- Research Article
- Open access
- Published:
A Kind of Estimate of Difference Norms in Anisotropic Weighted Sobolev-Lorentz Spaces
Journal of Inequalities and Applications volume 2009, Article number: 161405 (2009)
Abstract
We investigate the functions spaces on for which the generalized partial derivatives
exist and belong to different Lorentz spaces
, where
and
is nonincreasing and satisfies some special conditions. For the functions in these weighted Sobolev-Lorentz spaces, the estimates of the Besov type norms are found. The methods used in the paper are based on some estimates of nonincreasing rearrangements and the application of
,
weights.
1. Introduction
In this paper we study functions on
which possess the generalized partial derivatives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ1_HTML.gif)
Our main goal is to obtain some norm estimates for the differences
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ2_HTML.gif)
( being the unit coordinate vector).
The classic Sobolev embedding theorem asserts that for any function in Sobolev space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ3_HTML.gif)
Sobolev proved this inequality in 1938 for . His method, based on integral representations, did not work in the case
. Only at the end of fifties Gagliardo and Nirenberg gave simple proofs of inequality (1.3) for all
Inequality (1.3) has been generalized in various directions (see [1–6] for details). It was proved that the left hand side in (1.3) can be replaced by the stronger Lorentz norm, that is, there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ4_HTML.gif)
For the result follows by interpolation (see [7, 8]). In the case
some geometric inequalities were applied to prove (1.4) (see [9–13]).
The sharp estimates of the norms of differences for the functions in Sobolev spaces have firstly been proved by Besov et al. [1, Volume 2, page 72]. For the space Il'in's result reads as follows: If
and
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ5_HTML.gif)
Actually, this means that there holds the continuous embedding to the Besov space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ6_HTML.gif)
It is easy to see that inequality (1.5) fails to hold for , but, it was proved in [14] that (1.5) is true for
and
.
The generalization of the inequality (1.5) to the spaces was given in [12]. That is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ7_HTML.gif)
where and
the inequality is valid if
or
Using (1.7), we get the following continuous embedding:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ8_HTML.gif)
For this embedding was proved by Besov et al. [1, Volume 2, page 72]. The main result in [12] is the proof of (1.7) for
.
In [15], there was the sharp estimates of the type (1.7) when the derivatives belong to different Lorentz spaces
Before stating the theorem, we give some notations. Let
be the class of all measurable and almost everywhere finite functions
on
such that for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ9_HTML.gif)
Let and
for
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ10_HTML.gif)
Now we state the main theorem in [15].
Theorem 1.1.
Let , and
if
. Let
, and
be the numbers defined by (1.10). For every
satisfying the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ11_HTML.gif)
take arbitrary such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ12_HTML.gif)
and denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ13_HTML.gif)
then for any function which has the weak derivatives
there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ14_HTML.gif)
where is a constant that does not depend on
.
In many cases, the Lorentz space should be substituted by more general space, the weighted Lorentz space. In this paper, we will generalize the above result when the weighted Lorentz spaces take place of
, where
is a weight on
which satisfies some special conditions.
2. Auxiliary Proposition
Let be the class of all measurable and almost everywhere finite functions on
. For
, a nonincreasing rearrangement of
is a nonincreasing function
on
that is, equimeasurable with
. The rearrangement
can be defined by the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ15_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ16_HTML.gif)
If then the following relation holds [16, Chapter 2]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ17_HTML.gif)
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ18_HTML.gif)
Assume that A function
belongs to the Lorentz space
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ19_HTML.gif)
For , the space
is defined as the class of all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ20_HTML.gif)
We also let . Let
be a weight in
(nonnegative locally integrable functions in
).
If , we replace
with
. For
, or
and
, the weighted Lorentz space
is defined in [9, Chapter 2] by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ21_HTML.gif)
If , denote
It is well known that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ22_HTML.gif)
and if then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ23_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ24_HTML.gif)
In following part of this paper, we will always denote .
The weighted Lorentz spaces have close connection with weights of for
(see [9, Chapter 1]). Let
be the Hardy operator as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ25_HTML.gif)
The space is the cone of all nonnegative nonincreasing functions in
. We denote
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ26_HTML.gif)
is bounded and denote if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ27_HTML.gif)
is bounded.
Lemma 2.1 (Generalized Hardy's inequalities).
Let be nonnegative, measurable on
and suppose
and
is a weight in
,
then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ28_HTML.gif)
(with the obvious modification if ).
Proof.
It is easy to obtain this result applying Hardy's inequality [16].
Lemma 2.2.
Let be a nonnegative nonincreasing function on
,
be a nonincreasing weight on
and there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ29_HTML.gif)
Then for there exists a continuously differentiable
on
such that
(i)
(ii) decreases and
increases on
,
(iii)
where is a constant depends only on
, and
.
Proof.
Without loss of generality, we may suppose that . Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ30_HTML.gif)
Then decreases and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ31_HTML.gif)
Using the conditions which satisfy, it gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ32_HTML.gif)
Furthermore, noticing is nonincreasing and applying Lemma 2.1, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ33_HTML.gif)
now set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ34_HTML.gif)
Then increases on
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ35_HTML.gif)
Furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ36_HTML.gif)
where that is,
. Since
is decreasing function on
, thus
is decreasing and
is also decreasing on
.
Finally, using Lemma 2.1 and (2.19), we get (iii). The Lemma 2.2 is proved.
Let and
for
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ37_HTML.gif)
Then and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ38_HTML.gif)
To prove our main results we use the estimates of the rearrangement of a given function in term of its derivatives
We will use the notations (2.23).
Lemma 2.3.
Let for
and
is continuous weight on
. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ39_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ40_HTML.gif)
and suppose that are positive continuously differentiable functions with
on
such that
decreases and
increases on
. Set for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ41_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ42_HTML.gif)
Then
(i)there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ43_HTML.gif)
(ii)there exist continuously differentiable functions on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ44_HTML.gif)
(iii)for any such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ45_HTML.gif)
the function decreases on
.
Proof.
The proof is similar to [15, Lemma ]. All the argument holds true when we substitute the weight
in this lemma for
.
The Lebesgue measure of a measurable set will be denoted by
.
For any set
denote by
the orthogonal projection of
onto the coordinate hyperplane
. By the Loomis-Whitney inequality [17, Chapter 4]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ46_HTML.gif)
Let , and let
be a set of type
and measure
such that
for all
. Denote by
the
-dimensional measure of the projection
By (2.32), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ47_HTML.gif)
Lemma 2.4.
Let be nonincreasing, and
when
where
. Function
has weak derivatives
Then for all
and
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ48_HTML.gif)
where and
is a constant depending on
and
.
Proof.
Let then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ49_HTML.gif)
Due to the conditions of and (2.33), we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ50_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ51_HTML.gif)
So we immediately get (2.34).
Lemma 2.5.
If and
, then
Proof.
Let Since
so by [9, Chapter 1] we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ52_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ53_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ54_HTML.gif)
So
Lemma 2.6.
Let for
Assume that weight
on
satisfies the following conditions:
(i)it is nonincreasing, continuous, and ,
(ii)exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ55_HTML.gif)
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ56_HTML.gif)
Assume that a locally integrable function has weak derivatives
Then for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ57_HTML.gif)
where the constants
depends only on
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ58_HTML.gif)
Proof.
For every fixed we take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ59_HTML.gif)
Thanks to Lemma 2.5, and (for
is nonincreasing), we know
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ60_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ61_HTML.gif)
Next we apply Lemma 2.2 with defined as in Lemma 2.3. In this way we obtain the functions which we denote by
. Further, with these functions
we define the function
by (2.28). By Lemma 2.3, we have the inequality (2.44). Using Lemma 2.4 with
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ62_HTML.gif)
where . Taking into account (2.28), we get (2.43).
Corollary 2.7.
Let for
and
be the numbers defined by (2.42). Assume weight
on
satisfies the following conditions:
(i)it is nonincreasing, continuous, and ,
(ii)there exist two constants with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ63_HTML.gif)
and there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ64_HTML.gif)
Assume that a locally integrable function has weak derivatives
and
for some
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ65_HTML.gif)
Let and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ66_HTML.gif)
Then and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ67_HTML.gif)
Proof.
Let , with
and
. Applying Hölder's inequality and noticing
and
is nonincreasing, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ68_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ69_HTML.gif)
Let . Using (2.43) with
, which satisfies
(
are two constants in (2.49) for
), combining (2.49), (2.52), and Hölder's inequality, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ70_HTML.gif)
By (2.55), Furthermore, from (2.49), we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ71_HTML.gif)
Inequality (2.53) now follows from (2.44) and (2.55).
Remark 2.8.
If in Corollary 2.7, then it is easy to get
.
Remark 2.9.
Let for
. Let
, and
be the numbers defined by (2.42). Assume that
,
and
satisfies the conditions of Corollary 2.7 with
. Then for any function
with compact support we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ72_HTML.gif)
This statement can be easily got from Lemma 2.6. Inequality (2.58) gives a generalization of Remark of [15] when
because
satisfies the preceding conditions.
Remark 2.10.
Beyond constant weights, there are many weights satisfying conditions of Corollary 2.7. For example,
(i) where
-
(ii)
(2.59)
where
For weight in (i) or (ii), it is easy to see the weighted Lorentz space
for
does not coincide with any Lorentz space
.
3. The Main Theorem
Theorem 3.1.
Let for
Let
, and
be the numbers defined by (2.42). Suppose weight
on
satisfies the following conditions:
(i)it is nonincreasing, continuous, and
(ii)there exist two constants with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ74_HTML.gif)
and there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ75_HTML.gif)
For every satisfying the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ76_HTML.gif)
take arbitrary such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ77_HTML.gif)
and denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ78_HTML.gif)
Then for any function with the weak derivatives
there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ79_HTML.gif)
where is a constant that does not depend on
.
Proof.
First we can get by our conditions. denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ80_HTML.gif)
Further, assume that and set for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ81_HTML.gif)
For almost all we have [1, Volume 1, page 101]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ82_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ83_HTML.gif)
Indeed, for any subset with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ84_HTML.gif)
(3.10) then follows.
For is nonincreasing (
), we get
by Lemma 2.5. Thus from (3.10)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ85_HTML.gif)
It follows Furthermore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ86_HTML.gif)
Then due to Hardy lemma [16, page 56]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ87_HTML.gif)
It follows Analogically we get
Thus by Corollary 2.7 we have
.
Denote for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ88_HTML.gif)
Set (
are two constants in (3.1) for
), and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ89_HTML.gif)
where is the constant in Lemma 2.5. Then by (3.1)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ90_HTML.gif)
Therefore ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ91_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ92_HTML.gif)
Now for every by applying Lemma 2.2 with
We obtain
on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ93_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ94_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ95_HTML.gif)
For , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ96_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ97_HTML.gif)
We will estimate for fixed
and
By Lemma 2.4, (3.21), we have that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ98_HTML.gif)
where Applying Lemma 2.3, we obtain that there exist a nonnegative function
and positive continuously differentiable functions
on
satisfying the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ99_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ100_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ101_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ102_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ103_HTML.gif)
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ104_HTML.gif)
We will prove that for any and any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ105_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ106_HTML.gif)
By (3.24)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ107_HTML.gif)
For the inequality (3.32) follows directly from (3.26) and (3.33). If
then (3.32) is the immediate consequence of (3.10), (3.21), and (3.33).
Now, taking into account (3.26) and (3.32), we obtain that for and any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ108_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ109_HTML.gif)
and is defined by (3.33).
Further, we have (see (3.18)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ110_HTML.gif)
By (3.30), the function increases on
. It follows easily that
exists on
and satisfies
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ111_HTML.gif)
Furthermore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ112_HTML.gif)
Using Minkowsi's inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ113_HTML.gif)
Further, using Hölder's inequality and (3.38), we get when (the case
is obvious)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ114_HTML.gif)
Thus, by Fubini's theorem and (3.33)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ115_HTML.gif)
The same argument gives that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ116_HTML.gif)
By (3.33) the last integral is the same as one on the right side of (3.42). So, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ117_HTML.gif)
Now we apply Hölder's inequality with the exponents and
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ118_HTML.gif)
Therefore, we get, applying (3.27) and (3.34)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ119_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ120_HTML.gif)
we get the inequality (3.6). The theorem is proved.
Let be a rearrangement invariant space (r.i. space),
be an r.i. space over
and
. Set
integral part of
). The Besov space
is defined as follows (see [18, 19]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ121_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ122_HTML.gif)
and denotes the fundamental function of
with
being any measurable subset of
with
.
Then we have the following.
Corollary 3.2.
Let for
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ123_HTML.gif)
Let the weight be the same as that in Theorem 3.1. Take arbitrary
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ124_HTML.gif)
and denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ125_HTML.gif)
Then for any function which has the weak derivatives
there hold
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F161405/MediaObjects/13660_2009_Article_1902_Equ126_HTML.gif)
where is a constant that does not depend on
.
Proof.
We can easily obtain the similar result to Lemma in [20] by substituting
for
there. Now the corollary is obvious using the Hardy's inequality and Theorem 3.1.
Remark 3.3.
If there exists with
, whether Theorem 3.1 remains true is still a question now.
References
Besov OV, Il'in VP, Nikol'skiÄ SM: Integral Representations of Functions and Imbedding Theorems. Vol. I. V. H. Winston & Sons, Washington, DC, USA; 1978:viii+345.
Kolyada VI: Rearrangements of functions and embedding of anisotropic spaces of Sobolev type. East Journal on Approximations 1998,4(2):111–199.
Kudryavtsev LD, Nikol'skiÄ SM: Spaces of differentiable functions of several variables and imbedding theorems. In Analysis, Encyclopaedia Math. Sci.. Volume 26. Springer, Berlin, Germany; 1991:1–140.
Nikol'skià SM: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, New York, NY, USA; 1975:viii+418.
Triebel H: Theory of Function Spaces, Monographs in Mathematics. Volume 78. Birkhäuser, Basel, Switzerland; 1983:284.
Triebel H: Theory of Function Spaces. II, Monographs in Mathematics. Volume 84. Birkhäuser, Basel, Switzerland; 1992:viii+370.
Peetre J: Espaces d'interpolation et théorème de Soboleff. Université de Grenoble. Annales de l'Institut Fourier 1966,16(1):279–317. 10.5802/aif.232
Strichartz RS: Multipliers on fractional Sobolev spaces. Journal of Mathematics and Mechanics 1967, 16: 1031–1060.
Carro MJ, Raposo JA, Soria J: Recent developments in the theory of Lorentz spaces and weighted inequalities. Memoirs of the American Mathematical Society 2007.,187(877):
Faris WG: Weak Lebesgue spaces and quantum mechanical binding. Duke Mathematical Journal 1976,43(2):365–373. 10.1215/S0012-7094-76-04332-5
Kolyada VI: Rearrangements of functions, and embedding theorems. Uspekhi Matematicheskikh Nauk 1989,44(5):61–95. [English translation: Russian Math. Surveys, vol. 44, no. 5, pp. 73–118, 1989] [English translation: Russian Math. Surveys, vol. 44, no. 5, pp. 73–118, 1989]
Kolyada VI: On the embedding of Sobolev spaces. Matematicheskie Zametki 1993,54(3):48–71. [English translation: Math. Notes, vol. 54, no. 3, pp. 908–922, 1993] [English translation: Math. Notes, vol. 54, no. 3, pp. 908–922, 1993]
Poornima S: An embedding theorem for the Sobolev space
. Bulletin des Sciences Mathématiques 1983,107(3):253–259.
Kolyada VI: On the relations between moduli of continuity in various metrics. Trudy Matematicheskogo Instituta imeni V. A. Steklova 1988, 181: 117–136. [English translation: Proc. Steklov Inst. Math., vol. 4, pp. 127–148, 1989] [English translation: Proc. Steklov Inst. Math., vol. 4, pp. 127–148, 1989]
Kolyada VI, Pérez FJ: Estimates of difference norms for functions in anisotropic Sobolev spaces. Mathematische Nachrichten 2004, 267: 46–64. 10.1002/mana.200310152
Bennett C, Sharpley R: Interpolation of Operators, Pure and Applied Mathematics. Volume 129. Academic Press, Boston, Mass, USA; 1988:xiv+469.
Hadwiger H: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin, Germany; 1957:xiii+312.
MartÃn J, Milman M: Symmetrization inequalities and Sobolev embeddings. Proceedings of the American Mathematical Society 2006,134(8):2335–2347. 10.1090/S0002-9939-06-08277-3
MartÃn J, Milman M: Higher-order symmetrization inequalities and applications. Journal of Mathematical Analysis and Applications 2007,330(1):91–113. 10.1016/j.jmaa.2006.07.033
Kolyada VI: Inequalities of Gagliardo-Nirenberg type and estimates for the moduli of continuity. Uspekhi Matematicheskikh Nauk 2005,60(6):139–156. [English translation: Russian Math. Surveys, vol. 60, no. 6, pp. 1147–1164] [English translation: Russian Math. Surveys, vol. 60, no. 6, pp. 1147–1164]
Acknowledgments
This work is supported by NSFC (no. 10571156, 10871173), Natural Science Foundation of Zhejiang Province (no. Y606117), Foundation of Zhejiang Province Education Department (no. Y200803879) and 2008 Excellent Youth Foundation of College of Zhejiang Province (no. 01132047).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Chen, J., Li, H. A Kind of Estimate of Difference Norms in Anisotropic Weighted Sobolev-Lorentz Spaces. J Inequal Appl 2009, 161405 (2009). https://doi.org/10.1155/2009/161405
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/161405