Explicit Bounds and Sharp Results for the Composition Operators Preserving the Exponential Class

for some constant K ≥ 1. Here Df(x) stands for the differential matrix of f and J f (x) = detDf(x) denotes the Jacobian determinant of f. The norm |Df(x)| ofDf(x) in (1) is defined as |Df(x)| = sup{|Df(x)ξ| : ξ ∈ Rn, |ξ| = 1}. If Ω is a bounded domain of Rn with measure |Ω|, the space of exponentially integrable functions EXP(Ω) is the set of measurable functions u : Ω → R such that there exists λ > 0 for which


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  : Ω → R  defined on an open subset Ω of R  (with  ≥ 2) is a quasiconformal mapping if  ∈  1, loc (Ω, R  ) and      ()      ≤   () for a.e. ∈ Ω, for some constant  ≥ 1.Here () stands for the differential matrix of  and   () = det () denotes the Jacobian determinant of .The norm |()| of () in ( 1) is defined as |()| = sup{|()| :  ∈ R  , || = 1}.
If Ω is a bounded domain of R  with measure |Ω|, the space of exponentially integrable functions EXP(Ω) is the set of measurable functions  : Ω → R such that there exists  > 0 for which where the mean value notation − ∫ Ω = (1/|Ω|) ∫ Ω is used.One of the interesting properties of functions in EXP(Ω) consists in the fact that they may be characterized as BMO-majorized functions.Indeed, in [1] it is proved that  ∈ EXP(Ω) if and only if there exists V ∈ BMO(R  ) such that | ()| ≤ V () a.e. ∈ Ω. ( For the definition of the space BMO(R  ) of functions of bounded mean oscillation see Section 2 below.Quasiconformal mappings  : R  → R  and BMOfunctions  : R  → R are related by the fact that the composition operator   →  ∘  −1 maps BMO(R  ) into itself continuously, as stated by a result of Reimann [2]: there exists  = (, ) ≥ 1 such that, for every  ∈ BMO(R  ), one has In light of the connection between exponentially integrable functions and functions of bounded mean oscillation, composition operators acting continuously on EXP have been considered in [3,4], where it is proved that, given a -quasiconformal mapping  : Ω → R  , there exists 2 Journal of Function Spaces  = (, ) ≥ 1 such that, for every  ∈ EXP(Ω) and for every ball  ⊂⊂ Ω, one has 1  (, ) ‖‖ EXP() ≤       ∘  −1     EXP(()) ≤  (, ) ‖‖ EXP() . ( 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][8][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., [10] for an application).In the planar case, sharp estimates for the constant (2, ) appearing in (5) are given in [4].As a refinement of the result of Reimann, explicit estimates of the constant (, ) appearing in (4) are provided in [11].More precisely, a constant G1 (  ) depending on the Jacobian of  may be defined (see Section 2.4 below) in such a way that the results of [11] may be stated as follows for every  ∈ BMO(R  ) and for some suitable constant  = () ≥ 1 depending only on the dimension .The definition of the constant G1 (  ) strongly relies on the fact that the Jacobian of a quasiconformal mapping is a weight in  ∞ or equivalently in  1 (see Section 2.4 for the definitions of such classes of weights).In particular, it is well known (see [12]) that V ∈  1 if and only if for every ball  ⊂ R  and every measurable set  ⊂  it holds for some 0 <  ≤ 1 ≤  independent of  and .Equivalently (see again [12]) a weight  ∈  ∞ if and only if for every ball  ⊂ R  and every measurable set  ⊂  it holds for some 0 <  ≤ 1 ≤  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 V in  1 the constant Ĝ1 (V) as and similarly, we define for a weight  in  ∞ the constant Â∞ () as Our main result reads as follows.
The norm ‖ ⋅ ‖ EXP() is defined in ( 16) of Section 2.2 below.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  1 weight V is constant a.e. in R  if and only if Ĝ1 (V) = 1 and Â∞ (V) = 1.Moreover, the result of Theorem 1 extends the one of [3], where the case of planar principal quasiconformal mappings has been considered.We call principal any quasiconformal mapping  : R 2 → R 2 which is conformal outside the unit disk D = { ∈ R 2 : || < 1} and which satisfies the following normalization For this peculiar class of quasiconformal mappings, the following result has been previously established.
Theorem 2 (see [3]).Let  : R 2 → R 2 be a -quasiconformal principal mapping which maps D onto itself.Then, the following estimates hold true for every  ∈ EXP(D).
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  : R 2 → R 2 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 ‖ log   −1 ‖ EXP(()) and ‖ log   ‖ EXP() for any ball  ⊂⊂ Ω.

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  : Ω → Ω  .More precisely, if for every measurable  ⊂⊂ Ω (see, e.g., [13]).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., [14]); more generally the condition of being absolutely continuous on lines plays an important role, especially for planar mappings (see, e.g., [15]), since very often properties of this type of mappings are sufficient to prove statements which one would assign to general Sobolev homeomorphisms.

Exponentially Integrable Functions.
We recall (see, e.g., [16]) that EXP(Ω) is a Banach space equipped with the norm where and On the other hand, EXP(Ω) may be also equipped with the Luxemburg norm defined as This norm is equivalent to the one in (16).As observed in [17],  ∞ (Ω) is not a dense subspace of EXP(Ω).The distance to  ∞ (Ω) in the space EXP(Ω) is defined as Appealing to the results in [17] the distance to  ∞ (Ω) in EXP(Ω) evaluated with respect to the Luxemburg norm ( 19) is given by for every  ∈ EXP(Ω).

Functions of Bounded Mean Oscillation.
A locally integrable function  : Ω → R has bounded mean oscillation,  ∈ BMO(Ω), if The supremum in ( 22) is taken over all open balls  ⊂ Ω and the notation is used for averages.
Similarly, a weight V belongs to the Gehring class The suprema in ( 24) and ( 25) are taken over all balls  ⊂ R  .The link between Muckenhoupt and Gehring classes is given in [18,19] where it is proved that As a corollary of Gehring's Lemma [20], Jacobians of quasiconformal mappings are weights in the  ∞ (or equivalently  1 ) class.By virtue of the change of variables formula for quasiconformal mappings (see Section 2.1 above) and (7), for every ball  ⊂ R  and every measurable set  ⊂  it holds for some 0 <  ≤ 1 ≤  independent of  and .
For a weight V in  1 , in [21] (and also studied in [22]) the constant G1 (V) is defined as We briefly refer to G1 (V) as the G1 -constant of V.
As for (27), the change of variables formula for quasiconformal mappings and (8) for some 0 <  ≤ 1 ≤  independent of  and .
As done for the G1 -constant, in [21] a second auxiliary constant is defined as We briefly refer to Ã∞ () as the Ã∞ -constant of .
It is worth pointing out that to each function  ∈ BMO(R  ) there corresponds a weight in the  1 class of Gehring given by   , for some  > 0 depending on  and ‖‖ BMO (see [23]).Conversely (see again [23] In particular, log   is a BMO-function, whenever  : R  → R  is a quasiconformal mapping.
For the sake of completeness, we also recall the definition of the Muckenhoupt class   for 1 ≤  < ∞.A weight  belongs to the Muckenhoupt class   for 1 <  < ∞ if As a natural extension of the above definition, one can consider the Muckenhoupt class  1 which covers the limit case  = 1.A weight  belongs to the Muckenhoupt class  1 if The suprema in (31) and (32) are taken over all balls  ⊂ R  .
For each 1 ≤  < ∞ we call   () the   -constant of the weight .
We recall here the definition of the Gehring class   for 1 <  ≤ ∞.A weight V belongs to the Gehring class As a natural extension of the above definition, one can consider the class  ∞ which cover the limit case  = ∞.A weight V belongs to the Gehring class  ∞ if The suprema in (33) and (34) are taken over all balls  ⊂ R  .For each 1 <  ≤ ∞ we call   (V) the   -constant of the weight V.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 [20]).
For more details related to the Muckenhoupt and Gehring classes we refer to [19,23,24].

Explicit Bounds
We start by proving Theorem 1.

Proof of
for each ball  ⊂⊂ Ω and for every  ∈ EXP(Ω).
It remains to prove the first inequality in (11).Let us assume that  and  are arbitrary constants for which (29) holds, with 0 <  ≤ 1 ≤ .As before, we compare the distribution functions of  and  ∘  −1 .This time we make use of the estimate (29) instead of (27)  ) .
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 Â∞ (  ) and to the fact that  and  are arbitrary constants for which (29) holds, we obtain for each ball  ⊂⊂ Ω and for every  ∈ EXP(Ω).The proof is complete.
We are in a position to prove Theorem 2.
Appealing to the definition ( 9) we deduce that directly follows from (59).We introduce the auxiliary function Since  ≥ 1 we have   () = log  ≥ 0 and in particular Combining the latter identity with (61) we conclude that The proof is complete, since (11) infers (13) in case of any quasiconformal principal mapping  : R 2 → R 2 which maps D onto itself.

A Characterization of Constant Weights
We explicitly remark that, for a weight V in  ∞ , one always has This result is proved in [21].With a similar proof, one can show the same property for the G1 -constant, that is Our next result gives a similar characterization of constant weights in terms of the constant appearing in ( 9) and (10).It is crucial to prove sharpness of Theorem 1.
Proof of Proposition 3. Let us fix some integer  ≥ 1.By the definition (10) of Â∞ (), we may find two sequences {  } and {  } fulfilling and for every ball  ⊂ R  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  0 ∈ [0, 1].Since one clearly has 1/  ≥ 1 for every  ≥ 1, condition (67) implies  0 = 1.Conditions (67) and (68) also imply Appealing to the fact that   → 1 as  → ∞, the latter relation implies   → 1 as  → ∞.Therefore, we may pass to the limit as  → ∞ in (69) in order to get for every ball  ⊂ R  and every measurable set  ⊂ .In particular, for every ball  ⊂ R  and for a.e. 0 ∈  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 Ĝ1 (⋅)-constant.Proposition 4. Let V be an  1 weight such that Ĝ1 (V) = 1.Then V is constant a.e. in R  .

The Logarithm of the Jacobian of a Quasiconformal Mapping
Let  : Ω → R  be a quasiconformal mapping.In Section 2.4 we mentioned that log   is a BMO-function.Similarly (see, e.g., [4]) log   ∈ EXP(Ω).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 Ĝ1 (  ) and Â∞ (  ).Moreover, these results can be seen as the counterpart in the space of exponentially integrable functions of the results of Reimann [10] in the setting of the BMO-space.
In (76) we may replace  by  −1 , so that The desired inequality (74) is proved.
Next results provide quantitative estimates as in ( 73) and (74) where the norms are replaced by the distances to  ∞ .Corollary 7. Let Ω, Ω  be bounded domains of R  ,  ≥ 2. Let  : Ω → Ω  be a -quasiconformal mapping and let Ĝ1 (  ) be the constant defined in (9).Then for each ball  ⊂⊂ Ω.
Proof of Corollary 8. Let  ∈  ∞ (R  ) and let  = − ∘  −1 .Then, from (6), we have  introduced in [26] by Garnett and Jones.The supremum in (97) is taken over all balls  ⊂ R  .The main result of [26] states that (⋅) is equivalent to the distance to  ∞ (R  ) in the space BMO(R  ) defined as −     BMO(R  ) .(98) 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.(99)