Research Article | Open Access
Fernando Farroni, Raffaella Giova, "Explicit Bounds and Sharp Results for the Composition Operators Preserving the Exponential Class", Journal of Function Spaces, vol. 2016, Article ID 3769813, 9 pages, 2016. https://doi.org/10.1155/2016/3769813
Explicit Bounds and Sharp Results for the Composition Operators Preserving the Exponential Class
Let be a quasiconformal mapping whose Jacobian is denoted by and let be the space of exponentially integrable functions on . We give an explicit bound for the norm of the composition operator : and, as a related question, we study the behaviour of the norm of in the exponential class. The property of is the counterpart in higher dimensions of the area distortion formula due to Astala in the plane and it is the key tool to prove the sharpness of our results.
1. Introduction and Statement of the Main Result
This paper is concerned with the interplay between quasiconformal mappings and the space of exponentially integrable functions. Let us recall that a homeomorphism defined on an open subset of (with ) is a -quasiconformal mapping if and for some constant . Here stands for the differential matrix of and denotes the Jacobian determinant of . The norm of in (1) is defined as
If is a bounded domain of with measure , the space of exponentially integrable functions is the set of measurable functions such that there exists for which where the mean value notation is used. One of the interesting properties of functions in consists in the fact that they may be characterized as BMO-majorized functions. Indeed, in  it is proved that if and only if there exists such that For the definition of the space BMO of functions of bounded mean oscillation see Section 2 below.
Quasiconformal mappings and BMO-functions are related by the fact that the composition operator maps BMO into itself continuously, as stated by a result of Reimann : there exists such that, for every , one has
In light of the connection between exponentially integrable functions and functions of bounded mean oscillation, composition operators acting continuously on have been considered in [3, 4], where it is proved that, given a -quasiconformal mapping , there exists such that, for every and for every ball , one has The estimates above may be seen as the analogy of (4) in the framework of the space of exponentially integrable functions.
It is worth pointing out that spaces of functions of bounded mean oscillation and exponentially integrable functions are not the only ones which are stable under quasiconformal changes of variables. We recall that quasiconformal mappings and their generalizations provided by homeomorphisms of finite distortion or bi-Sobolev mappings (see, e.g., [5, 6]), turn to be the class of homeomorphisms for which the composition operator acts continuously between Sobolev, Lorentz-Sobolev, and Zygmung-Sobolev spaces (see [7–9] and the references therein).
Explicit estimates of the constants appearing in (4) and (5) have been considered a problem of its own interest (see, e.g.,  for an application). In the planar case, sharp estimates for the constant appearing in (5) are given in . As a refinement of the result of Reimann, explicit estimates of the constant appearing in (4) are provided in . More precisely, a constant depending on the Jacobian of may be defined (see Section 2.4 below) in such a way that the results of  may be stated as followsfor every and for some suitable constant depending only on the dimension . The definition of the constant strongly relies on the fact that the Jacobian of a quasiconformal mapping is a weight in or equivalently in (see Section 2.4 for the definitions of such classes of weights). In particular, it is well known (see ) that if and only if for every ball and every measurable set it holds for some independent of and . Equivalently (see again ) a weight if and only if for every ball and every measurable set it holds for some independent of and .
The goal of this paper is to seek a precise estimate as in (6) in the framework of the class of exponentially integrable functions. To this aim, we define for a weight in the constant as and similarly, we define for a weight in the constant as
Our main result reads as follows.
Theorem 1. Let be bounded domains of , . Let be a -quasiconformal mapping. Then for each ball and for every .
Our result is sharp, in the sense that equalities are attained in (11) for special choices of the mapping (e.g., when is the identity map). We are able to obtain such optimal result since we give a characterization of constant weights in terms of the constants in (9) and (10) in Section 4. In particular, a weight is constant a.e. in if and only if and . Moreover, the result of Theorem 1 extends the one of , where the case of planar principal quasiconformal mappings has been considered. We call principal any quasiconformal mapping which is conformal outside the unit disk and which satisfies the following normalization For this peculiar class of quasiconformal mappings, the following result has been previously established.
Theorem 2 (see ). Let be a -quasiconformal principal mapping which maps onto itself. Then, the following estimates hold true for every .
We will provide an alternative proof of the previous result, which is based on the fact that estimate (11) reduces to (13) for a principal quasiconformal mapping which maps the unit disk onto itself. In general, a principal quasiconformal map in the plane does not necessarily send the unit disk onto itself. However, there exist nontrivial examples of principal quasiconformal mappings sending the unit disk onto itself, such as the radial stretching of the form
The paper is organized as follows. Section 2 is devoted to the definitions of the function spaces object of our studies. In particular, the connection between BMO-functions and -weights is treated. In Section 3 we give the proof of Theorem 1. In Section 4 we give the aforementioned characterization of the constant weights, which allows us to conclude the sharpness of Theorem 1. Finally, in Section 5 applications of Theorem 1 are given; in particular we provide a precise estimate which relates and for any ball .
2.1. Properties of Quasiconformal Mappings
We report here some well-known facts about quasiconformal mappings. We recall that the change of variables formula holds for a quasiconformal mapping . More precisely, if then and for every measurable (see, e.g., ). More generally, for an arbitrary Sobolev homeomorphism, the validity of the change of variables formula depends on the set of the points where the homeomorphism is approximately differentiable (see, e.g., ); more generally the condition of being absolutely continuous on lines plays an important role, especially for planar mappings (see, e.g., ), since very often properties of this type of mappings are sufficient to prove statements which one would assign to general Sobolev homeomorphisms.
2.2. Exponentially Integrable Functions
We recall (see, e.g., ) that is a Banach space equipped with the norm where is the nonincreasing rearrangement of and is the distribution function of
On the other hand, may be also equipped with the Luxemburg norm defined as This norm is equivalent to the one in (16). As observed in , is not a dense subspace of . The distance to in the space is defined as Appealing to the results in  the distance to in evaluated with respect to the Luxemburg norm (19) is given by for every .
2.3. Functions of Bounded Mean Oscillation
A locally integrable function has bounded mean oscillation, , if
The supremum in (22) is taken over all open balls and the notation is used for averages.
2.4. and Classes
For our purposes, it is fundamental to introduce the Muckenhoupt class and the Gehring class . First of all, we say that a measurable function is a weight if is positive a.e. and locally integrable in . A weight belongs to the Muckenhoupt class if Similarly, a weight belongs to the Gehring class if The suprema in (24) and (25) are taken over all balls .
As a corollary of Gehring’s Lemma , Jacobians of quasiconformal mappings are weights in the (or equivalently ) class. By virtue of the change of variables formula for quasiconformal mappings (see Section 2.1 above) and (7), for every ball and every measurable set it holds for some independent of and .
We briefly refer to as the -constant of .
As done for the -constant, in  a second auxiliary constant is defined as We briefly refer to as the -constant of .
It is worth pointing out that to each function there corresponds a weight in the class of Gehring given by , for some depending on and (see ). Conversely (see again ) if (or equivalently if ) then . In particular, is a BMO-function, whenever is a quasiconformal mapping.
For the sake of completeness, we also recall the definition of the Muckenhoupt class for . A weight belongs to the Muckenhoupt class for if As a natural extension of the above definition, one can consider the Muckenhoupt class which covers the limit case . A weight belongs to the Muckenhoupt class if The suprema in (31) and (32) are taken over all balls . For each we call the -constant of the weight .
We recall here the definition of the Gehring class for . A weight belongs to the Gehring class for if As a natural extension of the above definition, one can consider the class which cover the limit case . A weight belongs to the Gehring class if The suprema in (33) and (34) are taken over all balls . For each we call the -constant of the weight . Each weight in the class satisfies a reverse Hölder inequality. This is a key fact in order to study the regularity of the Jacobian of quasiconformal mappings (see ).
3. Explicit Bounds
We start by proving Theorem 1.
Proof of Theorem 1. Let us assume that and are arbitrary constants for which (27) holds, with . We notice that for every We compare the distribution functions of and by means of the estimate (27) and we obtainLet and let be such that From (36) we get and therefore, from the definition of nonincreasing rearrangement (17), we obtain that We deduce directly from the definition of the norm (16) that Combining (39) and (40) we obtain Let us introduce the function so that (41) may be rewritten as and let us define Our aim is to show that is bounded in and we want to compute explicitly its upper bound. To this aim, we compute the derivative of so, the monotonicity of the function depends only on the sign of the constant and for every we deduce The definition of (see (44)) and (46) implyCombining the latter estimate with (43) we conclude that and, by virtue of the definition of the norm in (16), we conclude that Due to the definition of the constant and to the fact that and are arbitrary constants for which (27) holds, we obtain for each ball and for every .
It remains to prove the first inequality in (11). Let us assume that and are arbitrary constants for which (29) holds, with . As before, we compare the distribution functions of and . This time we make use of the estimate (29) instead of (27) and we argue as in the proof of estimate (36). Indeed, if we pick we obtain Let and let be such that From (51) we getwhich in turn implies It follows from the definition of nonincreasing rearrangement (17) that The argument which leads to (41) and the definition of the norm in (16), allows us to conclude that Arguing as in the proof of estimate (49) we have Due to the definition of the constant and to the fact that and are arbitrary constants for which (29) holds, we obtain for each ball and for every . The proof is complete.
We are in a position to prove Theorem 2.
Proof of Theorem 2. Any principal quasiconformal mapping in the plane, which maps the unit disk onto itself, satisfies for every measurable subset (see [25, Theorem 13.1.4]). Appealing to the definition (9) we deduce that therefore directly follows from (59).
We introduce the auxiliary function Since we have and in particular , that is Combining the latter identity with (61) we conclude that The proof is complete, since (11) infers (13) in case of any -quasiconformal principal mapping which maps onto itself.
4. A Characterization of Constant Weights
We explicitly remark that, for a weight in , one always has This result is proved in . With a similar proof, one can show the same property for the -constant, that is
Proposition 3. Let be an weight such that . Then is constant a.e. in .
Proof of Proposition 3. Let us fix some integer . By the definition (10) of , we may find two sequences and fulfilling and for every ball and every measurable set it holds Moreover, and are independent of and . Because of condition (67), we may assume (up to a subsequence) that converges to some . Since one clearly has for every , condition (67) implies . Conditions (67) and (68) also imply Appealing to the fact that as , the latter relation implies as . Therefore, we may pass to the limit as in (69) in order to get for every ball and every measurable set . In particular, for every ball and for a.e. we have which in turn implies that is a constant function.
Adapting suitably the proof of Proposition 3 one can prove next result featuring the -constant.
Proposition 4. Let be an weight such that . Then is constant a.e. in .
5. The Logarithm of the Jacobian of a Quasiconformal Mapping
Let be a quasiconformal mapping. In Section 2.4 we mentioned that is a BMO-function. Similarly (see, e.g., ) . Our next results are consequences of Theorem 1 and relate the logarithm of the Jacobian of a quasiconformal mapping with the one of its inverse taking into account the constants and . Moreover, these results can be seen as the counterpart in the space of exponentially integrable functions of the results of Reimann  in the setting of the BMO-space.
Corollary 5. Let be bounded domains of , . Let be a -quasiconformal mapping and let be the constant defined in (9). Then for each ball and for every .
Corollary 6. Let be a quasiconformal mapping with . Then, there exists a constant depending only on , such that
Proof of Corollary 5. We start by observing (see, e.g., ) that Therefore, we may write the following identity for the Jacobian of then, from (11), we have It remains to prove the first inequality in (73). Consider the first inequality in (11), namely We recall that and are both quasiconformal. So, we pick where is any function in , then (78) gives us In (76) we may replace by , so that Thus, (79) and (80) infer The desired inequality (73) is proved.
Proof of Corollary 6. As in the proof of Corollary 5, we know that (76) holds. Then from (6), we have It remains to prove the first inequality in (74). In (82) we may replace by . Hence The desired inequality (74) is proved.
Corollary 7. Let be bounded domains of , . Let be a -quasiconformal mapping and let be the constant defined in (9). Then for each ball .
Corollary 8. Let be a quasiconformal mapping with . Then, there exists a constant depending only on , such that
Proof of Corollary 7. Let and let . Then, from (11), we have It is clear that there exists a measurable set of zero measure such that is bounded in . Thus, is bounded in ; on the other hand, one has , since satisfies the Lusin condition (see, e.g., ). We proved that . Thus, from (88) we know that Since is an arbitrary function in , we conclude that Now we prove the inequality in (85). Thanks to (79) and arguing as before, we may conclude that Thus The inequality in (85) is proved.
Proof of Corollary 8. Let and let . Then, from (6), we have It is clear that there exists a measurable set of zero measure such that is bounded in . Thus, is bounded in ; on the other hand, one has , since satisfies the Lusin condition. We proved that . Thus, from (93) we know that Since is an arbitrary function in , we conclude that The inequality (87) follows by replacing by in (95), namely
Given a function , we consider the quantity introduced in  by Garnett and Jones. The supremum in (97) is taken over all balls . The main result of  states that is equivalent to the distance to in the space defined as We also refer to [17, 27] for the problem of finding a formula for the distance to in grand Sobolev and grand Orlicz spaces. The next result is an immediate consequence of the previous result.
Corollary 9. Let be a quasiconformal mapping with . Then, there exists a constant depending only on , such that
The authors declare that there is no conflict of interest regarding the publication of this paper.
Raffaella Giova has been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by Project Legge 5/2007 Regione Campania “Spazi pesati e applicazioni al calcolo delle variazioni”.
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