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 ,
at We will put
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 (). 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.
Let's put thus
where the top edge takes on all for which takes place the listed properties.
Lemma 1(). 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 (). Let Then takes assessment place
Theorem. Let , and let weight , and on meet conditions: there is a regular function that
where. Then the investment takes place
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:
and functions, satisfying the following conditions:
Let's copy representation (2) for and considering conditions and (5):
using integrated representation (8), (9), for a case we write out
where B (10) in conditions force (3), (7) , we will receive
where follows From a choice that for all From (11) and conditions (4), (5) follows that
where – the integrated operation with a care
From (13) follows that
Let Owing to a lemma 2 for any on
First composed in (15)
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
(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
In this case for any cube
Therefore in order that
it is enough to demand that the following conditions were satisfied:
Further we have
As well as
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
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.