Geometric characterization of generalized Hajłasz-Sobolev embedding domains
-
Ziwei Li
Abstract
In this article, the authors study the embedding properties of Hajłasz-Sobolev spaces with generalized smoothness on Euclidean domains, whose regularity is described via a smoothness weight function
1 Introduction
Throughout this article, let
It is known that the embedding property of function spaces on domains strongly depends not only on the analytical character of function spaces themselves but also on the geometric character of the underlying domain under consideration. There already exist a lot of works on the geometric characterizations of domains supporting various embeddings of function spaces. For instance, for any
where
To reflect the geometric character of a set
and, for any
If
with equivalent quasi-norms. In particular, if a domain
In particular, there also exist some related works on Hajłasz-Sobolev type spaces
On the other hand, the study of function spaces with generalized smoothness can be traced back to the mid-1970s [24,35] and has found various applications in harmonic analysis, probability theory and stochastic processes [18,19], fractal analysis and the related spectral theory [63, Chapters 18–23], and interpolations and both embedding of function spaces and entropy numbers [48,54]. As a special case, function spaces with logarithmic smoothness have received an increasing interest in the recent two decades [5–9,13–16,26,33]. Very recently, the Hajłasz-Sobolev spaces with generalized smoothness,
moreover, the inhomogeneous
If the inequality (1.4) holds only for almost every
then the related local homogeneous and local inhomogeneous
Motivated by the aforementioned works, the main purpose of this article is to find the geometric character of the underlying domain
if
Definition 1.1
Let
Clearly, if
and the
The main result of this article reads as follows, which gives an intrinsic geometric characterization of bounded
Theorem 1.2
Suppose that
and that the function
Then
The concepts of the number
Remark 1.3
Clearly, Theorem 1.2 generalizes the corresponding results in [66] for classical fractional Hajłasz-Sobolev spaces (namely in the case where
Let
Let
We also refer to [50, Examples 3.12 and 3.13] for two more examples.
Although Theorem 1.2 is on the embedding domain of generalized Hajłasz-Sobolev spaces, we also have some discussions on the related extension domains. Indeed, we prove in Theorem 4.10 that
The remainder of this article is organized as follows.
In Section 2, we state some basic notation and concepts used in this article, including some classical geometric concepts on domains such as uniform domains, cigar domains, and the slice property, as well as some basic assumptions on the function
In Section 3, we first introduce the concept of the modulus of continuity, which generalizes those introduced by Lappalainen in [47] in the sense that it contains more elements such as almost increasing but not increasing functions. Then we introduce the (local) weak
In Section 4, we discuss the relation between embedding and extension properties of
Section 5 is devoted to the proof of Theorem 1.2. We prove the sufficiency, that is, a weak
2 Preliminaries
We begin this section with some notational conventions and then recall some classical geometrical concepts and the Hajłasz-Sobolev spaces with generalized smoothness.
2.1 Basic notation and concepts
Let
For any
For any two nonempty subsets
and
For any given domain
In particular, when
For any curve
For any two points
We often use the following elementary inequality: if
2.2 Some classical geometric concepts
We now recall some geometric concepts.
Definition 2.1
A domain
Here,
A domain
(2.2)The smallest positive constant playing the role of
Remark 2.2
Uniform domains were introduced based on John domains by Martio and Sarvas [52]. It is known that bounded uniform domains are John domains and convex John domains are uniform domains; see [21–23,51,64] for some more details on these domains.
Definition 2.3
Let
In particular,
Remark 2.4
The weak
The following slice property was introduced by Buckley and Koskela [4].
Definition 2.5
Let
if
for any
for any
Remark 2.6
As pointed out in [4, p. 890], both (iii) and (iv) of Definition 2.5 imply that, for any domain
where the implicit positive constant only depends on the slice constant
Definition 2.7
A measurable set
Remark 2.8
The condition in Definition 2.7 is usually called the measure density condition.
We point out that changing “any
and hence, the measurable set
The value of
which competes the proof of the aforementioned claim.
2.3 The
ϕ
-Hajłasz-Sobolev spaces
We now recall the main function spaces under consideration. To this end, we begin with some basic concepts and properties on their generalized smoothness.
Definition 2.9
A function
This concept has been widely used [41, Section 2.2.1]. Some basic properties of almost increasing or decreasing functions are listed in the following two propositions. Since the proof of the first one is fundamental, we omit the details, and hence, we only show the second one. In what follows, when we prove a theorem (or the like), in its proof, we always use the same symbols as in the statement itself of that theorem (or the like).
Proposition 2.10
Let
Proposition 2.11
Let
and
where
Proof
Since, for any closed set
then, from the almost decreasing property of
By using the assumptions that
and hence,
and
which, combined with (2.7), further imply (2.5) and (2.6), respectively. This then finishes the proof of Proposition 2.11.□
Recall that a sequence
see, for instance, [19]. A continuous measurable function
We now recall the class
We also consider a larger class
Obviously, the only difference between
Remark 2.12
Clearly, if
where
For any given function
where
and
Remark 2.13
Let
By an argument similar to that used in the proofs of [50, (2.5) and (2.6)] and [49, (9) and (10)], we find that, if
(2.12)Notice that, for any
Similarly, if
If
which further implies that
By combining this and Definition 2.9, we find that
Similarly, if
In what follows, for simplicity, we always denote by the symbol
Definition 2.14
Let
For any given
(2.13)Let
(2.14)(2.13) holds. Especially, simply write
Definition 2.15
Let
The homogeneous
The homogeneous ball
The inhomogeneous
The inhomogeneous ball
Remark 2.16
There exists a positive constant
We now recall a Poincaré-type inequality related to the above spaces, which is precisely [49, (16)] in the Euclidean setting.
Lemma 2.17
Let
where the positive constant C is independent of B, u, and g.
Recall that the classical Hardy-Littlewood maximal operator
Below, for any
where
The following proposition gives the fact that, for any
Proposition 2.18
Let
Proof
Due to the symmetry, without loss of generality, we may assume that
To prove
For any
Then, by
which further implies that, for any
Notice that, by
Thus, using the Lebesgue differentiation theorem, (2.15), and Lemma 2.17 with
where
This finishes the proof of Proposition 2.18.□
Remark 2.19
For any
where the positive equivalence constants depend only on
Moreover, the space
Indeed, if (2.14) holds with
Conversely, if (2.17) holds, then
and hence
By this, we find that
Thus,
Obviously, by the definitions of both
Proposition 2.20
Let
Proof
By similarity, we only consider the homogeneous case. Since
Let
By using the uniformity assumption of
Since
Let
By applying this, the properties of
Remark 2.21
When
3 Generalized weak
F
-cigar domains and their weak self-improving property
In this section, we generalize the classical concept of weak
We begin with some notational convention. For any given function
Moreover, if the limit
Observe that, for any
3.1 Generalized weak
F
-cigar domains
We begin this subsection with the following definition.
Definition 3.1
A continuous function
Remark 3.2
Recall that the modulus of continuity
A modulus of continuity in Definition 3.1 belongs to both
where
Let
By an argument similar to that used in the proof of [47, Theorem 2.24], we find that every modulus of continuity
(3.3)where the infimum is taken over all rectifiable curves joining
We have the following properties of the modulus of continuity.
Proposition 3.3
Let
Proof
Let
This finishes the proof of Proposition 3.3.□
Proposition 3.4
Let
Definition 3.5
A function
Remark 3.6
Define
Any strictly decreasing function
There exist functions belonging to
Then
We now introduce another class of functions on
Definition 3.7
Let
Remark 3.8
Recall that an Orlicz function
Indeed, for any function
Thus, by
which further implies that
Let
Notice that (2.9) further implies that
where
which means that
Proposition 3.9
Let
If
Let
Proof
To prove (i), we argue by contradiction. Let
By using the admissible growth property of
Thus,
with the implicit positive constant independent of both
with the implicit positive constants independent of
Now, we show (ii). To this end, choose both
which means that
Now, we introduce the following concept of weak
Definition 3.10
Let
A domain
In the remainder of this article, for any given (locally) weak
Obviously, any weak
Recall that the class of weak
Proposition 3.11
Let F and G be two moduli of continuity such that
Remark 3.12
(i) Let
(ii) Recall that, when
There also exist some other functions
then, by Proposition 3.11, we find that all uniform domains are weak
Furthermore, by an argument similar to those used in the proofs of [47, Theorems 4.12 and 4.17], we conclude that, for any modulus of continuity
where
In addition, there also exist weak
(iii) Proposition 3.11 not only generalizes the classical result that
(iv) The class of weak
and, for any given
In this case,
When
which means that
When
that is, the union of three line segments
and
In the following, it suffices to show that
where
which proves (3.6) in this case. This finishes the proof of the aforementioned claim.
Let
(v) There also exist domains that are not weak
is not a
3.2 Weak self-improving property
It is known that the class of weak
Theorem 3.13
Let F be a modulus of continuity with
- (3.9)
- (3.10)
and, for any ball B centered in
(3.11) for any modulus of continuity
(3.12)
Inspired by the argument in [59, pp. 2222–2223], the proof of Theorem 3.13 relies on the following three main steps. First, find a family of
We begin with the following lemma, which, in the case where the two points
Lemma 3.14
Let F be a modulus of continuity and
Then the line segment
where
Proof
Without loss of generality, we may assume that
Since, for any
which, combined with the almost decreasing property of
This finishes the proof of Lemma 3.14.□
We also need the following two auxiliary lemmas, which generalize [59, Lemmas 2.2 and 2.7], respectively.
Lemma 3.15
Let F be a modulus of continuity,
then
for any given
and
there exists a positive constant
where
Proof
By Remark 3.2(ii), we find that
We first prove (i). The existence of
where
Now, we prove (ii). Since, for any
it follows that
which further implies that, whenever
Using both (2.5) with the almost decreasing function
with
Therefore, when
while, when
This finishes the proof of (ii), and hence, Lemma 3.15.□
Lemma 3.16
Let F be the same as in Definition
3.10,
where
and
- (3.19)
where
there exists a constant
(3.20)where
Proof
From Remark 3.2(ii), it follows that
We first show (i). By using (3.18), the assumptions of
To prove the remainder of (i), without loss of generality, we may assume that
Thus, by the definition of
where the implicit positive constants depend on both
Next, we prove (ii). We first claim that, for any
We show this by contradiction. Assume that there exist
By this, (3.3), and the triangle inequality for the metric
which contradicts (3.18). This proves the aforementioned claim.
Since, for any
where the implicit positive constant depends on both
Let
where, for any
Let
where the supremum is taken over all elements of the given family
Definition 3.17
A family
The following lemma is precisely [59, Corollary 2.10].
Lemma 3.18
Let
where
In the remainder of the proof of Theorem 3.13, as mentioned earlier, after finding a family of
Proof of Theorem 3.13
Notice that, for any
by setting
Let
and
where
From this, (3.27),
and
where
For
Below, we assume that the curve
By the definition of
Thus, by this, the almost increasing property of
which further implies that, for any
In addition, by both the almost increasing property of
By combining (3.30) and (3.31) and applying Lemma 3.16(ii) with
where
which, together with the arbitrariness of both
and hence,
From this, we infer that, for any
Thus,
Next, we prove that, for any
where the implicit positive constants are independent of
Indeed, on the one hand, for any given
by
where
On the other hand, the assumption
Notice that, for any
Recall that
which, combined with the almost decreasing property of
Thus, by the arbitrariness of
By using this and
which further implies that
From this, the almost decreasing property of
which shows (3.33).
Following the proof of [59, Theorem 1.5], we define
Here,
and
Write
where the implicit positive constant depends only on
Let
where the implicit positive constants depend on
By the assumption that
Using this, the almost increasing property of
Since, by the construction of
By using
which leads to a contradiction. Thus, we obtain
By letting
Finally, since the proof of the remaining inequality (3.11) depends only on the construction of both the family
Example 3.19
Let
Let
we deduce that
Let
For such a function
4 Relations between
ϕ
-Hajłasz-Sobolev embedding and extension domains
In this section, we discuss some embedding properties of
Let
Proposition 4.1
Let
Proof
Notice that, when
Now, we let
To show (i), let
similarly, we also have
Thus, by combining (4.1) and (4.2), we conclude that
which further implies
Next, we prove (ii). Let
which means that
Remark 4.2
Recall that the
which means that
where the last embedding naturally holds by (1.2). This means that
To characterize the extension and the embedding domains of
Lemma 4.3
Let
Proof
Let
where
Here, we find that, for any
and
This means that, if (4.3) holds for both
Now, we introduce an auxiliary function, which is supported in some ball. Let
Then, by
Next, we show that
Case 1. When
where the implicit positive constants depend only on
Case 2. When
Case 3. When
Case 4. In all other cases, one has
Altogether, we conclude that
where the implicit positive constant is independent of
where the implicit positive constant in the last quantity, denoted by
By
where
and, furthermore, from (4.8), it follows that
which further implies (4.3). This finishes the proof of Lemma 4.3.□
Lemma 4.4
Let all the assumptions be as in Lemma
4.3. Then
Proof
Let
with the implicit positive constant independent of both
Thus,
Remark 4.5
Let
We now have the following conclusion on the relations between
Proposition 4.6
Let
If
The domain
If
Proof
If
Next, we show the sufficiency of (ii). Let
By both the assumption of
where the implicit positive constant in the last quantity only depends on
Finally, we show (iii). We first prove its necessity. If
Conversely, to show the sufficiency, we let
Thus,
Let
When
Remark 4.7
Recall that a measure
Then, by the Hardy-Littlewood maximal function theorem for doubling measures [11], we find that, for any
We have the following technical lemma.
Lemma 4.8
Let
where, when
Proof
Let
Similarly to the proof of Lemma 4.3, without loss of generality, we may assume that, for any
By an argument similar to that used in the proof of Lemma 4.3 with
Thus,
where, when
with the implicit positive constant independent of both
Thus, when
Thus, by (4.11), we obtain
with the same modification made when
Remark 4.9
(i) In the proof of Lemma 4.8, for the given two points,
then, by an argument similar to that used in the proof of Lemma 4.3 and
with
(ii) If
We now establish the following relation between
Theorem 4.10
Let
To show Theorem 4.10, we need an auxiliary extension lemma as below. We refer the reader to [58, Theorem 1.3], [29, Theorem 6], and [66, Lemma 4.1] for some related results for the classical (Hajłasz-)Sobolev spaces.
Lemma 4.11
Let
Since the proof of [66, Lemma 4.1] was omitted, here, for the convenience of the reader, we give a complete proof of Lemma 4.11. The main strategy for proving Lemma 4.11 is to use the Whitney ball-covering, the reflected quasi-balls associated with this covering, and the fact that the measure of the boundary of any regular set is zero. To this end, we first recall some concepts.
Definition 4.12
Let
there exist two positive constants
there exists a positive constant
Remark 4.13
For any closed set
Definition 4.14
Let
is called a family of reflected quasi-balls with respect to
for any
for any
By [58, Theorem 2.6], we find that there exists such a family in Definition 4.14. Now, we are ready to prove Lemma 4.11.
Proof of Lemma 4.11
Applying both the regularity of
Let
We first consider the homogeneous case. In this case,
Let
and define
where, for any
To show that
if
one has
On the other hand, similar to the aforementioned arguments, from the almost increasing property of
Now, we prove that there exists a positive harmless constant
Case 1. When
which is a desired estimate in this case.
Case 2. When
which is a desired estimate in this case.
Case 3. When
Case 4. Finally, we consider the case where both
Case 4.1. If
we find that
on the other hand, by Definition 4.14(i) in the case of
Thus, using
which is a desired estimate in this case.
Case 4.2. If
on the other hand, using Definitions 4.14(i) and 4.12(ii) and using (4.24), we conclude that, for any
and, furthermore, for any
Thus, by
Altogether, we conclude that
Finally, from [58, Lemma 3.5], we deduce that
By this,
which, combined with (1.2), further implies
For the inhomogeneous case of this lemma, we introduce the same extension function
is a positive harmless constant multiple of an element in
which further implies that
Now, we show Theorem 4.10.
Proof of Theorem 4.10
If
We first prove
where
On the other hand, for almost every
with the implicit positive constants depending only on
where the implicit positive constants depend on
Altogether, we find that
Notice that
which further implies
Next, we show the homogeneous case, that is,
Thus,
5 Geometric characterization of
M
˙
ball
ϕ
,
p
-embedding domains
This section is devoted to proving our main theorem, Theorem 1.2. In Theorems 5.2 and 5.4, using the weak self-improving property of generalized weak cigar domains obtained in Theorem 3.13, we can show that a weak
In what follows, for any
Furthermore, if
Let
with
Lemma 5.1
Let
and the function
Proof
By both the almost decreasing property of
is almost decreasing. Furthermore,
The following theorem gives a better result than the sufficiency of Theorem 1.2.
Theorem 5.2
Let
Proof
From
where
We first claim that, for any
and, moreover, from
Thus, by Remark 2.13(i) with
Now, we turn to prove (5.3). Let
Let
Case 1.
and, furthermore, similar to (5.4), by
By combining (5.4) and (5.5), we conclude that
which shows (5.3) in the case of
Case 2.
and
Thus, from an argument similar to that used in the estimations of both (5.4) and (5.5) with
Let
In the following, we estimate
We first estimate
Notice that, for any
which, combined with (5.9), the almost decreasing property of
From
is almost decreasing. Similarly, by the assumption that
is almost increasing. From Remark 2.13(i), it follows that
Thus, by using
To estimate
with the implicit positive constants independent of
By using this, (5.15), and
Altogether, we conclude that (5.3) holds for almost every
The following gives an example, which confirms the existence of
Remark 5.3
Let all the assumptions be the same as in Theorem 5.2, and let both
Then there exists
Indeed, observe that the function
is continuous and decreases to
Then, by
which, together with Remark 2.13(i), further implies that
In particular, if
We now have the following conclusion.
Theorem 5.4
Let
To prove Theorem 5.4, it suffices to show that
Next, we turn the attention to the necessity of Theorem 1.2, where an additional assumption that
Theorem 5.5
Let
Then, for any modulus of continuity,
Proof
By Remark 2.13(ii), we have
Let
When
and, for any
Thus, from the almost increasing property of
where the implicit positive constants depend on both
In the following, we let
and
where
We first claim that there exists a positive constant
Without loss of generality, we may assume that
Case 1. When
which means that
Thus, by the definition of
Case 2. When
Case 3.
and hence,
Case 3.1. If
Case 3.2. If
From (5.21), we infer that
which, together with (5.24) and
However, notice that, by both Definition 2.5(ii) and (5.20), we have
which leads to a contradiction. Thus, this case does not exist, and hence, both
Altogether, we conclude that
Next, define
and
Then
Notice that, by
which further implies that
with the implicit positive constants depending only on both
and, moreover,
where
which, together with the definitions of both
From this, (5.26), and the definition of
Finally, notice that, for any
Thus, by applying the almost increasing property of
Now, we have already showed the above inequality for almost every
Finally, as an immediate consequence, we obtain Theorem 1.2 by combining Theorem 5.4 with
Remark 5.6
The assumption of the slice property is necessary for Theorem 5.5 and also for the necessity part of Theorem 1.2. To see this, let
where, for any
Acknowledgements
The authors would like to thank both referees for their carefully reading and useful comments which improve the quality of this article.
-
Funding information: This project is partially supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900), the National Natural Science Foundation of China (Grant Nos. 12431006 and 12371093), and the Fundamental Research Funds for the Central Universities (Grant No. 2233300008).
-
Author contributions: All authors developed and discussed the results and contributed to the final manuscript.
-
Conflict of interest: The authors state no conflict of interest.
-
Data availability statement: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
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