Spaces of Several Real Variables

and Applied Analysis 3 y 0 = (y 0 1 , y 0 2 , . . . , y 0 n ) such that y 0 M T = x 0 . Write y = xM for convenience as follows: y j = n ∑ i=1 x i a ji , j = 1, 2, . . . , n. (18) Consequently, ∂g ∂x 1 (y 0 ) = n ∑ j=1 ∂f ∂y j (y 0 M T ) a j1 = 󵄨󵄨󵄨󵄨∇f (x0) 󵄨󵄨󵄨󵄨 . (19) Similarly, ∂g ∂x i (y 0 ) = 0, i = 2, 3, . . . , n. (20) Thus ∇g (y 0 ) = ( 󵄨󵄨󵄨󵄨∇f (x0) 󵄨󵄨󵄨󵄨 , 0, . . . , 0) . (21) Note that g ∈ C(R). Then there exist a positive constant δ and a small cube I centered at y 0 on which ∂g(x)/∂x 1 > 2δ and ∂g(x)/∂x j < δ, j ≥ 2. Define D = {x = (x 1 , . . . , x n ) ∈ R n : 󵄨󵄨󵄨󵄨x2 󵄨󵄨󵄨󵄨 + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨󵄨xn 󵄨󵄨󵄨󵄨 < x1 < l (I) 4 } . (22) If x, y ∈ I and x − y ∈ D, using the mean value theorem, we get g (x) − g (y) > δ (x1 − y1) . (23) Thus ∫∫ I 󵄨󵄨󵄨󵄨g (x) − g (y) 󵄨󵄨󵄨󵄨 2 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨 2n K( 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨 l (I) ) dx dy ≥ ∫ I/2 dx∫ I−x 󵄨󵄨󵄨󵄨g (x + z) − g (x) 󵄨󵄨󵄨󵄨 2 |z| 2n K( |z| l (I) ) dz ≥ ∫ I/2 dx∫ D δ 󵄨󵄨󵄨󵄨z1 󵄨󵄨󵄨󵄨 2 |z| 2n K( |z| l (I) ) dz. (24) If z ∈ D, then |z| ≈ z 1 . Hence ∫∫ I 󵄨󵄨󵄨󵄨g (x) − g (y) 󵄨󵄨󵄨󵄨 2 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨 2n K( 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨 l (I) ) dx dy ≳ ∫ I/2 dx∫ D 󵄨󵄨󵄨󵄨z1 󵄨󵄨󵄨󵄨 2 󵄨󵄨󵄨󵄨z1 󵄨󵄨󵄨󵄨 2n K( 󵄨󵄨󵄨󵄨z1 󵄨󵄨󵄨󵄨 l (I) ) dz


Introduction
Recall that a locally integrable function  belongs to BMO(R  ) if         BMO(R  ) = sup where  denotes a cube in R  with edges parallel to the coordinate axes and || denotes the Lebesgue measure of  and Via the John-Nirenberg inequality [1], one can show an equivalent condition of BMO(R  ) as follows: C. Fefferman's famous equation, ( 1 ) * = BMO, describes a deep relation between BMO and the Hardy space (cf.[2,3]).This leads quite naturally to increased study of these functions from the point of real variable theory and complex function theory views in the recent fifty years.See [2][3][4][5][6][7][8][9] for more results about BMO(R  ) space.
As a generalization of BMO(R  ), the space Q  (R  ),  ∈ R, introduced by Essén et al. in [10], is defined to be the class of all locally integrable functions  ∈  2 loc (R  ) such that where ℓ() = || 1/ denotes the edge length of the cube .
It is easy to see that Q  (R  ) is always a subclass of BMO(R  ) and Q  (R  ) = BMO(R  ) by choosing  = −/2.Moreover, we know by [10] that Q  (R  ) = BMO(R  ) if and only if  < 0. Also, we see that Q  (R) is trivial (containing a.e.constant functions only) if and only if  > 1/2 and Q  (R  ),  ≥ 2, is trivial if and only if  ≥ 1.
In this paper, we introduce and develop a more general space Q  (R  ) of several real variables, which can be viewed as an extension and improvement of Q  (R  ) spaces as well as BMO(R  ).A theory of Q  (D) spaces on unit disc D has been extensively studied for recent years in the context of a wide class of function spaces; see, for example, [11][12][13][14][15]. Motivated by the theory of analytic Q  (D) spaces, we define the following.Definition 1.Let  : [0, ∞) → [0, ∞) be a nondecreasing function.A function  ∈  2 loc (R  ) is said to belong to the space Q  (R  ) if If we take () =  −2 , for  ∈ R, then Q  (R  ) = Q  (R  ).Modulo constants, Q  (R  ) is a Banach space under the norm defined in (5).
Our paper is organized as follows.
In Section 2, we investigate the relationship between Q  (R  ) and BMO(R  ) and give a sufficient and necessary condition for space Q  (R  ) which is nontrivial.
In Section 3, we give several results about the weight function  on which Q  (R  ) obviously depends.In the study of Q  (R  ), the auxiliary function   defined by still works well as the analytic Q  (D) spaces.
In Section 4, we define the -Carleson measure on R +1 + .By establishing a Stegenga-type estimate, we obtain a characterization of Q  (R  ) spaces in terms of the -Carleson measure.
Throughout this note,  ≲  means that there is a positive constant  such that  ≤ .Moreover, if both  ≲  and  ≲  hold, then one says that  ≈ .For the convenience of calculation, in this paper, we always assume that  : [0, ∞) → [0, ∞) is nondecreasing and (2) ≈ ().

Basic Properties of Q 𝐾 (R 𝑛 )
Our first observation is that Q  (R  ) is invariant under the conformal mappings and rotations; that is, for any conformal map () =  +  0 ,  ̸ = 0 and  0 ∈ R  , or any rotation () =  for an orthogonal matrix  of order , hold for any  ∈ Q  (R  ).
By assumption on  we have Hence, then Q  (R  ) is trivial.We will prove the necessity by two steps.
Step 2. Note that where In particular, if  ∈  ∞ 0 , the class of smooth functions with compact support, then  *  ∈ Q  (R  )∩ 1 (R  ).Thus  *  is constant by Step 1.By [10], there exits a sequence {  } ⊂  ∞ 0 with   ≥ 0, ∫   = 1, and supp   shrinking to 0 such that  *   →  a.e.It follows that  is constant a.e.Thus, we complete the proof of necessity.

Sufficiency
For any cube  and ,  ∈ , if  > 0 is small enough, we know that the Lebesgue measure of the set Consequently, For a small enough  > 0, we obtain (a) Note that For a cube  and for every  ∈ R  with || < √(), Therefore, Thus BMO(R  ) ⊆ Q  (R  ), and this deduces that for all multi-indices  = ( 1 , . . .,   ) and  = ( 1 , . . .,   ) of nonnegative integers, where Let  ∈ S(R  ) be a fixed function such that the Fourier transform φ of  has support in the unit ball and  ̸ = 0 on the cube [−3, 3]  .Let {  } be a sequence of real numbers and define where  1 is the first coordinate of .By [10], For any cube  of R  , when ,  ∈ , we have that |−| ≤ √ℓ().By the definition, the Q  (R  ) space depends on () when 0 ≤  ≤ √.In fact, Q  (R  ) depends only on () when  is near origin, which can be found by the following theorem.Here the proof of the theorem is left to the reader.(a) Suppose that () > 0 for some  > 0. One defines (44) ) is trivial for  ≥ 2, we only pay attention to the case (0) = 0.

Weighted Functions
The characterization of Q  (R  ) depends on the properties of the weight function  obviously.In this section we give several results about the weight functions that are needed for the next section.
In the analytic Q  (D) spaces, the auxiliary function plays a key role; see [12,14,15], for example.
(d) If we repeat the construction   →  * , then we can make the new weight function differentiable up to any desired order.
(e) Note that if  > 0 is sufficiently small, then we have This means that  * ()/ − is nonincreasing.The proof is complete.

Lemma 8. Let 𝐾 satisfy (49). Then one can find another nonnegative weight function 𝐾
and the new weight function  1 has the following properties.
Proof.Assume that () = (1) −1 for  ≥ 1. Define Since  is a -Carleson measure, This together with (48) yields Let  be a measurable function on Its Poisson integral is defined by where The gradient of (, ) is It is known that (101) holds for  ∈ BMO(R  ) (see [9]).
The following main theorem generalizes the result of Q  (R  ) in [10].
In order to prove Theorem 10, we need the following Hardy-type inequalities.
The following Stegenga-type estimate will be used in the proof of Theorem 10.
Lemma 12. Suppose that (81) holds for  and Let  and  be cubes in R  centered at  0 with ℓ() = 3ℓ() and let  ∈  1 loc (R  ) satisfy (101).Then, there is a constant  independent of , , and , such that Proof.Without loss of generality, we may assume that  0 = 0.
Let  be a function with 0 ≤  ≤ 1 such that  = 1 on (2/3), supp  ⊆ (3/4), and Following Stegenga [19], we write Then we have for the corresponding Poisson integrals.Since  1 is constant, it contributes nothing to the integral with the gradient square.Note that We obtain Hence, We write  =  ∈ R  with  = || and || = 1.Let Then           (, ) Thus, By Lemma 11, The proof of Theorem 10 is complete.

Theorem 4 .
The following statements are true.

Lemma 7 .
The following are equivalent.