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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 16 Выпуск 2 (2015)
УДК 511.6
О СЛЕДАХ И ДИСТАНЦИЯХ В АНАЛИТИЧЕСКИХ ФУНКЦИОНАЛЬНЫХ ПРОСТРАНСТВАХ В Сп И ИНТЕГРАЛАХ МАРТИНЕЛЛИ - БОХНЕРА
Р. Ф. Шамоян, С. М. Куриленко (г. Брянск)
Аннотация
В этой работе мы приводим аналоги наших многочисленных результатов о следах и дистанциях в аналитических функциональных пространствах в Сп, полученных ранее, в терминах интегралов и ядер Мартинелли - Бохнера. Это первые результаты такого типа в терминах этих интегралов и ядер. Также нами будут обсуждаться некоторые новые утверждения для интегралов типа Мартинелли - Бохнера, связанные с классами типа Гельдера и точками Лебега.
В последние годы в большом цикле работ первого автора был получен ряд новых точных результатов, связанных со следами и расстояниями в различных функциональных пространствах. Во всех этих работах важную роль играют свойства ядер типа Бергмана и интегральные представления типа Бергмана. В этой статье мы получим некоторые аналоги этих результатов в терминах более общих интегральных представлений и более общих ядер в аналитических функциональных пространствах большей размерности. Это так называемое интегральное представление Мартинелли - Бохнера и ядра Мартинелли - Бохнера в Сп.
Наша работа состоит из трех частей. В первой части мы обобщаем полученные ранее результаты по следам. Во второй части мы получаем оценки функции расстояния в терминах ядер Мартинелли - Бохнера и интегралов Мартинелли - Бохнера. В третьей части представлены результаты для интегралов Мартинелли - Бохнера, связанные с классами Гельдера и точками Лебега. Эти вопросы естественно возникают из недавней серии работ первого автора о многофункциональных аналитических пространствах и связанными с ними вопросами.
Наши доказательтсва модифицируют методы и рассуждения известных ранее результатов и теорем для случая интегралов и ядер типа Мар-тинелли - Бохнера.
Ключевые слова: Интегралы и ядра Мартинелли - Бохнера, аналитическая функция, следы, дистанции, псевдовыпуклые области.
Библиография: 20 наименований.
UDC 511.6
TRACES AND DISTANCES IN ANALYTIC FUNCTION SPACES IN Cn AND MARTINELLY — BOCHNER INTEGRALS
R. Shamoyan, S. Kurilenko (Bryansk)
Abstract
In this note we provide some analogues of our numerous recent results on traces and distances in terms of Martinelly — Bochner integrals and kernels. These are first results of this type in terms of such kernels. Some assertions for Martinelly — Bochner integrals related with Holder classes and Lebegues points will be also discussed.
In recent years various new sharp results on traces and distances were provided in a big series of papers of the first author. In all these papers properties of Bergman-type kernels and Bergman-type integral representations are playing a critical role. The intension of this paper to find some analogues of these results in terms of or with the help of more general integral representations and more general kernels in analytic function spaces in higher dimension so-called Martinelly — Bochner integral representations and Martinelly — Bochner kernels in Cn.
Our work consists of three parts. In the first part we partially generalize our results on traces. In the second part we provide estimates of distance function in terms of Martinelly — Bochner kernels and Martinelly — Bochner integrals. In the third part we present results on Martinelly — Bochner integrals related with Holder classes and Lebegues points. This type of issues arise naturally in view of recent series of papers and new results of the first author on multifunctional analytic spaces and related issues.
In our proofs we modify the methods of earlier results and theorems for the case of Martinelli-Bochner integrals and kernels.
Keywords: Martinelly — Bochner integrals and kernels, analytic function, traces, distances, pseudoconvex domains.
Bibliography: 20 titles.
Introduction
In recent years various new sharp results on traces and distances were provided in a big series of papers of the first author (see [4], [9], [10], [11], [12], [13]). In all these papers properties of Bergman type kernels and Bergman-type integral representations are
1This work was supported by the Russian Foundation for Basic Research (grant 13- 353 01-97508) and by the Ministry of Education and Science of the Russian Federation (grant 1.1704.2014K).
playing a critical role. The intension of this paper to find some analogues of these results in terms of or with the help of more general integral representations and more general kernels in analytic function spaces in higher dimension so-called Martinelly — Bochner integral representations and Martinelly — Bochner kernels in Cn (see [5]). Some new results on Martinelly — Bochner integrals related with the Lebegues points and Holder functional classes will be also presented.
1. Traces in pseudoconvex domains and Martinelly — Bochner integrals.
In this section we partially generalize our results on traces. The goal of this section is to provide a class of analytic functions on products of bounded strictly pseudoconvex domains D x- ■ -x D so that their traces allows Matrinelly-Bochner integral representation. To be more precise we first need standard definitions (see [5] and references there).
We define the unit ball in Cn by B. Let U be an open set in Cn, we say f € Ck(U) if f is complex valued function and f(k) € C°(U) = C(U). We denote by H(U) the set of all analytic in U functions. Um = U x ■ ■ ■ xU, m > 1 is a product domain and H (Um) is a space of all analytic function on (Um) (see [4], [9], [10], [11], [3]) or Dm. Using definitions of analytic function spaces in a domain, we define analytic function spaces in product domains in a natural way (see [10]).
We denote by D in Cn a region with dD boundary of Ck class in some neighborhood of closure of D and dp = 0 on dD. The p function we call as usual the defining function of D (see [10]). We consider an outer differential form (Bochner-Martinelly kernel) U(Z,z) of (n,n — 1) type of the following form (see [5])
U (Z, z) = Y ("I)'-1 ,Zfc fk d([K] A d(
vs ! (2ni)n ' |Z - z\2n SL J s
where df[K] = df1 A ■ ■ ■ A dZk-1 A dZ,k+1 A ■ ■ ■ A dfm, d( = d(1 A ■ ■ ■ A d(n. For n = 1 we have Causchy kernel U(Z,z) = t^i (z—z). Obviously coefficients of U are harmonic in Cn\{z} and dçU(Z,z)=0.
Moreover (see [5]) if g(Z, z) is a fundamental solution of Laplace equation then we have
nn
U(Z,z) = E("1)fc-1Tdgdf[K] A dZ = ("1 )n-10cg A £ dff[K] A dZ[K] k= 1 °Zk k=1
where d = En=1dZkdzk.
We denote various constants in estimates in this paper by C or by C with indexes, these constants are independent of functions in estimates.
Theorem 1 (A. (see [5]).). Let D be a bounded region in Cn with partially-smooth boundary and let f be holomorphic in D of C(D) class then
if (Z )U (Z.^if'-f (1)
JdD [0, z z / D.
This is a Martinelly — Bochner integral representation.
This theorem was obtained by Bochner then by Bochner then by Martinelly independently and by different methods. This formula is now classical and can be seen in various textbooks on complex analysis. Note for n = 1 we have classical Causchy formula, but for n > 1 we can easily see that the U(Z,z) is not analytic by z or Z. We will need the mentioned formula for Hardy spaces. We need some definitions. Below we omit the index of space if it is zero. Let D be a bounded region, dD € C1+a, a > 0. We say that analytic function f belongs to Hardy space Hp(D), p > 0 if
sup/ |f (Z - ev (Z ))|p da(Z) < x, e>0 JdD
where da is a Lebegues measure on dD. And v(Z) vector field of outer normals to dD (see [5]). For bounded pseudoconvex D domains we have another definition, for a defining function p, D = {z € Cn : p(z) < 0}; let De = {z € D : p(z) < -e} for e > 0. Then
Hp(D) = {f € H(D) : sup / |f(Z)|pdae < <x>};
e>0 JQDs
0 < p < x>.
We define the same space in product domains in a natural way as we define Hardy spaces in polydisk using definitions in the unit disk see [3], [4].
Proposition 1 (B. (see [5]).). Let D be bounded strongly pseudoconvex domain with smooth boundary. For allp >1 and all f, f € Hp(D) Martinelly — Bochner representation is valid.
Proof of proposition B.
Note first (see [5]) our proposition is valid for all bounded domains with dD € C1+a, a > 0. For each f, f € Hp, p > 1, f has boundary values almost everywhere on dD so that these values are in Lp(dD). This is a classical fact together with representation via boundary values
f (z)= i f (Z )P (Z,z)da,
JdD
where P is Poisson kernel of D domain.
Note the G Green function admits representation G(Z,z) = g(Z,z) + h(Z,z), where g is a solution of Laplace equation and where h for all fixed z € D is a harmonic function in D of C 1+a(D) class hence
n dh
P(Z, z)da = U(Z, z)/dD + Et-l)'-1^-^] A dZ/dD = $1 + $2
k=1 °ZkdZ
but note
n .k_ 1 dh
r " dh -
(f(0)^2(0= / f(oEt-^VdCK] a dz
k=i
= jd f (Z)d(p((-1)k-1dhd<K a d^j =0.
This follows directly from the fact that
t/(-i)k-1dhd({K ] A dZ k=1 dZk
is a closed form (see [5]). And this gives directly what we need for each f, f € Hp(D), p > 1 Martially-Bochner represention is valid. □
The intention of this section to show such type results for other spaces in unit ball or general bounded strictly pseudoconvex domains D, based on our previous results and on embeddings from [1].
The natural question is the following:
Let X be a space of analytic functions, X c H(Dm). Let also
TrX = {f(z,...,z),f € X}.
Can we say there is a function g0, g0 € Tr X so that it admits Martinelly — Bochner integral representation (1). Below we provide a general scheme of a solution for case of unit ball and bounded domains.
We define for a Da — differential operator in pseudoconvex domain D (see [1]), 0 < p,q < x Hardy-Sobolev and Bergman-Besov spaces of analytic functions
Ha ß(D) = f € H(D) :Sup( \Daf (Z)|pdac < < , ß >0, a > 0.
£>0 \JOD
rro ' - * *
A?k(D) = { f <= H(D) : £ / ( / iDViZWdacYr^drKx , 5 > 0>
{ \a\<kJ0 \JdDr /a> 0
We define same spaces in product domains in a natural way as we did in polydisk using definitions of spaces in the unit disk see [2], [3], [4].
We give a typical result in this direction on traces. Note for case of unit ball we have (see [2], [3], [4]).
Theorem 2 (C.).
1. Letp <1, 0 > a >0, then Ap'p +(p ) (B) c Trace Hpa p(Bm);
2. Letp < 1 a, >0,t > ^mp then A™^—n—a, (B) c Trace (A™(&"));
= aj + 1, j = 1,... ,m.
Obviously we have also (see [1], [3], [4]) the following embeddings (0<p < x, p > q and s1, s2 > 0).
AP(S2—s1),s2 (B) CK (B) CHP(B) CHq(B)
then we have (Ap>p = Ap, Hp0 = Hp)
APa,p (B) C APa^(B) = APa(B ),0 >0.
Now according to proposition B for any analytic f function, f € Hp(B), 1 < p < oo the Martinelly — Bochner integral representation is valid. This in combination with embeddings above gives
Theorem 3.
ai ß then 'I ' r ace H (
1. Let p < 1, a > P then TraceH^, ~(Bm) contains a f function for which
f (z)=/ f (Z)U(Z,z);z € D (M)
JdD
2. Let p <1, aj > 0, Pj =aj + 1, t > , j = 1,---,m then Trace (Bm) contain a f function for which (M) integral representation is valid.
Remark 1. Note previously this assertion was known for Bergman representation and Bergman kernels.
The case of general D domains (pseudoconvex) can be considered similarly. It is based on our recent results from [11] and [12] and similar embeddings for pseudoconvex domains from [1]. Analytic Herz type spaces can be considered similarly (see [13]).
We need some definitions to formulate a theorem we need. Let K(w) be entire function of one variable w = u + iv,K : R ^ R, K(u) = 0 for all u € C. Let
V(T w^ ^ dn-2 T K(ia + Im zn) CnK(Im Zn)$(C,z) = n n_2 Im-
dsn 2 a(ia + Im (Zn — zn))'
n— 1
s = E IZk — Zk|2, a2 = s + Re(Zn — Zn), Cn = (—1)n—1(2ni)n. k=1
Theorem 4 (D1. (see [5]).). Let n> 1 then $(Z, z) = g(Z,z) +h(Z,z) where g is fundamental solution of Laplace equation, h(Z,z) is harmonic by Z in Cn for all fixed z.
We need an extension of Martinelly — Bochner integral to unbounded domains D in Cn then for tubular domains the same problem can be posed and solved based on our recent results on traced in Bergman type spaces in tubular domains over symmetric cones with smooth boundary.
If f € H(D) n C(D) and if h € C'(D) and h is harmonic by Z for all z € D. Then jljl form is closed in D
m
jh = k=i( —1)^ dkdZ[K] A dZ
and
/ f (Z)jh(Z) = 0, f € H(D) n C(D).
JdD
Note in bounded domain then we have (see [5])
f (z)=/ f (Z)[U(Z,z)+ (Z)],z eD (Gl)
JdD
with coefficients (K x K) matrices x,t € Rm.
Remark 2. Note if K = 1 then $ = g.
Theorem 5 (D2. (see [5]).). If D is unbounded domain in Cn, n > 1 with smooth boundary, f € H(D) n C(D) where B(0, R) is a ball of radius R then if
Yimii f (Z )Q(Z,z) )=0 (K0)
yJdD\B(0,R) J
for all fixed z € D then
f(z)=i f(Z)Q(Z,z) (K)
JdD
where
n(Z,z) = ±(-1)k—1 (H) dZlk] A dZ.
Proof.
Just use G1 for domain D n B(0, R) then pass R ^ <x>.
To formulate our theorem 6 we will need basic facts of theory of analytic function spaces in tubular domains over symmetric cones (see [20]).
A subset Q of Rn or V, so that dim V = n to be a cone if Ax € Q, for all x € Q, X > 0, if Ax + ¡iy € Q for all x,y € Q, X,^ > 0 then it is convex. Let in addition Q* = {y € Rn : (y/x) > 0, yx € Q\{0}} and Q* = Q. This type open cone is selfdual (Q* is dual cone).
Let G(Q) = {g € Gl(Rn) : gQ = Q}, where Gl(Rn) denotes the group of all linear invertible transformation of Rn. If for all x,y € Q, y = gx, for some g € G(Q) then our open convex cone Q is homogeneous, if also Q* = Q then it is symmetric cone.
If the equation Q = Q1 + Q2 is not possible for each Vi C Rn, V2 C Rn, then our cone is irreducible, here Vi = 0, i = 1,2 (Q1, Q2 are symmetric cones), where also Qi C Vi, i = 1, 2.
We need also determinant Al(Im z), z € Cn, t € (0, x). We fix V — a simple Euclidean Jordan algebra with rank r.
If x € V, m(x) = min{k > 0 : (l,x,x2,... ,xk) are linearly dependent}, then 1 < m(x) < dim V and r = max{m(x) : x €V}, we say rank of V is r.
According to spectral theorem if V has rank r, then x = YZ=1 Xici; Xj, € R; c — are elements of so called Jourdan frame, and {Xi} are determined uniquely by x (with their multiplicities). We fix now a Peirce decomposition of V = ®1<i<j<rVij; (we formally look at V as a space of symmetric matrices (Vij), Vu = Rci, where R is a special mapping, Vij = V(d, 1/2) n V(d, 1/2) = {x € V : CiX = CjX = f}, i < j, dim Vij =d = We denote by Pj the orthogonal projection of V onto Vij for i < j. Finally we denote by Aj(x), j = 1,... ,r, the principal minors of x € V with respect to the fixed Jordan frame {c1}... ,cr}. That is Ak(x) is the determinant of the projection Pkx of x in the Jordan subalgebra V(k) = ®1<i<j<rVij.
It is well-known that Q = {x €V : Ak(x) > 0 : k = 1,... ,r}. We have also Ak(mx) = Ak(x), x €V, m € Z+, m > 0. See other properties of Ak in [20].
We define As(x) = n As/—sj+1(x) = A11—s2(x)... A?r(x), x € Q, s € Cr. We have j=1
r
that |As| = A(Im z) and As ^r=1 a^ = Yl a!si; ai > 0, i = 1,... ,r.
i=1
Let Q be an irreducible symmetric cone in the Euclidean space V, and Tq = V + iQ the corresponding tube domain in the complexified space VC, TQ7" = Tq x .. .Tq. As combination of this theorem D2 and theorems on traces [16] in tube we have
Theorem 6. Let Tq be tubular domain over symmetric cone, then let
Apa(Tm) = If €H(Tm):/ .../ |f(zi,...,zm)|p n ^a(Imz3)dv(z3) < to I ,
I JTq JTQ j=1 I
0 < p < to, a > —1. Then TraceAa = {f (z,... ,z), f € AOt(Tm)} contain a function, so that representation (K) is valid if (K0) holds.
Similar results can be obtained based on integral representation based on second Green formula (see [5]) and our recent results on traces of Bergman harmonic function spaces in the unit ball (see [17]) B = {|x| < 1} c Rn and R++1. We give more detail on this integral representation.
m
Let in Rm, m > 1 us consider a differential operator in Rm A = E Ajdj, where Aj
j=1 J
are matrices of order l x K on C, Sm = {|x| = 1,x € Rm} if
AJAj + Aj Ai ={» ^ =J [2Ik ifi =j.
where i,j € [1, n], A* = AT complex conjugate of Ai; Ik is unit matrix. Then A is Dirac operator. For solutions of Dirac operator we have analogue of Martinelly — Bochner theorem
For definition of vector function space in theorem E we refer the reader to [5].
Theorem 7 (E. (see [5]).). Let A be Dirac operator, D c Rn is a bounded domain with smooth boundary then if f € [C(D) n C 1(D)]k, Af = 0 then
LUA(^x)f(t)={0(x) fx H
where Ua is a special Martinelly — Bochner type kernel in Rm see [5] a differential form type of (n — 1) type
mm
uA^ = vowm) SB-1)'-1'A*A. j—j ^
3=1'=1
Details of proofs of these assertions analogues of previous theorems 3, 6 for harmonic function spaces based on theorem E we leave to readers (see [12], [13]).
2. Distance functions and Martinelly — Bochner integrals
The goal of this section to provide estimates of distance function in terms of Martinelly — Bochner kernels and Martinelly — Bochner integrals based on methods and results of [9], [10] and [11] and our earlier work cited there.
Note previously we obtained such type theorems in various spaces of analytic and harmonic functions of one and several variables in various domains in terms of Bergman kernel and Bergman-type projections (see [9], [10], [11] and various references there). The line of our proof is rather similar however to those simpler cases but it is based on some results on more general kernels as Martinelly — Bochner kernels and Martinelly — Bochner integrals (see [5] and references there).
Let D be bounded domain in Cn. X, Y be quazinormed subspaces of H (D), where H(D) is the space of all analytic functions in D. Let X cY, let f € Y. We search for those estimates of a function distY (f,X) which generalize previously known such type estimates obtained via similar Bergman type kernels.
We have the following result in this direction. First if X c Hp, p > 1 then the problem of estimates of distHp (f,X), f € Hp appears naturally. In our previous papers to role of Bergman type kernels for this problem was critical.
Let Hp c X, X c H (Dm), then let f € X. We also can consider a problem of finding estimates of function distX(f,Hp), 1 < p < oo, Hp = Hp(Dm) on product domains in terms of Martinelly — Bochner kernels or integrals.
We have following results following arguments of proofs of our previous papers [10],
[11], [14], [15].
Let X c Hp(Dm), 1 < p < o, m € N, we assume X is a quazinormed subspace of H(Dm). To estimate distHp(f,X), f € Hp, 1 <p < oo we follow our arguments from [14] in the unit disk {|z| < 1} and unit ball.
Since f € Hp(Dm), then applying Martinelly — Bochner representation by each variable
/» a A
f(z1,...,zm)= f(()U(C,z)= ... f((i,...,(m)U U (Zj, Zj);
J(dD)m J (dD) JdD j=
Zj € D, j = 1,... ,m where
Hp(Dm) = {f € H(Dm) :sup [ ... ! If((n)lpda < o}, 1<p< o
R>0JdD JdD
is a Hardy space on products of D domains, m € N, D is bounded in Cn, dD € C1+a, a > 0 (note these are Banach spaces) and where ZR = ^ — Rv(Z1), ■ ■ ■ ,Zm — Rv(Zm)), v is a vector field of outer unit normals to dD.
Let X*f = {Z € d(Dm) : *(Z)lf (Z)| > 4; * € C 1(d(Dm)) fixed. Then f = f 1 + f2
- m „ m
fi(z)= f (Z )n U(Zj,z); f2(z)= f (Z )n U(Zj,z).
Jx*f j=i Jd(Dm)\X*f j=i
Assuming f2 € Hp and it is norm is less than e, we get
distHP (f,X) < c inf i e > 0:\\ [ f (Z U (Zj ,z)\\x (Dm) < to i Jxf,f j=1
Since distHP (f, X) < c\\f — f1\HP = \\f2\HP.
theorem 8. Letp >1. Let tf € C 1(d(Dm)), ^ > 0, be fixed function, let f € Hp(Dm), J f (Z) 11 m=1 U (Zj, z) € Hp and it is norm is less than e. Then d(Dm)\Xf/
distHp (f,Y) < c inf { e > 0 :
lx *
f(0 UU(Zj,z)
,f j=i
< TO
Y (Dm)
where
Xtf = {Z ed(Dm):f(Z)|f(Z)|> e},
where Y c Hp is a quazinormed subspace of Hp(Dm) for any bounded D domain with dD € Ca+1, a > 0 boundary.
Remark 3. Similar results are valid for Martinelly — Bochner type integrals and harmonic function spaces in R++1 and unit ball in Rm. See [17] where similar problems where solved via Bergman kernel.
Theorem 8 is a typical assertion, other assertions of such type in terms of U(z, Z) kernels can be also formulated similarly.
3. Some remarks on Martinelly — Bochner integrals related with Holder classes and Lebegues points.
The goal of this section is among other things from [5] related with expressions like (z1) —(z2)| to the case of two functional similar type expressions like (z1) —(z2)|, where $ is a certain operator, f, g are certain functional class members, z1, z2 € D, where D is a fixed domain in Cn.
This type of issues arise naturally in view of recent series of papers and new results of the first author on multifunctional analytic spaces and related issues (see [8], [17] and various references there).
To obtain this natural tranfsormation of known results from one functional case to two functional case we follow the proof of [5] and at the same time modify proofs from there to our case. First we provide certain equalities (A), (B), (C), (D), which can be checked directly, then use them in our proofs. We alert the reader that in those places where arguments are close or the same as in [5], we omit details making the exposition of this note rather concise we note also that our problems we considered below is an attempt to provide natural extensions of known one functional results to two functional case.
Let
(Mf )(z+)=/ f (Z )U (Z,z+);
JdD
(Mf )(z-)=/ f (Z )U (Z,z-); JdD
then we have following equalities
(Mf)(z+) — (Mg)(z-) — f(z) — g(z) = / (f(Z) — f(z))(U(Z, z+) — U(Z, z-)) —
JdD
— f g(Z)U(Z,z-) — f g(z)(U(Z,z+) — U(Z,z-))+/ f(Z)U(Z,z-)= (A)
./dD ./dD ./dD
= / (f(Z) — f(z))(U(Z,z+) — U(Z,z-)) + M(f — g)(z-) — g(z); JdD
(Mf)(Z+) — (Mg)(z-) = / (f(Z) — f(z0))U(Z,Z+) —
JdD
I (f(Z) — f(z0))U(Z,Z-) + f(z0)/ (U(Z,Z+) — U(Z,z-)) — / g(Z)U(Z,Z-) +
,/dD ./dD ./dD
(B)
+ / f(Z)U(Z,Z-)=I (f(Z) — f(z0))u(Z,z+) — f (f(Z) — f(z0))u(Z,z-)+
,/dD JdD ./dD
+f(z0)/ (U(Z, Z+) — U(Z, Z-)) + M(f — g)(Z-); JdD
we omit technical calculations leaving them to readers. Note all equalities are based on simple properties of Martinelly — Bochner integrals (see [5]). It can be checked directly that
f (f(Z) — f(0))U(Z, z) — f (g(Z) — g(0))U(Z, 0) =
JdD JdD\B(0,e)
= f (f (Z) — f (0))(U (Z, z) — u (Z, 0)) + f (f (Z) — f (0))U (Z, z)—
JdD\B(0,£) JdD nB(0,e)
f (g(Z) — g(0))u (Z, 0) + f (f (Z) — f (0))u (z, 0) =
' dD\B(0,£) JdD\B(0,£)
/ (f (Z) — f (0))(U (Z,z) — u (Z, 0))+ (C)
' dD\B(0,£)
W (f (Z) — f (0))U (Z, z) + / ((f — g)(Z) — (f — g)(0))U (Z, 0);
JdDnB(0,£) JdD\B(0,£)
a))U (Z z1) , f f (Z) g(z2))U (Z z2) =
f (f (Z) — f (z1))U (Z, z1) + f (f (Z) — g(z2))U (Z, z2)
JdD\ox JdD\ox
'dD\as JdD\as
f (g(Z) - 9(z2))(U(z, z2) - U(Z, z1)) + (f (z1) - f (z2)) f U(Z, z1)+ (D)
JdD\as JdD\as
+ / (g(Z) - g(z2))U(z, z1) - i f (z)u(z, z1) + f (z2) i u(z, z1) =
JdD\as JdD\as J dD\as
= i (g(z) - g(z2))(u(z,z2) - u(z,z1))+
JdD\as
+(f (z1) - f (z2)) i U(z, z1) + Af,g + Bfg + Cfg = JdD\as
= / (g(z) - g(z2))(U(z, z2) - u(z, z1)) + (f (z1) - f (z2)) i u(z, z1)+
JdD\as JdD\as
+ f ((g - f)(z) - (g - f )(z2))u(z, z1);
JdD\ox
'dD\as
where as is arc on dD and B(0,e) is a standard ball in a domain.
Remark 4. For those cases when g = f all equalities (A), (B), (C), (D) can be seen in [5] (see lemma 5.3, theorem 6.1, 6.2).
Let Q be a domain, we say f is Holder class a if If (Z) — f (z)| < c|Z — z|a, Z,z ^ a > 0. We consider bounded domains with smooth boundary, f € Ll(dD), D = {z € Cn,p(z) < 0}, p € Cl(Cn), dp = 0 on dD. Let V(dD) be a neighborhood of dD. We assume f can be extended to V(dD). Consider integrals
*f(z)=f (f(Z) — f(z))U(Z,z).
J dD
Note if z € dD then the integral has no singularity, if z € dD, then
If (Z) — f (z)I x IU(Z, z)I < cIZ — zIa+l-2nda(Z) (2)
if f is a Holder class function. The natural question is let f is of Holder class a in V(dD) then can we say §f (z) is of same class a. The answer is positive (see [5]).
The next more general question is if f, g are of Holder class a then what can we say on I$f (z\) — &g(z2)I, zl,z2 € V(dD). We will use formula (D) for our proof. But first note that the following simple observation is valid.
Let zl, z2 € V(dD), Izl — z2I =5,5 is small, if B(zl, 25) C V(dD),as = dDnB(zl, 25). Then (see [5])
I / (f(Z) — f(zj)U(Z,zj)I< clf IZ — zj Il+a-2nda <C25a,j = 1,2.
J ax J ax
This follows from (2) directly and the fact that as is smooth. So for f = g case it is remains to estimate integrals by D\as to get estimate for I$f(z\) — §f(z2)I. But for I$f(z\) — <&g(z2)I we have formula (D) and estimates
Af = I i (f(Z) — f(zl))U(Z,zl)I< ~d5a, 0<a< 1
J ax
Bg = | / (g(Z) - g(z2)U(Z, z2)| < CiSa, 0 < a < 1.
Since obviously we have
IdD
|$f(zi) - $g (Z2)| =
(f (Z) - f (zi))U(Z,zi) - f (g(Z) - g(z2))U(Z^)
JdD
<Af +Bg +C
f,g ;
i (f (Z) - f (zi))U(Z, zi) - / (g(Z) - g(z2))U(Z, Z2) JdD\as JdD\az
dD\as
Using (D) we have now that cf,g < c1 + c2 + c^ + f + f Note c^ + f + f
W,((g — f)(Z) — (g — f)(z2); u(Cz1) = f(z1,z2).
Remark 5. Note we also have
So we have that
f,g (zi,z2)| < №g-f (z2) | + C^-f, i-f = f ((g - f )(Z) - (g - f)(z2))(u(Z, zi) - u(Z, z2)).
|$f(z1) — $g(z2)| < c] + Z]-f + f + c&a + Z|$g-f (z2)|.
We will stop on estimate with $g,f. We have c2 < cba since | JdD\as(U(Z,z1))| < 1, see [5]. We estimate c1 and Z)g-f. This is based on simple estimates of U(Z,z2) — U(Z,z1), z1,z2 € V(3D).
We have that (see [5])
/ (g(Z ) - g(z2))(U(Z,z2) - U(Z,zi)) < cS / ^ - zi\a-2nda <c(5a), JdD\a* JdD\as
f ((f - g)(Z) - (g - f)(z2))(u(Z, zi) - u(Z, z2)) < JdD\as
r ^ - zi|a-2nda < C(Sa).
< CS
ldD\as
Since JdD\ag ^ — z1|a-2ndCT < cSa-1 (see [5]). So finally we have an estimate
and
|$f(zi) - $g(z2)| < cöa + |$g,f(zi,z2)|, 0 < a < 1 |$f(zi) - $g(z2)|-|$g,f(zi,z2)|< cSa.
Note if f = g this obtained in [5] and [6]. We now formulate the final result.
Theorem 9. Let f, g be of Holder class a, 0 < a < 1 in V(dD). Then in V(dD) we have the following estimate
(zi) — (z2)\< c5a + №gtf (Z1,Z2)I; zi,z2 £ V(dD),
Iz1 — z2l = 5, 5 < 50, for some fixed 50, 50 > 0.
We assume Mf (z) = fdD f (Z)U(Z,z), z £ dD, f £ C 1(D) below. Now we use (A), (B), (C) to obtain such type results for other mentioned assertions from [5]. Note to get the proof we must follow one functional proof from [5] modify it to use mentioned equalities (A), (B), (C).
Let D be bounded domain with C1 boundary and f £ L1(dD) and z0 £ dD and Vzo be a cone with z0 as peak (vertex), with conic angle equal or less to n/2. Let z £ D n Vzo. We say z0 is a Lebegues point for f if (see [5])
H™e1-2n i If(() — f(z°)\da = 0.
e—+0 JdDnB(z0,e)
We will be using (C) and arguments from proof of theorem 5.2 [5]. We note first (see
[5])
/ if (z) - f mu (z,z)
<K if (Z) - f(0)ida ^ 0;
JdDnB(0,£)
jr C2 n
K = —r—- as £ ^ 0.
£2n—l
This and (C) lead to theorem. So based on (C) we have the following.
Theorem 10. Let z £ D n Vzo be Lebegues point for f and g, f,g £ L1(dD). Then
imj (f(Z) — f(z0))u(z, z) — f (g(Z) — g(z0)U(Z, z0)) =
z^z0 Jqd JdD\B(z0,\z-z0\)
z&Vzo
= Um0 i ((f — g)(Z) — (f — g)(z0))U(Z,z0).
z \z JOD\B(z0,\z-z0\)
zevz0 u
Using (A) and (B) and following the proof of theorem 6.1, 6.2 (see [5]). We can have assertions concerning
lm Mf(z+) — Mg(z-) — f(z0) — g(z0),
z±—>z0
where, f,g £ L1(dD), z0 is a Lebegues point of f and g simultaneously and
i \Mf (z+) — Mg(z-) — f (z) — g(z)\pda
J dD
if D is bounded, dD £ C1, f £ Lp(dD), p >1, v(Z) is a unit vector of outer normal to dD on Z, z+ = z — ev(z), z- = z + ev(z). We omit details here. So based on (B) we have the following
Theorem 11. Let z0 be Lebegues point of f and g, f € Li(dD), g € Li(dD), z+ € Vz0n D, z- = Vz0n (Cn\D), a|z+- z0| < |z- - z0| < b|z+- z0|, a,b € R, a < b, zo € dD, D is bounded, dD € C1, z0 is a peak of cone Vz0, cone angle is fi < n/2. Then
lim (Mf (z+) - Mg(z-)) = f (zo) + lim (M(f - g)(z-)).
z±—>z0 z±—>z0
Remark 6. For f = g this is a theorem from [5]. Based on (A) we have the following theorem 12.
Theorem 12. Let D be a bounded domain with smooth boundary. Let f,g € Lp(dD), p >1. Let z+, U_ belong to D and Cn\D for e small enough. Then
Af,g = f |Mf (z - ev(z)) - Mg(z + ev(z))|pda < Jdd
< c[ f ^da + / M(f - g)(z_)|pda; JdD JdD
lim / |Mf (z - ev(z)) - Mg(z + ev(z)) - f (z) - g(z)|pda < +0 JdD
<lim||M (f - g)(z-)\\lp + M\lp . £—0
Remark 7. At the end of this paper we find interesting to suggest another idea related from one hand to Martinelly—Bochner integrals from the other hand to our recent work on traces in analytic functions Bergman spaces. In [7] the role of expanded Bergman projection based on expaned Bergman kernel was crucial for solution of certain problems related with traces of anlytic functions in polyballs. It will be nice to study their analogues expanded Martinelly—Bochner kernels and based on them expanded Martinelly—Bochner integrals based in particular on the following simple idea.
Let as usual g((, z) = cn ^_zpn-2, n > 1, p be defining function of D c Cn, D = {p < 0}, dp = 0 on dD, p € C
Pk,s be a complete orthonormal system of homogeneous harmonic polynomials in L2(S),
n
* is well-known Hodze operator for differential forms see [5] a(() = E (- 1)k-iZkd([k] Ad(.
k=i
We suggest to study kernels of type on product domains (expanded Martinelly — Bochner kernels)
n Q m
U(Z,zi,...,zm) = ]T(-1)k_i— ng(Z,z3)dZ[K] adZ k=i dZk j>i
U(Z ) (n - 1)! ^ dp m (Uk - U) d
U(Z, zi,...,zm) = — k=l QZ kjPi da
U(Z,zi,...,zm)= |- £ Pk's(zi)
n + k - 1
k,s
,d Pks(Z)
|Z |2n+2k_2
E
k,s
Pk,s(zm) П + k — 1
Pk,s(Z )
2n+2k-2
IZI
Note for m = 1 these kernels coincide with classical Martinenelly—Bochner integrals (see [5]). Same idea used for Bergman kernel in series of papers of first author (see [7] and references there).
B(zi,... ,zm,w) = Y\
(i -ЫУ
У i1 ~
<xj +2 '
aj > —1, Zj,w € D, j = 1,... ,m.
These types of extensions of Bergman kernel was considered by author in [7].
j
4. Conclusion.
In our paper we got some analogues of our numerous recent results on traces and distances in terms of Martinelly — Bochner integrals and kernels. These are first results of this type in terms of such kernels. We also discussed some assertions for Martinelly — Bochner integrals related with Holder classes and Lebegues points.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. J. Ortega, J. Fabrega Mixed-norm spaces and interpolation // Studia Math. 1994. Vol. 109, № 3. P. 234-254.
2. R. Shamoyan On some characterizations of Carleson type measure in the unit ball // Banach J. Math. Anal. 2009. Vol. 3, № 2. P. 42-48.
3. R. Shamoyan, O. Mihic On traces of holomorphic functions on the unit polyball // Appl. Anal. Disorete Math. 2009. Vol. 3. P. 198-211.
4. R. Shamoyan, O. Mihic On traces of Qp type spaces and mixed norm analytic function spaces on polyballs // Siauliau. Math. Seminar. 2010. Vol. 5, issue 13. P. 101.
5. А.М. Кытманов Интеграл Мартинелли - Бохнера и его приложения. Новосибирск: Наука. 2002. 240 с.
6. Е. М. Чирка Аналитическое представление CR-функций // Мат. сборник. 1975. Т. 98(140), № 4(12). С. 591--623.
7. R. Shamoyan, O. Mihic, On traces of holomorphic functions on the unit polyball // Appl. Anal. Discrete Math. 2009. Vol. 3. P. 198--211.
8. M. Arsenovic, R. Shamoyan On embeddings, traces and multipliers in harmonic function spaces // Kragujevac Math. Journal. 2013. Vol. 37. No. 1. P. 45-64.
9. R. Shamoyan, M. Arsenovic On distance estimates and atomic decompositions in spaces of analytic functions on strictly pseudoconvex domains // Bulletin of the Korean Mathematical Society. 2015. Vol. 52, No. 1. P. 85-—103.
10. R. F. Shamoyan, S. M. Kurilenko On a sharp estimate for a distance function in Bergman type analytic spaces in Siegel domains of second type // Mathematica Montisnigri. 2014. Vol. XXX. P. 5-16.
11. R. F. Shamoyan, S. M. Kurilenko On extremal problems in tubular domains over symmetric cones // Issues of Analysis. 2014. Vol. 3(21), N. 1. P. 44-65.
12. R. Shamoyan, O. Mihic On Traces in Some Analytic Spaces in Bounded Strictly Pseudoconvex Domains // Journal of Function Spaces. 2015. Vol. 2015, 10 p.
13. R. Shamoyan, S. Kurilenko On traces of Herz type and Bloch type spaces on pseudoconvex domains // Issues of Analysis, Petrozavodsk, 2015.
14. R. Shamoyan, O. Mihic On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball // Boletin de la Asociacion Matematica Venezolana. 2010. Vol. XVII, No. 2. P. 89-103.
15. R. Shamoyan, O. Mihic, On new estimates for distances in analytic function spaces in unit disk, polydisk, and unit ball // Сиб. электрон. матем. изв. 2009. Т. 6. С. 514-517.
16. R. Shamoyan, E. Povprits Sharp theorems on traces in Bergman type spaces in tubular domains over symmetric cones // Журн. СФУ. Сер. Матем. и физ. 2013. Т. 6(4). С. 527-538.
17. M. Arsenovic, R. Shamoyan Trace theorems in harmonic function spaces on (R++1 )m, multipliers theorems and related problems // Kragujevac Journal of Mathematics. 2011. P. 411-430.
18. Dariush Ehsani, The d-Neumann problem on product domains in Cn // Math. Ann. 2007. Vol. 337, № 4. P. 797-816.
19. Debraj Charabarta, Mei-Chi Shaw, The Cauchy-Riemann equations on product domains // Math. Ann. 2011. Vol. 349, issue 4. P. 977-998.
20. D. Bekolle, A. Bonami, G. Garrigos, C. Nana, M. Peloso, F. Ricci Lecture notes on Bergman projectors in tube domain over cones, an analytic and geometric viewpoint. Proceeding of the International Workshop on Classical Analysis, Yaounde. 2001. 75 p.
REFERENCES
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3. Shamoyan, R. & Mihic, O. 2009, "On traces of holomorphic functions on the unit polyball" , Appl. Anal. Disorete Math, vol. 3, pp. 198-211.
4. Shamoyan, R. & Mihic, O. 2010, "On traces of Qp type spaces and mixed norm analytic function spaces on polyballs" , Siauliau. Math. Seminar, vol. 5, issue 13, p. 101.
5. Kytmanov, A. 2002, "Martinelly — Bochner integral and it is applications" , Nauka, Novosibirsk, 240 p. (in Russian)
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8. Arsenovic, M. & Shamoyan, R. 2013, "On embeddings, traces and multipliers in harmonic function spaces" , Kragujevac Math. Journal, vol. 37, no. 1, pp. 45-64.
9. Shamoyan, R. & Arsenovic, M. 2015, "On distance estimates and atomic decompositions in spaces of analytic functions on strictly pseudoconvex domains", Bulletin of the Korean Mathematical Society, vol. 52, no. 1, pp. 85-—103.
10. Shamoyan, R. F. & Kurilenko, S. M. 2014, "On a sharp estimate for a distance function in Bergman type analytic spaces in Siegel domains of second type", Mathematica Montisnigri, vol. XXX, pp. 5-16.
11. Shamoyan, R. F. & Kurilenko, S. M. 2014, "On extremal problems in tubular domains over symmetric cones" , Issues of Analysis, vol. 3(21), no. 1, pp. 44-65.
12. Shamoyan, R. & Mihic, O. 2015, "On traces of Bergman spaces in pseudoconvex domains, preprint" , Jornal of Function Spaces, vol. 2015, 10 p.
13. Shamoyan, R. & Kurilenko, S. 2015, "On traces of Herz type and Bloch type spaces on pseudoconvex domains" , Issues of Analysis, Petrozavodsk.
14. Shamoyan, R. & Mihic, O. 2010, "On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball" , Boletin de la Asociacion Matematica Venezolana, vol. XVII, no. 2, pp. 89-103.
15. Shamoyan, R. & Mihic, O. 2009, "On new estimates for distances in analytic function spaces in unit disk, polydisk, and unit ball", Siberian Mathematical Reports, vol. 6, pp. 514-517.
16. Shamoyan, R. & Povprits, E. 2013, "Sharp theorems on traces in Bergman type spaces in tubular domains over symmetric cones " , Journal Siberian Federal University, vol. 6(4), pp. 527-538.
17. Arsenovic, M. & Shamoyan, R. 2011, "Trace theorems in harmonic function spaces on (R++1 )m, multipliers theorems and related problems" , Kragujevac Journal of Mathematics, pp. 411-430.
18. Ehsani, Dariush 2007, "The d-Neumann problem on product domains in Cn", Math. Ann., vol. 337, no. 4, pp. 797-816.
19. Charabarta, Debraj & Shaw, Mei-Chi 2011, "The Cauchy-Riemann equations on product domains" , Math. Ann., vol. 349, issue 4, pp. 977-998.
20. Bekolle, D., Bonami, D., Garrigos, G., Nana, C., Peloso, M. & Ricci, F. 2001, "Lecture notes on Bergman projectors in tube domain over cones, an analytic and geometric viewpoint" , Proceeding of the International Workshop on Classical Analysis, Yaounde, 75 p.
Bryansk State University Received 22.03.2015
