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Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 2(22). C. 20-27. ISSN 2079-6641
DOI: 10.18454/2079-6641-2018-22-2-20-27 MSC 32A07, 432A10, 32A07
ON SOME NEW ESTIMATES RELATED WITH DISTANCE FUNCTION FOR DIFFERENTIAL FORMS
R. F. Shamoyan, A. N. Shipka
Bryansk State University, 241036, Bryansk, Bezhitskaya st., 14, Russia. E-mail: [email protected],[email protected]
In this short note we consider a new extremal problem for some general spaces of differential forms and provide some estimates for related distance function.
Key words: extremal problems, differential forms, Hodge norms
© Shamoyan R. F., Shipka A.V., 2018
УДК 517.53+517.55
О НЕКОТОРЫХ НОВЫХ ОЦЕНКАХ СВЯЗАННЫХ С ДИСТАНЦИЯМИ В ПРОСТРАНСТВАХ ДИФФЕРЕНЦИАЛЬНЫХ ФОРМ
Р. Ф. Шамоян, А. Н. Щипка
Брянский государственный университет, 241036, Брянск, ул. Бежицкая., 14, Россия E-mail: [email protected],[email protected]
В этой короткой заметке мы рассматриваем новую экстремальную задачу в некоторых пространствах дифференциальных форм и приводим некоторые новые оценки, связанные с функцией дистанции в этих пространствах. Эти результаты обобщают ранее полученные нами оценки для функциональных классов.
Ключевые слова: экстремальные проблемы, дистанции, дифференциальные формы, норма Ходжа.
© Шамоян Р.Ф., Щипка А. Н., 2018
We consider in this short note some new extremal problems in various spaces of differential forms related with distance function. Using known integral representations for differential forms from such type spaces some general new (not sharp) estimates will be provided.
We consider in this section only bounded D domains in Cn with dD e C™, so that Cn \ D is connected. We put as usual D = {z e Cn : p(z) < where p is a real valued function of C™(Cn) class and dp = 0 on (dD), and we choose p, so that |dp | = 1 on (d D).
In our previous papers we solved many extremal problems related with distances in various spaces of functions on various type of domains. It is very natural to try to solve similar type problems in general spaces consisting of differential forms. Note various similar type problems related with approximation of various forms were provided by various authors (see [1,2] and various references there).
We denote positive constants in this paper by c1,c2,c3,— To formulate our results we need various standard definitions from [1-3].
The root of our approach is to try to replace Bergman kernel and Bergman representation formula which was heavily used in our previous papers on distances in various functional classes by similar reproducing formula, but for differential forms.
For that reason we apply general so-called Bochner-Martinelly-Koppelman formulas to be more precise their versions in some functional classes consisting of differential forms taken from [1,2]. Our paper is based on some results on differential forms taken from [1,2].
We first remind the reader some basic facts on mentioned reproducing formula. Then we define spaces,and formulate our problem and then give some new results on extremal problems. Approaches we used here are very similar to those we used previously in various functional classes via Bergman reproducing formula (see [4,5,6-7] and references there also).
We provide first Koppelman formula.
Let I =(i1,...,ip) and J = (j1,...,jq); be growing multiindexes
1 < i1 < ••• < ip < n; 1 < j1 < ••• < jq < n,0 < p < n,0 < q < n. For q < n — 1 we consider Koppelman kernels
Upq(% , Z) = (—1) Pn—q—1)(n—f! X
X £' £ a (J, k)a (I) jr—^d^ [J, k] A d % [I]dz] A dzi,
I,J k/J
where dzj = dz1 A-•A dj, dzI = dzi1 A —A dzip, where to get d % [I] we have to remove d%i1...d%ip and
a (J, k)dz = dzk A dzj A dz [J, k] a (I)dz = dzi A dz[I].
The Upq(%,z) kernel is a double differential form of (n — p,n — q — 1) type by % and of (p, q) type by z variable.
We consider (p, q) type differential form y = £ 'Y//dz/ Adz/ and we put as usual
dy = £ £ Y^^^^^dzk A dz/ A dz/
Theorem A. (Koppelman) (see [1,2]). Let D be a bounded region in Cn, with a boundary dD partially smooth. Let y be a differential form of (p, q) type with coefficients of C!(D) class, then we have
J Y (£ ) A Up,q (É , z) - J dY(ï ) A Up,q(«§ , z)-d D D
-djY (<§ ) A Up,q-!(^ , z) =
D
If Y is a differential form of (p, q) type
Y(z), z_G D 0, z £ D
7 = £ ' (7/, J (z))dz/ A dzj,
/,J
I = (ii,...,ip);J = (ji,..., jq),0 < p,q < n
where summation moves via 1 < i^ < ■ ■ ■ < ip < n, 1 < j < ■ ■ ■ < jq < n. Then the Hodge operator on forms * is defined as follows
(*y) = £(Y/,j(z)) * (dz/ A dzj) * (dz/ A dzj) =
= (2p+q-n)(-1)np(i'n)(a (/))(c (/))dz([J]) A dz[/]
where d(z/) Adz[/] = (c[J])dz, so (*y) is a form of (n — q, n — p) type. For properties of * operator we refer the reader to [1,2].
If y, ^ are two forms of (p, q) type with coefficients of L2(D) class then scalar
product of Hodge (y, ^) can be defined as (y, ^) = ( / YA*^), and ||y|| = \j(y, Y) (norm
of a differential form). Note also that
D
¿>7 = £ £ "dz^dzk A dz/ A dzj
k=i /,J
dY = £ £ '^r^dzk A dz/ A dz/
Let Ws(D) = W|(D),s = 0,1,2,3,... be a Sobolev space with f functions whose derivatives up to s order are in L2(D) with the usual scalar product. We define Wqs(D) as a spaces of forms of (0,q) type with coefficients from Ws(D), W0s = Ws(D).
We consider here bounded D domains in Cn, so that d D g and so that Cn \ D is connected.
In this paper we put always q is less or equal than 2. And s is a natural always. This with natural inclusions of Wqs spaces allows us to use representation formulas in our
theorems. We need some more definitions and assertions now to formulate our results below.
Let X be a closed subspace of W^,X c (Wqp. The natural question is to estimate in Hodge norms
(dist)(y,X) = inf (||y — g||),ye Ws,
geX H
Note in a recent series of papers of the first authors such type problems were solved in various functional spaces in various domains (see [4,5-7] and various references there also).
We first provide a short scheme for function spaces in the unit disk D. Let H(D) is a space of all analytic functions in D.
Let B—t = {f e H(D): (supzeD)|f (z)|(1 — |z|)—t < ™},t < 0. Then we have for f e B—t
(distS—t )(f, Bq) < c inf{e > 0 : J ( j (1 —zwS+p dA(w))q x
d a£ ,—t (f) zw
X(1 — |z|)—sq—1 dA(z) < ™}, where dA is a Lebegues measure in D, 0 < q < s < 0,p > p0, p0 is large enough, and
(Bq) = {f e H(D) : J |f (z)|q(1 — lzl)—sq—1dA(z) < ~},
D
Ae,—t = {z e D : | f (z)|(1 — |z|)—t > e},t < 0,e > 0.
This is a known result (see [4,6,7]). Here is the short proof of this result which we wish to extend to much larger spaces of differential forms in this note, using Koppelman's formulas repeating same type arguments.
For p > p0 by Bergman representation formula we have.
f(z) = c(p)( i dA(w)+ J «*)) =
D\Ae—t Ae —t
= f1(z)+ fi(z), z e D
For t < 0
If1(z)l< c / ]f0*—wfdA(*) < ce^;z e D.
D\A£—
So we have (supzeD)|f1(z)|(1 — |z|) t < ce. For s < 0,t < 0 we also have ||f2||Bq < c, and hence finally we have
distB—t(f,Bq) < c|| f — f2||b—t = c||f1 ||b— t < ce
We turn here to the same problem, but in more general spaces of differential forms. The main idea here in this paper is to use Koppelmans formulas (reproducing formulas) instead of Bergman reproducing formulas as we did in our previous papers on this topic previously.
We refer for all these to [1,2]. We consider only kernels for forms of type (0,q),q e [0,n — 1] in reproducing formula of Bocher-Martinell-Koppelman that is
u0,q(^,z) = L' L c(J,k)d^gd^ [/[Jk] Ad^dzJ, where JUk means J(J1,..., Jq) with k within and c(J,k) we define as dA d£/ A d^ [JU k] = c(J,k)d^ where ,z) is a standard fundamental solution of Laplace equation (see [1,2]).
We consider now the following bounded operators acting on spaces of differential forms (see [1,2])
M : WqS(D) ^ WqS(D),
where
MY(z) = / Y (^) A Uo,q(<§, z);
d D
P : WqS(D) ^ Wqs+}(D),
where Py(z) = — / y(^) AU0,q—1 (^,z); (see [1,2]). Here is version of Bochner-Koppelman
dD
which formula we need for Wqs spaces.
Theorem A. If ye Wqs(D),s > 1, then Y = My + dPy + Pd/ in D.
This formula is exactly the one we need for our purposes as substitution of Bergman integral reproductiong formula discussed above, we will need also it is versions for various subspaces of Wqs (see [1,2]).
We define Vqs(D) spaces and pose a problem. Let Vqs(D) = {ye Wqs(D): My = 0,MPy = 0,..., MPq y = 0}i'nD.
Then Vq is closed subspace of Wq(D). (see [1,2])
Our idea is to generalize a distance problem for function spaces to more general spaces of differential forms and to find concrete estimates for distances di'stX (y, Y), in Hodge norms in spaces of forms, where X c Y,and X, Y are various spaces of differential forms, and y e X.
1) We pose a new problem to find estimates for di'st(y, Vqs), y e Wqs(D), Vqs c Wq in Hodge norm using theorem A above.
2) Let next ^ e W0s(D) then ^ = M^ + (Pd^) ([1,2]) by theorem A and we can consider a problem of estimates of (di'st,G), for closed subspaces G of W0s,G c W0s.
3) For all Y-functions (see [1,2]) y e V0s : ^ = (Pd^)(*). We consider problem of estimates of (di'st)(y,X),X c V0s, Ye (V0s),X c V0s using equality (*).
4) Also we can look at closed subspaces X, X c Vqs, since theorem A with My = 0 for Vqs is valid also (see [1,2]).
5) Let X be a closed subspace of Q0,X c Q0(D) = W0s \ V0s(D) then we look at (disi)(f,X), f e Q0(D), note f = Mf for all f e Q0(D) (see [0,2]).
All these questions are interesting enough and precise estimates are from our point of view an important issue, which may have various applications also.
We give some partial answers following our previous papers on extremal problems in various functional spaces in various domains.
We note that our results probably can be sharpened. The answer for us is unknown at this moment.
We fix indexes /0, J0. Then we put
Xy,£ = xY0/° = {<§ e dD : |Y/0,J0(<§)||0(<§)l > e}, Y^0 = {z e D : |Y/o/o(z)||0(z)| > e},
where we choose 0 e C^(D), we omit /0, J0 below.
We have the following results following argument of our previous papers on distances.
We have the following theorem for the fifth problem.
Theorem 1. Let Y e (Q0)(D), let X be a closed subspace of (Q0)(D). Then
(distQ0)(Y, X) < ciinf{£ > 0:|| J y($ ) A U0jq(<§, z)||^ <
X
£,Y
if
Y(ï ) A Uo,q (<§ , z)
D\X
Q0
< £
E,Y
For the second problem we have finally the following.
Theorem 2. Let Ye (W0s). Let X be a closed subspace of (W0)(D). Then
(dist)w.(Y,X) < cs{£ > 0 : 11 y (y)) A (U0q(S,z)) - j dy(S) A Uo,q+i(Z,z)| | < ~}
X
£,Y
Y,£
if
J Y(ï ) A (Uo,q(ï , z)) - j dY(ï ) A Uo,q+1(ï )
dD\X£Y d\Yy,e
W^
<£
for some positive constants c3.
We leave the formulation of completely similar theorems for the first and fourth problems to readers though it is more complicated technically.
Similar results are valid for other embeddings and problems similar to those we put as for as appropriate integral representation of a differential form exists. We add some lines of short proofs of these assertions below.
Indeed the simple argument for the proof of the first theorem is the following.
We have f = f1 + f2, where
fl(z)= J Y(ï ) A Uo,q(<§, z), z G D
X£,Y
f2(z)= j № )) A Uo,q(Ç , z), z G D
d D\X£Y
by theorem A.
It remains to note that under conditions of our theorem, we have obviously the following
(dist)Qo(f,X) < c||f -fi||Qo = c||f21|Q0 < ce
Other our theorems have similar proofs. It will be nice to show if these estimates are sharp or not.
Similar results are valid for so -called eq(D) type spaces of differential forms. To formulate related version of Koppelman-Martinelly-Bochner formula, we will need some definitions in bounded strongly convex domains in Cn with C boundary.
Here we consider domains which are given as follows D = {Z g Cn : p(Z) < 0} and we denote by eq(D) the space of all differential forms of type (0,q) in D with coefficients in C in D.
By theorem from [1,2] each differential form u e eq(D), q > 0 can be represented as
U = FqU + Pq+l(du) + d (PqU).
(see [1,2])
where Pq = Tq + Lq, for 1 < q < n,Pq = 0, for q = 0,q = n + 1
(FqU)(Z) = (-1)q l-f^ / (u(| )) A ^ ( ^g) ) A rf«
d D
(TqU) (z) = (-1)q /(u(| )) A «q-1 (il—2) A dI
D
(L,U)(zz)=(-1)q/ (u(|))A»q-1((-1 -*)()+)A
d Dx [0,1]
where we put
n do
0(l,z) =< P(|), l - z >= £ dp(|j - Zj)
j=1
for (z, I) e Cn x Cn, and dp = 0 on dD, and P(|) = ,..., Ig), P real valued and is
n
of class and ©'(n) = £ (-1)jnjdn[j] is Lere form (see [1,2]).
j=1
Using this representation we can following arguments of proof of our previous theorems find upper estimates for dz'si(eq)(f,X); f e eq, for X closed subspaces of eq(D). We leave this to readers.
References
[1
[2
Kytmanov A.M., Martinelly-Bochner integral and it is applications, Novosibirsk, 2002, 240 c.
Kytmanov A.M., Martinelly-Bochner integral and it is applications, Berkhauser Verlag, Basel-Boston-Berlin, 1995, 305 c.
[3] Kytmanov A.M., "On sharp calculations of Martinelly-Bochner integral in the unit ball of Cn", Uspekhi Math. Nauk, 36:219 (1981), 217-218.
[4] Shamoyan R., Mihic O., "On new estimates for distances in analytic function spaces in higher dimension", Sib. Elektron. Mat. Izv., 2009, 514-517.
[5] Shamoyan R., Mihic O., "On a distance function in some New analytic Bergman type spaces in higher dimension", Journal of Function spaces, 2014, 268-287.
[6] Shamoyan R., Mihic O., "On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball", Boll. Asoc. Mat. Venez., 17:2 (2010), 89-103.
[7] Shamoyan R. F., Arsenovi'c M., "Some remarks on extremal problems in weighted Bergman spaces of analytic function", Commun. Korean Math. Soc., 27:4 (2012), 753-762.
References (GOST)
[1] Kytmanov A.M. Martinelly-Bochner integral and it is applications, Nauka, Novosibirsk, (2002), 240 p.
[2] A.M. Kytmanov, Martinelly-Bochner integral and it is applications eds., Berkhauser Verlag, Basel-Boston-Berlin, (1995), 305 p.
[3] A.M. Kytmanov, On sharp calculations of Martinelly-Bochner integral in the unit ball of Cn, Uspekhi Math. Nauk, 36, 219, (1981), 217-218.
[4] R. Shamoyan and O. Mihic, On new estimates for distances in analytic function spaces in higher dimension, Sib. Elektron. Mat. Izv. (2009), 514-517.
[5] R. Shamoyan, O. Mihic, On a distance function in some New analytic Bergman type spaces in higher dimension, Journal of Function spaces, Vol. (2014), 10
[6] R.Shamoyan and O. Mihic, On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball, Boll. Asoc. Mat. Venez. 17 (2010), no 2, 89-103.
[7] R.F.Shamoyan and M. Arsenovi'c, Some remarks on extremal problems in weighted Bergman spaces of analytic function, Commun. Korean Math. Soc. 27 (2012), no. 4, 753-762.
Для цитирования: Shamoyan R. F., Shipka A. V. On some new estimates related with distance function for differential forms // Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 2(22). C. 20-27. DOI: 10.18454/2079-6641-2018-22-2-20-27
For citation: Shamoyan R. F., Shipka A. V. On some new estimates related with distance function for differential forms, Vestnik KRAUNC. Fiz.-mat. nauki. 2018, 22: 2, 20-27. DOI: 10.18454/2079-6641-2018-22-2-20-27
Поступила в редакцию / Original article submitted: 04.04.2018
В окончательном варианте / Revision submitted: 07.06.2018
