Embedding theorems of multi-weighted multi-parametric Sobolev spaces on domains with arbitrary shapes are obtained. Conditions on weight functions 
, 
and at which the inequality of an investment is fair are received 
.
Examples with proofs are resulted.
Let 
area in 
, 
vectors with the whole coordinates 
, 
Below we will use designations: for
, 
, 
, 
Let 
, 
record of pokoordinatny comparison,
For
, sets 
, and 
let
, 
.
Let 
, area 
,
Let further 
at 
We will put
,
and let
.
Through 
, 
, 
will be designated, respectively 
, 
, 
For a multiindex 
, for
;
.
Through 
, the weight Lebesgue space with norm will be designated
.
Below record 
will mean that
.
Definition 1 ([1]). Domain 
we will call if area with a condition of a flexible 
horn (a flexible cone at 
) if at some 
, 
for 
there is a curve
, with the following properties:
(a) for all 
it is absolutely continuous on 
; 
for the item of century. 
(b)
.
Let's put thus
where the top edge takes on all 
for which takes place the listed properties.
Lemma 1([2]). Let 
. Then from family of parallelepipeds 
it is possible to take 
a covering 
a set 
parallelepipeds 
Thus family 
also forms 
The Frequency rate covering a covering, 
depend only from 
respectively 
Lemma 2 ([2]). Let 
Then takes assessment place
where 
Theorem. Let 
, and let weight , and on meet conditions: there is a regular function 
that
and
where
. Then the investment takes place
(1)
from an exact constant
.
Proof. In these work [1, page 17] for functions 
on area with a condition of a flexible 
horn was received in particular, the following integrated representation:
(2)
where
. (3)
definite care, 
and 
functions, satisfying the following conditions:
function (4)
(5)
(6)
(7)
where 
Let's copy representation (2) for 
and considering conditions 
and (5):
(8)
where 
and
(9)
using integrated representation (8), (9), for a case 
we write out
(10)
where 
B (10) in conditions force (3), (7) 
, we will receive
. (11)
, (12)
where 
follows From a choice that 
for all 
From (11) and conditions (4), (5) follows that
(13)
where 
– the integrated operation with a care
From (13) follows that
(14)
where
Let 
Owing to a lemma 2 for any 
on
(15)
where
(16)
First composed in (15)
(17)
In (17) we will apply a lemma 1 in which for a kernel 
the following estimates are fair to an assessment of each integral:
Owing to a lemma 1
(18)
(17), (18) follows from estimates that
Having taken
, at 
for 
from 
we remove that
Example 1. Let's consider theorem conditions for permission of a question about existence of an inclusion
where 
at
In this case for any cube 
Therefore in order that
and 
it is enough to demand that the following conditions were satisfied:
(19)
Further we have
As well as
that 
Let's say as 
let 
Then owing to (19)
Let's consider more general case now, namely, let 
and for any 
the surface 
has the area 
Then
So, in these conditions on 
, the inclusion takes place
References:
1. O. V. Besov. Integrated representations of functions and the theorem of an inclusion for area with a condition flexible roga.//Works of Mathematical institute of Academy of Sciences of the USSR, 1984. T.170. Page 12–29.
2. Kusainova L. K. About limitation of one class of operators in weighted spaces of Lebega.//Works of inter@ konf. Semipalatinsk. 2003. Page 94–95.
