .




It can be shown that relation (<ref>) is also satisfied
for φ∈ (-π, -3π/4).

Let α_3 and β_3 be f-liftings of I and L starting
x and y, correspondingly. As before, the existence of such
liftings is due to <cit.>. Set
Γ=Γ(α_3, β_3,  B^n). Further reasoning is
similar to what was carried out in case 1, namely, the existence of
continua I and L with distance ε_0 implies that



    M(Γ)⩾c_n·log(1+δ_1/2 dist(α_3,
    β_3))⩾c_n·log(1+δ_1/2|x-y|)


and 
    M(Γ)⩽1/ε_0^n∫_
    B^n Q(y) dm(y)=‖ Q‖_1/|f(x)-f(y)|^n .


Hence we obtain the inequality


    c_n·log(1+δ_1/2|x-y|)⩽‖ Q‖_1/|f(x)-f(y)|^n .


From the last inequality we obtain the statement of the theorem due
to the passage to the limit as y→ x_0. 

The analog of Theorem <ref> is also valid for mappings with a
fixed point of the unit ball. In order to formulate and prove the
corresponding statement, we introduce the following definition. For
elements a, b∈ B^n and the Lebesgue measurable function
Q: B^n→ [0, ∞] denote by F_a, b,
Q the family of all open discrete and closed mappings f of the
unit sphere onto itself, such that f(a)=b. The following statement
is true.


 Let Q∈ L^1( B^n). Then any mapping f∈
S_δ, A, Q has a continuous extension f:
B^n→ B^n, in addition, for any
x_0∈∂ B^n there exist C_n>0 and r_0=r_0(x_0)>0
such that the relation (<ref>) holds. 



The possibility of a continuous extension of the mapping f to
S^n-1 follows from Theorem 3.1 in <cit.>. We prove
the logarithmic Hölder continuity of the family of extended
mappings. Let E=B(0, 1/2). The following two cases are
possible:

1) there exists δ>0 such that dist (f^ -1(E),
∂ B^n)⩾δ for any f∈
S_δ, A, Q . In this case, the desired statement follows
from Theorem <ref>;

2) there exist f_m∈ S_δ, A, Q and x_m, y_m∈ B^n, m=1,2,… , such that f_m(x_m)=y_m, y_m∈ E
and dist (x_m,  S^n-1)→ 0 as
m→∞. Then, arguing exactly as in the proof of
Lemma <ref>, we come to the conclusion that the family of
mappings S_δ, A, Q is not equicontinuous at least
at one point x_0∈ S^n-1, which contradicts the
assertion of Theorem 7.1 in <cit.>.

Thus, a case 2) is impossible, and a case 1) gives the desired
statement of the theorem. 


In addition to our article, we will illustrate our results with some
simple examples.


Consider a family of plane mappings f_n(z)=z^n, n=1,2,… ,
z∈ B^2={z∈ C: |z|<1}. The mappings f_n have
a bounded distortion as smooth mappings whose dilatation is equal to
one. So f_n satisfy inequality (<ref>) for Q(z)=N(f_n,
 B^2), where, as usual, N is a multiplicity function
determined by the ratios


    N(y, f,  B^2) = 
    card {z∈ B^2: f(z)=y} ,      N(f, 
    B^2) = sup_y∈ C N(y, f,  B^2)


(see, e.g., <cit.> or
<cit.>). All mappings f_n are discrete and
open, in addition, preserve the boundary of the unit disk and,
therefore, are closed (see, e.g., <cit.>). The
mappings f_n also fix the point 0, so they satisfy all the
conditions of Theorem <ref> except one: the existence of the
integrable function Q in (<ref>) independent on n. As a
result, the family f_n is not equicontinuous at the boundary of
the unit disk.



A simple example of conformal automorphisms
f_n(z)=z-n-1/n/1-zn-1/n of the unit disk
on itself, for which Q(z)≡ 1 (see, e.g.,
<cit.> or <cit.>), shows
that the violation of the condition  dist (f^ -1(A),
∂ B^n)⩾ δ in Theorem <ref> is
also an obstacle to performing the desired inequality (<ref>)
on the boundary of the unit disk. We observe that the family f_n
is not even equicontinuous, and thus more, logarithmically
Hölder continuous at the boundary points of the unit disk.



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Evgeny Sevost'yanov 

1. Zhytomyr Ivan Franko State University,  

40 Bol'shaya Berdichevskaya Str., 10 008  Zhytomyr, UKRAINE 

2. Institute of Applied Mathematics and Mechanics

of NAS of Ukraine, 

1 Dobrovol'skogo Str., 84 100 Slavyansk,  UKRAINE

esevostyanov2009@gmail.com



