A Generalization of Exponential Class and its Applications

A function space, $L^{\theta,\infty)}(\Omega)$, $0 \leq \theta<\infty$, is defined. It is proved that $L^{\theta,\infty)}(\Omega)$ is a Banach space which is a generalization of exponential class. An alternative definition of $L^{\theta,\infty)}(\Omega)$ space is given. As an application, we obtain weak monotonicity property for very weak solutions of $\mathcal{A}$-harmonic equation with variable coefficients under some suitable conditions related to $L^{\theta,\infty)}(\Omega)$, which provides a generalization of a known result due to Moscariello. A weighted space $L^{\theta,\infty)}_w(\Omega)$) is also defined, and the boundedness for the Hardy-Littlewood maximal operator $M_w$ and a Calder\'{o}n-Zygmund operator $T$ with respect to $L^{\theta,\infty)}_w(\Omega)$ are obtained.

consists of all functions f (x) ∈ 0<ε≤p−1 L p−ε (Ω) such that in the paper [1] by Iwaniec and Sbordone in 1992 where they studied the integrability of the Jacobian under minimal hypotheses. For p = n in [2] imbedding theorems of Sobolev type were proved for functions f ∈ W 1,n) 0 (Ω). The small Lebesgue space L (p (Ω) was found by Fiorenza [3] in 2000 as the associate space of the grand Lebesgue space L p) (Ω).
Fiorenza and Karadzhov gave in [4] the following equivalent, explicit expressions for the norms of the small and grand Lebesgue spaces, which depend only on the non-decreasing rearrangement (provided that the underlying measure space has measure 1): In [5], Greco, Iwaniec and Sbordone gave two more general definitions than (1.1) and (1.2) in order to derive existence and uniqueness results for p-harmonic operators. For 1 < p < ∞ and 0 ≤ θ < ∞, the grand L p space, denoted by L θ,p) (Ω), consists of functions where (1.4) (1.5) Grand and small Lebesgue spaces are important tools in dealing with regularity properties for very weak solutions of A-harmonic equation as well as weakly quasiregular mappings, see [6,7].
The aim of the present paper is to provide a generalization L θ,∞) (Ω), 0 ≤ θ < ∞, of exponential calss EXP (Ω), and prove that it is a Banach space. An alternative definition of L θ,∞) (Ω) is given in terms of weak Lebesgue spaces. As an application, we obtain weak monotonicity property for very weak solutions of A-harmonic equation with variable coefficients under some suitable conditions related to L θ,∞) (Ω). This paper also consider a weighted space L θ,∞) w (Ω), and some boundedness result for classical operators with respect to this space.
In the sequel, the letter C is used for various constants, and may change from one occurrence to another. §2 A Generalization of Exponential Class Recall that EXP (Ω), the exponential class, consists of all measurable functions f such that Ω e λ|f | dx < ∞ for some λ > 0. It is a Banach space under the norm In this section, we define a space L θ,∞) (Ω), 0 ≤ θ < ∞, which is a generalization of EXP (Ω), and prove that it is a Banach space.
There are two special cases of L θ,∞) (Ω) that are worth mentioning since they coincide with two known spaces.
Case 1: θ = 0. In this case, From the fact (see [8, P12])  Proof. In order to realize that a function in the L 1,∞) (Ω) space is in EXP (Ω), it is sufficient to read the last lines of [2]. The vice-versa is also true, see e.g. [9, Chap. VI, exercise no. 17].
Proof. In the proof of Theorem 2.1 we always assume θ > 0. Let f (x) ∈ L ∞ (Ω), then there exists a constant M < ∞, such that |f (x)| ≤ M , a.e. Ω. Thus, The following example shows that L ∞ (Ω) ⊂ L θ,∞) (Ω) is a proper subset. Since we have the inclusion (2.2), then it is no loss of generality to assume that θ ≤ 1. Consider In fact, for m a positive integer, integration by parts yields By L'Hospital's Law, one has This equality together with (2.3) yields Recall that the function where we have used the assumption θ ≤ 1, and [pθ] is the integer part of pθ. The proof of Theorem 2.1 has been completed.
Proof. This theorem is easy to prove, we omit the details.
We drop the subscript Ω from · θ,∞),Ω when there is no possibility of confusion.
Proof. (1) It is obvious that f θ,∞) ≥ 0 and f θ,∞) = 0 if and only if f = 0 a.e. Ω; (2) For any , and for any positive integer p, It is no loss of generality to assume that the Ω m s are disjoint. (2.4) implies that for any positive integer p, Thus, by the completeness of Hence for any positive integer m, there exists a subsequence {f If we let It is no loss of generality to assume that the subsequence {f We now prove f (x) ∈ L θ,∞) (Ω) and f n − f θ,∞) → 0, (n → ∞). In fact, by (2.6), for any ε > 0, there exists N = N (ε), such that if n > N , then for every t > 0, where |E| is the n-dimensional Lebesgue measure of E ⊂ R n , and f * (t) = |{x ∈ Ω : |f (x)| > t}| denotes the distribution function of f .
For p > 1, we recall that if f ∈ L p weak (Ω), then f ∈ L q (Ω) for every 1 ≤ q < p, and f ∈ L p weak (Ω) if and only if for every measurable set E ⊂ Ω, the following inequality Recall also that (3.5) The following theorem shows that L θ,∞ weak (Ω) = L θ,∞) (Ω), thus L θ,∞ weak (Ω) can be regarded as an alternative definition of the space L θ,∞) (Ω).
Proof. We divided the proof into two steps.
If 1 ≤ s < p, for each a > 0, one can split the integral in the right-hand side of (3.3) to obtain This implies here we have used (3.6) and the definition of L ∞ weak (Ω).
Since for any t > 0, This implies The proof of Theorem 3.1 has been completed. §4 An Application In this section, we give an application of the space L θ,∞ (Ω) to monotonicity property of very weak solutions of the A-harmonic equation where A : Ω × R n → R n be a mapping satisfying the following assumptions: (1) the mapping x → A(x, ξ) is measurable for all ξ ∈ R n , (2) the mapping ξ → A(x, ξ) is continuous for a.e. x ∈ R n , for all ξ ∈ R n , and a.e. x ∈ R n , Conditions (1) and (2) insure that the composed mapping x → A(x, g(x)) is measurable whenever g is measurable. The degenerate ellipticity of the equation is described by condition (3). Finally, condition (4) guarantees that, for any 0 ≤ θ < ∞ and any ε > 0, A(x, ∇u) can be integrated for u ∈ W θ,p (Ω) against functions in W  can be found in [14], see also [6,7]. says we want the same condition in B, that is the maximum and minimum principles.
Manfredi's paper [14] should be mentioned as the beginning of the systematic study of weakly monotone functions. Koskela, Manfredi and Villamor obtained in [15] that A-harmonic functions are weakly monotone. In [16], the first author obtained a result which states that very weak solutions u ∈ W 1,p−ε loc (Ω) of the A-harmonic equation are weakly monotone provided ε is small enough. The objective of this section is to extend the operator A to spaces slightly larger than L p (Ω).
is a very weak solution to (4.1), then it is weakly monotone in Ω provided that θ 1 + θ 2 < 1.
Proof. For any ball B ⊂ Ω and 0 < ε < 1, let It is obvious that ∇u, otherwise, say, on a set E ⊂ B.
Consider the Hodge decomposition (see [6]), The following estimate holds (4.6) The condition τ ∈ L θ 1 ,∞) (Ω) implies Since u ∈ W θ 2 ,p) (Ω), then By   we have w(2Q) ≤ Cw(Q), where 2Q denotes the cube with the same center as Q whose side length is 2 times that of Q. When w satisfies this condition, we denote w ∈ ∆ 2 , for short.
A weight function w is in the Muckenhoupt class A p with 1 < p < ∞ if there exists C > 1 such that for any cube Q Let w be a weight. The Hardy-Littlewood maximal operator with respect to the measure w(x)dx is defined by We say that T is a Calderón-Zygmund operator if there exists a function K which satisfies the following conditions: T f (x) = p.v. R n K(x − y)f (y)dy.
For w a weight and 0 ≤ θ < ∞, we define the space L The following lemma comes from [18].
Lemma 5.1. If 1 < p < ∞ and w ∈ ∆ 2 , then the operator M w is bounded on L p w (Ω).
Proof. By Lemma 5.1, since for 1 < p < ∞ and w ∈ ∆ 2 , the operator M w is bounded on The following lemma can be found in [19].
The following lemma can be found in [20,21].
Lemma 5.3. If 1 < p < ∞ and w ∈ A p , then a Calderón-Zygmund operator T is bounded on L p w (Ω).