SCIENTIFIC ABSTRACT NIKOLSKIY, S. M. - NIKOLSKIY, V. G.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001137220002-3
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: January 3, 2017
• Document Release Date: August 1, 2000
• Sequence Number: 2
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

On inequalities between
3/5 17/6 1/064/000/0 03 11/0;06
D299/D301
G is a region of the n-dimensional apace x = (XV... xn); the par-
tial derivatives are of order k: (1,< k *r). The problem io ,oaed,
which of the intermediate partial aeri~atives of f (mixed or non-
mixed) have finite norm in the sense of L P(G) and whether they can
be estimated by the norm (3). Several inequalitieng related to this
problem, are obtained. From these inequalitica, It follows t-hat if
the two-dimensional region G is bounded and has a sufficiently
smooth contour, then the inequality
I of
+ L G ) I
OX c 11W(r,r)
2 (G)
2
(4)
holds. This inequality, in conjunction with results obtained in
tho referenceag lead to a theorem about the region G, for which
(wider certain conditiona) the inequality
Card 2/7

On inequalities between
3/517/61/064/000/003/006
D299/ ft301
)If il
L (G)< C ',IIW(2#2
p 2 )(G),
where
a2f
~2f
f +
I'W(2 2)
(G) I'L (G) IL (G) (5)
iL
2 2(G
4 2 2
holds. Further, twi possible cases are conaidered of the two-di-
Menuional region G meeting the contour Fin the nei6hborhood of the
point P 00 By Imposing certain conditions on rg it is possible to
find (by racans of the Heine-Borel lemma ) a finite number of open
sets of type A I or A 2. whose Bum Meato /-" If these sets are uub-
tracted frota G, then a set GI is left. In the following, 0 is ex-
pressed as the sum
Card 3/7

On inequalities between
3
V517/6 1/0054/000/003/006
D299/D301
G = G I + 1+ ~A2*
The inequalities are proved for each summand separately; hence they
hold for G. The function f(x,y) on G is considered. This function
is exdanded in a Taylor series; thereupont a linear system of equa-
tions in obtained. The determinant of the system in denoted by W.
One obtains:
PI Lr-_~f rl (9. 0
-W Ws I (X, V.) (4-1r, dy de
Ob
+
the integrals in the right-hand aide of E.q. (6) are eatimated from
above. After calculations, one obtains
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S/517/61/064/000/003/006
On inequalities between D299/D301
r
6kf k r-k 3r f
p f p
< c mp +
~yk ilL (,),\ p.r~ 1i L W) ~yrij L G)j~
p p p
(12)
(k. = Is 20 ... r - 1)
Vnere)',(x) is an arbitrary measurable function, and the constant
c depends only on r and p. B. -r/P, one obtains
p,r y netting
L-f 1~ (11,1:kf k 6'f
k Kep r ~'L (G)+ ayr (13)
dy 11L (Gf p Lp ( G
p
Card 5/7

On inequalities between *as
3/517/61/064/OJO/003/006
D299YD301
Further, the region A is considered, coneisting of the points
(x,y) for which
0