A Hardy-Littlewood Maximal Operator Adapted to the Harmonic Oscillator

This paper constructs a Hardy-Littlewood type maximal operator adapted to the Schr\"{o}dinger operator $\mathcal{L} := -\Delta + |x|^{2}$ acting on $L^{2}(\mathbb{R}^{d})$. It achieves this through the use of the Gaussian grid $\Delta^{\gamma}_{0}$, constructed by J. Maas, J. van Neerven and P. Portal with the Ornstein-Uhlenbeck operator in mind. At the scale of this grid, this maximal operator will resemble the classical Hardy-Littlewood operator. At a larger scale, the cubes of the maximal function are decomposed into cubes from $\Delta^{\gamma}_{0}$ and weighted appropriately. Through this maximal function, a new class of weights is defined, $A_{p}^{+}$, with the property that for any $w \in A_{p}^{+}$, the heat maximal operator associated with $\mathcal{L}$ is bounded from $L^{p}(w)$ to itself. This class contains any other known class that possesses this property. In particular, it is strictly larger than $A_{p}$.


Introduction and Preliminaries.
The Hardy-Littlewood operator is ubiquitous in classical harmonic analysis. From the Lebesgue differentiation theorem to Calderón-Zygmund theory, the importance of this averaging operator can hardly be overstated. Classical harmonic analysis can be thought of as being intricately linked to the Laplacian ∆. Many of its fundamental objects, including the Hardy-Littlewood operator, are closely related to the functional calculus of the Laplacian. A current area of active research is the study of the harmonic analysis associated with differential operators other than the Laplacian. At present, there is no suitable candidate for the Hardy-Littlewood operator in this setting. It is quite possible that such an operator would play a fundamental role in extending the theory even further. In this paper, our aim is the construction of a Hardy-Littlewood type maximal operator adapted to the Schrödinger operator L := −∆ + |x| 2 on L 2 (R d ). In order to outline the details of this construction, we must first present some motivating theory.
Note that throughout this paper, we will be working in the Euclidean space R d endowed with the Lebesgue measure dx. The dimension d will be considered to be fixed. Let V : R d → R ≥0 be a potential that is non-identically zero and satisfies, for some q > d/2 and C > 0, the reverse Hölder inequality, for every cube Q ⊂ R d . Consider the Schrödinger operator L V := −∆ + V on L 2 (R d ). An important step in the comprehension of the harmonic analysis of such an operator was made by Shen through the introduction of the critical radius function, see [She95]. This is defined by, where B(x, r) is the ball in R d , centered at x and of radius r. At a scale smaller than this critical radius, the operators associated with L V behave "locally" like their classical counterparts for the Laplacian. This indicates that if we are to construct a Hardy-Littlewood type maximal operator for L, then our construction should resemble the classical Hardy-Littlewood operator at this local scale. What should it look like at a larger scale? In order to answer this question, we must briefly delve into some Gaussian harmonic analysis.
As is quite frequent in mathematics, when studying a particular object, it can be fruitful to change perspective by studying an isomorphic object in a different setting. Let dγ(x) := π −d/2 e −|x| 2 dx denote the Gaussian measure on R d . Gaussian harmonic analysis is the study of the Ornstein-Uhlenbeck operator, O := −∆ + 2x · ∇, on the space L 2 (γ) and its associated harmonic analysis. Its relevance to the study of L is that through the isometry U : L 2 (dx) → L 2 (γ), defined by for f ∈ L 2 (dx) and x ∈ R d , the operators L and O become, more-or-less, similar. See [AFT06] for further details. This similarity allows for the transfer of geometric ideas between the Gaussian and the harmonic oscillator setting.
A measure µ on R d is said to be doubling if there exists some C > 0 such that µ (B(x, 2r)) ≤ Cµ (B(x, r)) , for all x ∈ R d and r > 0. Many of the constructions from classical harmonic analysis directly rely on the fact that the Lebesgue measure is doubling. A fundamental obstruction in the development of Gaussian harmonic analysis is that, due to the non-doubling nature of the Gaussian measure, many of these constructions do not directly translate to the Gaussian setting. In their seminal paper [MM07], Mauceri and Meda made a crucial step in this development by transposing the critical radius over to Gaussian harmonic analysis. They introduced their concept of admissibility.
Let us introduce ρ as shorthand notation for the critical radius function of L, ρ |x| 2 . It is not too difficult to see that ρ(x) = min {1, 1/ |x|}. A ball, B(x, r), is then said to be admissible if r ≤ ρ(x). The collection of all admissible balls in R d , B, possesses the desirable property that there exists some C > 0 such that the Gaussian measure satisfies the doubling condition (1) for all balls in B. As such, by restricting their attention to the collection B, Mauceri and Meda were able to construct Gaussian analogues of the spaces BM O and H 1 . A similar construction for the harmonic oscillator, also based on the distinction between local and non-local scales, was developed by Dziubanski and Zienkiewicz in [DZ99] and subsequent papers.
In [MvNP12], Maas, van Neerven and Portal extended the idea of admissibility by constructing an admissible dyadic grid ∆ γ . It is this grid that will form the foundation for our construction. We recall some pertinent details. For m ∈ Z, let ∆ m denote the collection of cubes, The standard dyadic grid is then the union ∆ = ∪ m∈Z ∆ m . Define the layers, for l ≥ 1. Then define, for k ∈ Z and l ≥ 0, The collection ∆ γ is called the Gaussian grid and will be used extensively throughout this paper. Lets introduce some notation that can be used in conjunction with this grid. For any x ∈ R d , R x will be used to denote the unique cube in ∆ γ 0 that contains the point x. For any R ∈ ∆ γ 0 , j(R) is defined to be the unique integer such that R ⊂ L j(R) . The more commonly used notation, c Q and l(Q), representing the center and side-length of a cube Q respectively, will also be used. Next we will define what will be considered to be our local region in the Gaussian grid.
Definition 1 For a cube R ∈ ∆ γ 0 , fix a subcollection N (R) ⊂ ∆ γ 0 that satisfies the following two properties.
is a cube of sidelength 2 2 l(R).
It is obvious that such a subcollection must exist for each cube. There might even be more than one such example. This, however, is unimportant. What is important, is that we fix N (R) from the outset. Examples of subcollections that satisfy these properties are illustrated below. Figure 1: In each of the above illustrations, a black cube R ∈ ∆ γ 0 is depicted together with the region N (R).
As our operator is expected to behave differently at large scales than at local scales, it is desirable to split it up into local and non-local components. For any sub-linear operator B, define , and for f ∈ L 1 loc (R d ) and x ∈ R d . Notice that due to sub-linearity, for any weight w on R d , to bound the quantity Bf L p (w) , it is both sufficient and necessary to bound B loc f L p (w) and B f ar f L p (w) . Now that sufficient preliminaries have been discussed, the details of our construction will be outlined. As noted previously, ∆ γ 0 acts as a mediator between the local and non-local worlds. It is then appropriate to consider maximal functions of the below general form as candidates for an adapted maximal function for L.
Definition 2 For Q ∈ ∆, let G(Q) be the collection of cubes Then for c : Notice that if c(Q, R, R ) = 1 for all R ∈ G(Q) and Q ∈ ∆, then the operator M c is identical to the classical dyadic Hardy-Littlewood operator.
This looks promising but how do we determine what the right c-coefficients are? Any candidate for an adapted Hardy-Littlewood should share similar properties to the classic Hardy-Littlewood. We will determine appropriate coefficients from one of these properties. Let M and T * denote the classical Hardy-Littlewood and heat maximal operator respectively. That is, for f ∈ L 1 loc (R d ) and x ∈ R d . Also recall that the A p class of weights is defined to be the collection of all weights, w on R d , for which there exists a constant C > 0 that satisfies for all cubes Q in R d . The following theorem is a well-known result from weighted theory.
Theorem 1 Let w be a weight on R d and 1 < p < ∞. Then, Refer to [Gra09] for proof. The above theorem indicates that if we are to construct a Hardy-Littlewood type maximal operator for L, then the correct c-coefficients should satisfy the below equivalence for each 1 < p < ∞, where T * is the semigroup maximal operator associated to L, The coefficients for our generalised maximal function will be optimised in an attempt to produce the above equivalence.
A significant source of inspiration for this investigation stemmed from [BHS11]. In this paper, Bongioanni, Harboure and Salinas defined a new class of weights, A ∞ p , for which T * was bounded on L p (w) for all weights w ∈ A ∞ p . What was so interesting about this class was that it was strictly larger than the classic Muckenhoupt class. It seems that by including the potential |x| 2 , the weight class A p effectively increases in size. It can be inferred from this, that in order to produce a maximal function smaller than M and therefore a larger weight class, the coefficients for our maximal function must be smaller than unity. The A ∞ p class is defined to be A ∞ p := ∪ θ≥0 A θ p , where w ∈ A θ p if and only if there exists some constant C > 0 such that for all cubes Q ⊂ R d , In [LRW16], the authors developed a maximal function, M θ , adapted to the class A θ p , in the sense that M θ : This operator is defined through, where Notice that this maximal function is also an example of the general class from definition 2 with c(Q, R, R ) = ψ θ (Q) −1 < 1. This allows for more weights in the class A θ p . However, it does not take into account the fact that if the cubes R and R are far apart, then the potential should have a larger effect and therefore the coefficient c(Q, R, R ) should be smaller. The coefficients that we define for our maximal function take this into account. The main theorem of this paper is stated below.
Theorem A There exists maximal functions, M − f ar and M + f ar , of similar form to definition 2 that satisfy the chain of implications for any weight w on R d and 1 < p < ∞.
For a precise definition of the above maximal functions, M − f ar and M + f ar , and a proof of this statement, refer to section 3. A secondary result of this paper that characterises the local behaviour of an adapted maximal function is stated below.
Theorem B For any weight w on R d and 1 < p < ∞, This theorem will be proved in section 2. Together, these two statements demonstrate that for any weight in the class It is then natural to ask how our weight class compares with the class A ∞ p ? Section 4 provides an answer to this question in the form of the following proposition.
Proposition C The following chain of inclusions holds for any 1 < p < ∞, The above inclusion indicates that our coefficients serve as an improvement upon the constant coefficients of (3).
Finally, in section 5, the techniques developed throughout this paper will be used to show that the heat maximal operator for L can be safely truncated when considering weighted questions.
This paper is part of my PhD thesis, supervised by Pierre Portal at the Australian National University. It is inspired by discussions of my supervisor with Paco Villarroya and follows Villarroya's philosophy (see e.g. [Vil15]): a refined dyadic analysis can detect strong properties of Calderón-Zygmund operators, such as compactness or boundedness on weighted L p -spaces for non A p weights. This research was partially supported by the Australian Research Council through the Discovery Projects DP12010369 and DP160100941.

The Local Class.
In this section, a local version of the A p class is introduced, A loc p . This class is a dyadic variation of a similar class introduced in [BHS11]. Through this class, and a few preliminary lemmas, Theorem B will be proved.
In the usual manner, this cube can be divided into 2 d congruent disjoint cubes with half the side-length of the original cube. These cubes can themselves be divided into 2 d disjoint cubes each and so on ad infinitum. If a cube Q ⊂ R d can be obtained in this manner from Q 0 , then it is called a dyadic subcube of the cube Q 0 . Note that we did not require our initial cube Q 0 to be a member of the standard dyadic grid and that Q 0 is a dyadic subcube of itself.
for all dyadic subcubes Q ⊆ Q 0 . The smallest such C is denoted [w] Ap(Q0) .
A variation of the next statement was originally proved in [Har84]. It is an extension lemma for weights that satisfy the A p property when restricted to a cube.
Proof. Our proof proceeds by construction. Let D Q0 denote a dyadic system of cubes on R d for which Q 0 is a member. This will simply be the standard dyadic grid scaled and translated appropriately. Let D Q0 0 denote the subcollection that consists of all cubes in D Q0 of the same size as Q 0 . A weight, w Q0 on R d , will be constructed for which there exists B > 0 such that for all Q ∈ D Q0 . As the dyadic description of A p (R d ) is scale and translation invariant, this criteria will be sufficient to determine that w Q0 ∈ A p (R d ).
As the cubes in D Q0 0 partition R d , this description defines a unique function w Q0 on R d . Moreover, it is clear that this function will be a weight that coincides with w on Q 0 . By definition, as w ∈ A p (Q 0 ), it follows that there must exist a C > 0 such that (5) is satisfied for all dyadic subcubes Q ⊂ Q 0 . Fix a cube Q ∈ D Q0 . Suppose that Q is a dyadic subcube of a cube from D Q0 0 . Then (5) must be satisfied automatically with constant C. So suppose that Q is not a dyadic subcube of any cube in 0 . Then there must exist finitely many cubes The subsequent lemma will be used numerous times throughout this investigation. It states the exact form of the heat kernel corresponding to L. Its proof can be found in [Sim79] in dimension 1. Higher dimensions follow from this case by taking tensor products of Hermite functions.
where h t is the classic heat kernel, and α is defined by for all x and y in R d . The operator T * is then given by Note that the fundamental solution for L is actually k sinh 2t . We have chosen to rescale the kernel for simplicity. An expanded version of theorem B is presented and proved below.
Theorem B Let T * and M denote the classic heat maximal operator and Hardy-Littlewood operator respectively. Let w be a weight on R d . For any 1 < p < ∞, the following statements are equivalent.
(1) ⇒ (2). Fix a cube R ∈ ∆ γ 0 , Q a dyadic subcube of N (R) and f ∈ L 1 loc (R d ). Define C := M loc L p (w)→L p (w) . Then, using standard techniques from weighted theory, for each ε > 0. An application of the Lebesgue monotone convergence theorem then produces the desired result.
(2) ⇒ (3). Lemma 1 states that for any cube R ∈ ∆ γ 0 , the restriction w| N (R) can be extended to an A p weight w N (R) . As w N (R) ∈ A p , we know from classical theory that T * where the final inequality was obtained from the bounded overlap property of the cubes {N (R)} R∈∆ γ 0 .
(3) ⇒ (4). This follows trivially from the inequality k t (x, y) ≤ h t (x, y) for all x, y ∈ R d and t > 0.
(4) ⇒ (1). Fix f ∈ L 1 loc (R d ) and x ∈ R ∈ ∆ γ 0 . Let Q be any cube containing x that satisfies Q ⊆ N (R). We first observe that for any y ∈ Q, This implies that and therefore, Moreover, we trivially have Note that for any x, y ∈ Q, since l(Q) ≤ 4l(R), we have the bound .
This then implies that It is easy to show that the bound is satisfied for all t > 0. This then gives us For t := l(Q) 2 , we then have, On taking the supremum over all such Q, we obtain M loc f (x) T * loc f (x).

The Far Class.
In this section, the adapted operators M − f ar and M + f ar are defined and theorem A is proved. With this, a sufficient condition for the boundedness of T * L p (w)→L p (w) is obtained. Prior to presenting these definitions, it is necessary to introduce a collection of cubes that represent the regions over which our averaging operators will act.
Definition 5 For each R ∈ ∆ γ 0 , define the following subsets of R d .
• Q 0 (R) is the smallest cube containing the region y ∈ R d : |y| ≤ 2 16 d 4 2 j(R) , that can be decomposed into cubes from the grid ∆ γ 0 .
• For t ≥ 2 4 d 2 , Q t (R) is the smallest cube containing the region y ∈ R d : |y| ≤ 2 8 t 2 2 j(R) , that can be decomposed into cubes from the grid ∆ γ 0 .
For sets A and B contained in R d , introduce the notation k + t (A, B) and k − t (A, B) to denote respectively the supremum and infimum of k t (x, y) over all x ∈ A and y ∈ B.
Definition 6 For f ∈ L 1 loc (R d ) and x ∈ R ∈ ∆ γ 0 , define the operators M + f ar and M − f ar through, With the introduction of our maximal functions, it is a straightforward matter to define their corresponding weight classes.  In order to verify our main result, a string of technical lemmas must first be proved. The first two of these provide some valuable estimates concerning the maximum of the function t → k t (x, y) for fixed x and y in R d .
Lemma 3 Fix points x ∈ R ∈ ∆ γ 0 and y / ∈ Q 0 (R). There is precisely one maximum for the function t → k t (x, y). Denote this point by t m (x, y). Then for R not contained in the first layer, t m (x, y) must satisfy For R contained in the first layer, t m (x, y) will satisfy Proof. On differentiating expression (6) with respect to t, we obtain ∂ ∂t k t (x, y) = 1 2t 2 g(t)k t (x, y), where the function g is defined to be, As the kernel k t (x, y) is always positive, it follows that the sign of the derivative will be identical to the sign of the function g(t). Suppose that g is negative. Then we must have (d · t + 2 x, y ) 1 + t 2 > |x| 2 + |y| 2 That is, the derivative of the kernel will be negative if and only if the above inequality holds. Likewise, the derivative of the kernel will be positive if and only if and the derivative will vanish if and only if equality holds.
It is simple to show that |x − y| 2 /d serves as the only maximum of the function t → h t (x, y). This implies that h t (x, y) is decreasing for t > |x − y| 2 /d. As the function α(t) is strictly increasing, we have that exp −α(t) |x| 2 + |y| 2 is strictly decreasing for all t. This shows that k t (x, y) is strictly decreasing for t > |x − y| 2 /d. It then follows that any maximum for t → k t (x, y) must be less than |x − y| 2 /d. As this function must approach 0 as t approaches 0, continuity of the derivative then implies that there must exist at least one maximum in the interval 0, |x − y| 2 /d .
Let t m (x, y) denote the largest maximum in the above interval. It will be shown that t m (x, y) is the only maximum. From our previous argument, equality will hold in (11) for the value t m (x, y). Suppose that t 0 < t m (x, y). Then t 0 = t m (x, y) − a for some a > 0. We then have, As equality holds in expression (11) for t m (x, y), it follows that the factor (d · t m (x, y) + 2 x, y ) must be positive. Therefore, This demonstrates that the derivative must be positive for any t 0 < t m (x, y).
Lets now show the lower bound for t m (x, y). First suppose that R is not contained in the first layer. It will be shown that for any t 1 < |y| / (9 · d |x|), inequality (11) holds. From our previous argument, this will then imply that the function is increasing on the interval [0, |y| / (9 · d |x|)). As y / ∈ Q 0 (R), it follows that y satisfies the bound |y| > 3 |x|. We know that, 1 + t 2 1 < 1 + We also have, This demonstrates that Now suppose that R is in the first layer and y / ∈ Q 0 (R). Then |y| ≥ 2 16 d 4 . Let t 2 < |y| / (9d). Then, On noting that |x| ≤ √ d, This finally leads to, which validates our lower bound.
Lemma 4 Fix cubes R and R in ∆ γ 0 with R ⊂ Q 0 (R) c . Fix points x ∈ R and y ∈ R . The maximum t m (x, y) satisfies the inequality, Proof. As y / ∈ Q 0 (R), we have |y| ≥ 2 16 d 4 2 j(R) and also |y| ≥ 2 j(R )−1 . The upper inequality then follows from, As for the lower bound, first consider when R is not in the first layer. On applying Lemma 3 and recalling that |y| ≥ |x|, Then on applying the bounds |x| ≤ √ d2 j(R) , |y| ≤ √ d2 j(R ) and |y| ≥ 2 16 d 4 2 j(R) successively we obtain, Next, consider when R is in the first layer. Once again apply Lemma 3 and |y| ≥ |x| to obtain, Then on successively applying the bounds |y| ≤ √ d2 j(R ) and |y| ≥ 2 16 d 4 , This concludes the proof.
The next lemma obtains an estimate on ratios of the form k t (x, y) · k tm(x,y) (x, y) −1 for fixed x and y. It will play a key role in the proof of theorem A.
Lemma 5 Fix cubes R and R in ∆ γ 0 with R ⊂ Q 0 (R) c . Fix the points x ∈ R and y ∈ R . Introduce the shorthand notation t m := t m (x, y). Define Then we must have the bound Proof. According to Lemma 4, t m /M ≤ t m . As t → k t (x, y) is increasing for t ≤ t m (x, y), it follows that it is sufficient to show (13) for the value t m /M . We then have, Lets find a bound on the function α(t m ) − α(t m /M ) in terms of t m and M . Define the function β : R >0 → R through, For any u ≤ 1, perform a Taylor expansion about the origin for β to obtain, According to Lemma 4, both t m and t m /M are greater than 1. The above formula will therefore apply to these values.
It is easy to see that there must exist some A ≥ 0, independent of both R and R such that This would then give M d/2 2 (j(R)+j(R ))(d+1) e 2 j(R)+j(R ) .
On applying (14) and the above, On applying M/t m ≤ 1/ 2 4 d 2 , From which the definition of M then provides, The next result is a direct analogue for A + p of the defining condition for the classic A p class. It is unlikely that this condition is enough to completely characterise A + p .
Lemma 6 Let w be a weight on R d and suppose that M + f ar : L p (w) → L p (w) is bounded for some 1 < p < ∞. Fix cubes R and R in ∆ γ 0 with R ⊂ Q 0 (R). Then there must exist some constant C > 0, independent of both R and R , such that for allx ∈ R andỹ ∈ R .
Proof. It shall first be shown that R ⊂ Q tm(x,ỹ) (R).
Fix any point y ∈ R . From the definition of Q t (R), it will be sufficient to show that |y| ≤ 2 2 t m (x,ỹ) 2 2 j(R) .
Finally enough machinery is in place to prove our main result.
Theorem A Let w be a weight on R d and 1 < p < ∞. Then we have, Proof. The second implication follows quickly from the pointwise bound, for any f ∈ L 1 loc (R d ) and x ∈ R ∈ ∆ γ 0 .
As for the first implication, suppose that M + f ar L p (w)→L p (w) < ∞. Then, The heat operators can be expanded dyadically to obtain, On applying Minkowski's inequality, It remains to bound the tail end term on the right hand side of the above expression. On expanding dyadically once more, Let x t R and y t R denote points contained in R and R respectively that satisfy On applying Hölder's property and Lemma 6 we obtain, Note that since |y t R | ≥ 2 8 t 2 2 j(R) , 1 8 This implies that Lemma 5 can be applied to obtain, since the number of cubes in a layer L k is bounded by a constant multiple of 2 kd .
Theorems A and B, together with the fact that T * L p (w) < ∞ if and only if both T * loc L p (w)→L p (w) < ∞ and T * f ar L p (w)→L p (w) < ∞ for any weight w on R d , lead to the below corollary.
Corollary 1 The following chain of inclusions holds for any 1 < p < ∞, The class of weights in the middle of the above chain of inclusions is a natural candidate for the A p class associated with the harmonic oscillator. The above corollary indicates that our A p classes are honing in on what should be the correct class.

Relation to the A ∞ p Class.
Recall the definitions of the classes A ∞ p and A θ p from section 1. This section is devoted to the proof of the inclusion in proposition C. This will be accomplished by first proving, for any θ ≥ 0, the pointwise bound The following upper bound for the heat kernel k will be utilised. Refer to [Kur00] for proof.
Lemma 7 For any N > 0, there exists a constant C N > 0 such that, for all x, y ∈ R d .
Recall that the sinh 2t factor in the above expression is due to the kernel rescaling introduced in section 2.
Proposition 1 For any θ ≥ 0, there exists some C θ > 0 so that for every locally integrable function f on R d and x ∈ R d .
Proof. For R ∈ ∆ γ 0 and k ≥ 0, define C k (R) to be the collection of cubes, R ∈ ∆ γ 0 , that satisfy d(R, R ) < 2 k l(R). As F(R) ⊂ ∆ γ 0 /C 0 (R), the operator M + f ar can be decomposed as, Lemma 9 Let w be a weight on R d and suppose that M − f ar : L p (w) → L p (w) is bounded for some 1 < p < ∞. Fix cubes R and R in ∆ γ 0 with R ⊂ Q 0 (R). Then there must exist some constant, C > 0, independent of both R and R , such that for any x 0 ∈ R and y 0 ∈ R .
Proof. Recall that R ⊂ Q tm(x0,y0) (R). Refer to the proof of Lemma 6 for why this statement is true. Then, Then from arguments identical to that of Lemma 6, our result is obtained.
Theorem D Fix 1 < p < ∞. For any weight w on R d , the following equivalence holds, Proof. It is trivially true that the equivalence holds for the local components of these operators. That is, for any weight w on R d , This leaves the far equivalence. The forward implication of the far equivalence follows from the bound T # f (x) ≤ T * f (x) for all f ∈ L 1 loc (R d ) and x ∈ R d .
It remains to show that for any weight w on R d , Fix a weight w and suppose that T # f ar : L p (w) → L p (w) is bounded. Fix f ∈ L 1 loc (R d ). Then, We know from Lemma 5 that, Lemma 8 can then be applied to acquire, k t (x t R , y t R ) k tm(x t R ,y t R ) (x,ỹ) · 2 −(j(R)+j(R ))(d+1) , for allx ∈ R andỹ ∈ R . Therefore, tm(x t R ,y t R ) (R, R )2 −(j(R)+j(R ))(d+1) This can be applied to (24) to obtain, which concludes our proof.