Библиографическое описание:

Искакова Г. Ш. About one multiweight anisotropic inequality of the investment // Молодой ученый. — 2015. — №5. — С. 8-14.

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.


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


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:



.                                   (3)

 definite care,

and functions, satisfying the following conditions:

function                                                                    (4)





Let's copy representation (2) for  and considering conditions  and (5):


where  and


using integrated representation (8), (9), for a case we write out


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


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.

Основные термины: Theorem. Let, multi-weighted multi-parametric Sobolev, conditions, flexible horn, weight Lebesgue space, following conditions, lemma, O. V. Besov, condition flexible roga.//Works, Kusainova L. K., area, pokoordinatny comparison, following properties, arbitrary shapes, weight functions, flexible cone, following estimates, definite care, regular function, theorem conditions


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