Journal of Function Spaces
Volume 2019 (2019), Article ID 5234507, 9 pages
https://doi.org/10.1155/2019/5234507

Research Article

An Orlicz-Besov Poincaré Inequality via John Domains

Hongyan Sun

Department of Sciences, China University of Geosciences, Beijing 100083, China

Correspondence should be addressed to Hongyan Sun; sun_hy@cugb.edu.cn

Received 14 October 2018; Accepted 17 November 2018; Published 1 January 2019

Academic Editor: Stanislav Hencl

Copyright © 2019 Hongyan Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Denote by the intrinsic Orlicz-Besov space, where , is a Young function, and is a domain. For and optimal , via John domains, we establish criteria for bounded domains supporting an Orlicz-Besov Poincaré inequality. This extends the known criteria for bounded domains supporting Sobolev-Poincaré inequality and its fractional analogue.

1. Introduction

Let in this paper. Recall that a bounded domain is called a John domain with respect to some and if, for each , there is a rectifiable curve parameterized by arc-length (called a John curve for ) such that , , and for all .

It is well known that John domains essentially characterize bounded domains of supporting Poincaré-Sobolev inequality and some fractional analogue in the literature. Precisely, Bojarski [1] proved that, for any , a John domain always supports the following Sobolev-Poincaré inequality: there exists a constant such that

whenever the distributional derivative . In this paper, is the average of in set with . When and , as proved by Hurri-Syrjänen and Vähäkangas [2] (see also Dyda, Ihnatsyeva, and Vähäkangas [3]), a John domain always supports the following fractional Sobolev-Poincaré inequality: there exists a constant such that

whenever . Moreover, as proved by Zhou [4], when and , a John domain always supports the following Hajłasz-Sobolev Poincaré inequality: there exists a constant such that

whenever is an intrinsic -order Hajłasz upper gradient of ; that is,

Conversely, suppose that is a bounded simply connected planar domain and a bounded domain that is quasiconformally equivalent to some uniform domain when . Buckley-Koskela [5] proved that if satisfies (1), then is a John domain. If either (3) or (2) holds, then is also proven to be a John domain; see [2–4].

The main purpose is to prove that John domains essentially characterize bounded domains of supporting an Orlicz-Besov Poincaré inequality as below. Let be a Young function in ; that is, is convex and satisfies , for and . Given any and domain , as motivated by the Orlicz-Besov spaces considered in [6] and also semi-norms that appeared in the right hand side of (2) and (3), we consider the intrinsic Orlicz-Besov space , which is defined as the collection of all measurable functions in whose (semi-)norms

are finite. Modulo constant function is a Banach space. In the spirit of (1), (2), and (3), we say that a bounded domain supports an Orlicz-Besov Poincaré inequality with respect to and if there exists a constant depending on , , , and such that

Theorem 1. Let and be a Young function satisfying

(i) If is a John domain, then supports the Orlicz-Besov Poincaré inequality (6).
(ii) Assume that is a bounded simply connected planar domain, or a bounded domain that is quasiconformally equivalent to some uniform domain when . If supports the Orlicz-Besov Poincaré inequality (6), then is a John domain.

The assumption (7) above guarantees and hence the nontriviality of ; see Lemma 4. For the optimality of (7), see Remark 5. The assumption (8) always fails if . Indeed, since is convex, , and for all , there exists such that for all . Then we always have

For , Young functions satisfying (7) and (8) include with and or , with and also their convex combinations. For , Young functions satisfying (7) and (8) include with and or and also their convex combinations.

Moreover, for any and , let and . Then , and satisfies (7) and (8). In this case, we always have

We see that (6) coincides with (2). Thus, Theorem 1 extends above criteria for bounded domains supporting fractional Poincaré inequality given in [2, 3] (see also [4]).

We prove Theorem 1 in Section 3. To prove Theorem 1 (i), we use Boman’s chain rule for John domains and an Orlicz-Besov imbedding in cubes under (7) and (8) as proved by [7] (see Lemma 8) and also borrow some ideas from [1, 4]. To prove Theorem 1 (ii), we use (7) and (8) to calculate the -norms of some cut-off functions in a precise way; see Lemmas 6 and 7. This allows us to prove the LLC(2) property of domains supporting inequality (6); see Proposition 9. By borrowing some ideas from [4, 5, 8], we then obtain Theorem 1 (ii).

Here we make some conventions on the notations or notion used in this paper. Throughout the paper, denotes positive constants, which depends only on but whose value might be changed from line to line. We write if there exists a constant such that . For any and , denotes the distance from to and denotes the diameter of the set .

2. Some Basic Properties

The following properties of Young functions are given in [7, ].

Lemma 2. Let and be a Young function.
(i) If satisfies (7), then

(ii) If satisfies (8), then as for any and

Next we show that intrinsic Orlicz-Besov functions are locally integrable.

Lemma 3. Let and be a Young function. For any bounded domain , we have as sets.

Proof. Let . For , we have

By Fubini’s theorem, for almost all we have

For any ball with , one may choose such satisfying above inequality so that

Thus this implies that

By Jensen’s inequality, we have

Notice that as . Then

which implies that

That is, as desired. The proof of Lemma 3 is complete.

The assumption (7) guarantees the nontriviality of intrinsic Orlicz-Besov spaces in bounded domains.

Lemma 4. Let and be a Young function satisfying (7). For any bounded domain , we have as sets.

Proof. Let and . By (7) we have

Since the right hand side is less than when is sufficiently large enough, we know that ; that is, as desired. The proof of Lemma 4 is complete.

Remark 5. We remark that the assumption (7) is optimal to get or the nontrivial of in the following sense. For and , let . By a direct calculation, the Young function satisfies (7) if and only if . On the other hand, let . If , then , and observe that (10) also holds for all measurable functions in . For any ball , write as the space of all functions satifying

By [9, &4.2] and , we know that only contains constant functions. If , then for any ball with . Thus is a constant on every ball with , and hence by the connectedness of , is a constant function in .
To end this section, we calculate -norms of some special functions, which will be used in Sections 2 and 3.
Let and . If is connected, we write ; if is disconnected, we let be any component of . For with , define a function in by

Lemma 6. Let and be a Young function satisfying (7) and (8). For any bounded domain and with , if is a component , with , we have

Here is a constant depending only on , , and .

Proof. Write for simplicity. For any and , implies that at least one of lies in . Thus

Since for all with , and for all with , we further write

By a change of variables and applying (7), we obtain

Similarly, by a change of variables and applying (8), we obtain

Therefore,

If

by the convexity of , we have Thus as desired. The proof of Lemma 6 is complete.

Let , , and such that . Define

where the infimum is taken over all rectifiable curves joining and .

Lemma 7. Let and be a Young function satisfying (7) and (8). There exists a constant such that, for all bounded domains , , and satisfying , we have

Proof. Assume that is disconnected. One has . Note that if and , we have . Indeed, for such , we have

and hence

Therefore, . Letting be the line segment joining contained in , we have . For any , we know that gives a curve joining and and

Thus we have . Similarly, .
For any , we have

Indeed, letting be the line segment joining , we have . Since connects and for any , we have

Similarly, by changing the roles of , we also have

Therefore, we obtain

Note that, for , one has

By (7), we obtain

for some . Thus for any , by Jensen’s inequality, we have , which implies that as desired.

3. Proof of Theorem 1

With the aid of the following result given in [7, ], we are able to prove Theorem 1 (i) as below. For any cube and any measurable functions in , we write

Lemma 8. Let and be a Young function satisfying (7) and (8). Then there exists a constant depending on such that

Proof of Theorem 1. (i) Assume that is a bounded John domain. By Boman [10] and Buckley et al. [11], enjoys the following chain property: for every integer , there exist a positive constant and a collection of cubes such that
(a) for all , and

(b) is fixed cube; for any other , one can find a sequence from satisfying that, for all , we have

Set . By given in (a), we write

By Lemma 8, we have

and hence

Let , for with . For any , by as in (a), we have . Thus by (12) and convexity of , we get

We then obtain

and hence .
To estimate , for each write

By (b), Hölder inequality, and Lemma 8, we have

Since , given in (b), we have

Therefore, by (a),

Recall that by Lemma 4.2 in [1] (see also [12]), for any and , we have

Thus,

Since (a) implies the bounded overlaps of cubes in , by , we have

for all . From this, it follows that

By an argument same as , we have
Combining the estimates for and together, we arrive at

Since this yields and

we obtain (6) as desired. This completes the proof of Theorem 1 (i).

To prove Theorem 1 (ii), we need the following result.

Proposition 9. If is a bounded domain supporting the -imbedding, then has the LLC(2) property; that is, there exists a positive constant depending only on , and such that, for all and , any pair of points in can be joined in .

Proof. Fix a point so that

We claim that there exists such that are contained in the same component of whenever for some and .
Note that this claim gives LLC(2) properties of . Indeed, assume that this claim holds for the moment. First, letting , we show that are contained in the same component of whenever for some and . Indeed, if , then, for any , we have

and hence, by , we have ; that is, are contained in the connected set . If and , are contained in the connected set . If and , letting , by , we see that . By the above claim, are contained in the same component of . Noting that

we see that are contained in the same component of as desired.
Let for some and . If , then . Since , by the above argument, we know that for some component of . Similarly, since , . Thus . If , then , which implies , and hence . Thus,

which implies is a connected set. Note that . This gives the desired LLC(2) properties of .
In the following, we prove above claim by finding . Assume that for some and . Set

Without loss of generality, we assume that ; otherwise we take . denotes the component of containing . To find , it suffices to prove

where is a constant independent of . Indeed, set By (65), we have , and hence are contained in the same component of as desired.
To prove (65), write . We also denote by the component of containing . Then

Indeed, by the definition of , Note that So
Proof of . Define

where the infimum is taken over all rectifiable curves joining and .
Since is disconnected, by Lemma 7, we have and hence by the imbedding assumption, we have

On the other hand, by (66) one has in . Thus

Since for all , we have . By this, (69), and (68), we have

which gives as desired.
Proof of For integer , take such that

Define a function in by

By Lemma 6, we have

Hence, by the imbedding assumption, we obtain

By (66), we have for all . Similarly to (69), we have Thus

Therefore

This further yields that

By Lemma 2, as , and hence, there exists a constant such that

If , then (78) implies that as desired.
If , then for , define

Let . Applying Lemma 6, (71), and (66), by an argument similar to (78), we get

This implies that

as desired.

Using Proposition 9, we are ready to prove Theorem 1 (ii).

Proof of Theorem 1. (ii) Suppose that is a simply connected planar domain, or a bounded domain that is quasiconformally equivalent to some uniform domain when . Assume that supports the Orlicz-Besov Poincaré inequality (6).
By [5, 13], we know that has a separation property with respect to and some constant , that is, for every , there exists a curve with such that for each , either or each belongs to a different component of than .
For any , let be a curve as above. It then suffices to prove

Indeed, by the arguments in [14, pp.385-386] and [15, pp.7-8], (82) allows us to modify to get a John curve for . Here we omit the details.
To prove (82), by Proposition 9, we know that enjoys the LLC(2) property. Let , where is the constant in LLC(2). For any , if , then

which implies

Assume . It is enough to prove that

Indeed, this gives

This and (84) give (82) as desired. We prove (85) by contradiction. If (85) is not true, then there exists . Since

by Proposition 9, and are contained in the same component of . Noting that , we have that and are contained in the same component of , which is contradicted with the separation property. Thus, (85) holds. This finishes the proof of Theorem 1 (ii).

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author would like to thank Professor Yuan Zhou for valuable discussions. The author also would like to thank the supports of National Natural Science of Foundation of China (No. 11601494).

References

B. Bojarski, “Remarks on Sobolev imbedding inequalities,” in Complex analysis, Joensuu 1987, vol. 1351 of Lecture Notes in Math., pp. 52–68, Springer, Berlin, 1988.

R. Hurri-Syrjänen and A. V. Vähäkangas, “On fractional Poincaré inequalities,” Journal d'Analyse Mathématique, vol. 120, pp. 85–104, 2013.

B. o. Dyda, L. Ihnatsyeva, and A. V. Vähäkangas, “On improved fractional Sobolev-Poincaré inequalities,” Arkiv fOr Matematik, vol. 54, no. 2, pp. 437–454, 2016.

Y. Zhou, “Criteria for optimal global integrability of Hajł asz-Sobolev functions,” Illinois Journal of Mathematics, vol. 55, no. 3, pp. 1083–1103 (2013), 2011.

S. Buckley and P. Koskela, “Sobolev-Poincaré implies John,” Mathematical Research Letters, vol. 2, no. 5, pp. 577–593, 1995.

M. C. Piaggio, “Orlicz spaces and the large scale geometry of Heintze groups,” Mathematische Annalen, vol. 368, no. 1-2, pp. 433–481, 2017.

H. Sun, Orlicz-Besov imbedding and Ahlfors n-regular domains, arXiv:1810.03796.

P. Hajł asz, P. Koskela, and H. Tuominen, “Sobolev embeddings, extensions and measure density condition,” Journal of Functional Analysis, vol. 254, no. 5, pp. 1217–1234, 2008.

A. Gogatishvili, P. Koskela, and Y. Zhou, “Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces,” Forum Mathematicum, vol. 25, no. 4, pp. 787–819, 2013.

J. Boman, “Lp-estimates for every strongly elliptic systems,” Tech. Rep., Department of Mathematics, University of Stockholm, Sweden, 1982.

S. Buckley, P. Koskela, and G. Lu, “Boman equals John,” in XVI Rolf Nevanlinna Colloquium (JOEnsuu, 1995), pp. 91–99, de Gruyter, Berlin, 1996.

P. Koskela, Lectures on quasiconformal and quasisymmetrical mappings, University of Jyväskylä, 2011.

S. M. Buckley and P. Koskela, “Criteria for imbeddings of Sobolev-Poincaré type,” International Mathematics Research Notices, vol. 1, p. 8, 1996.

O. Martio and J. Sarvas, “Injectivity theorems in plane and space,” Suomalaisen Tiedeakatemian Toimituksia. Sarja A. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, vol. 4, no. 2, pp. 383–401, 1979.

R. Näkki and J. Väisälä, “John disks,” Expositiones Mathematicae. International Journal for Pure and Applied Mathematics, vol. 9, no. 1, pp. 3–43, 1991.