Centered Hardy-Littlewood maximal function on product manifolds

: Let X be the direct product of X i where X i is smooth manifold for 1 ≤ i ≤ k . As is known, if every X i satisfies the doubling volume condition, then the centered Hardy-Littlewood maximal function M on X is weak (1,1) bounded. In this paper, we consider the product manifold X where at least one X i does not satisfy the doubling volume condition. To be precise, we first investigate the mapping properties of M when X 1 has exponential volume growth and X 2 satisfies the doubling condition. Next, we consider the product space of two weighted hyperbolic spaces X 1 = ( H n +1 , d , y α − n −1 dydx ) and X 2 = ( H n +1 , d , y β − n −1 dydx ) which both have exponential volume growth. The mapping properties of M are discussed for every α , β ≠ n 2 . Furthermore, let X = X 1 × X 2 × · · · X k where X i = ( H n i +1 , y α i − n i −1 dydx ) for 1 ≤ i ≤ k . Under the condition α i > n i 2 , we also obtained the mapping properties of M .


Introduction
Let S be a smooth manifold and L a second order di erential operator. Furthermore, we assume that there is a σ-nite measure µ on the Borel sets of S such that L is self-adjoint on L (µ). Then there is a canonical distance d associated with the operator L. Denote by B(x, r) the ball centered at x with radius r > and by V(x, r) the volume of B(x, r) with respect to µ. The centered Hardy-Littlewood maximal function is de ned by: Mf (x) = sup r> V(x, r) B(x,r) |f (y)|dµ(y), ∀f ∈ L loc (S, d, µ).
As is well known, the maximal function plays an important role in harmonic analysis and is closely related to the theory of singular integral operators, square functions( [23]). Now we recall some known facts. The classical way to study the mapping properties of M is by various covering lemmas. If the manifold S supports the Besicovitch covering lemma, one can prove M is weak (1,1) bounded. However, the Besicovitch covering lemma is not easy to verify. Meanwhile it does not hold on some common manifolds. For instance, counterexamples are constructed on Heisenberg groups equipped with Carnot-Carathéodory or Korányi metric (see [9,21]), and also on Symmetric spaces of noncompact type (see [4]).
We say (S, d, µ) satis es the doubling volume condition if there exists a constant C > such that V(x, r) ≤ CV(x, r), ∀ r > , x ∈ S. (1.1) According to (1.1), there exist ν, C > such that If S satis es the doubling volume condition, Vitali covering lemma holds on S. Then it is well known that M is weak (1,1) bounded.
On the other hand, we say a noncompact manifold (S, d, µ) has the exponential volume growth if there exist constants c , c , C , C , C > such that C e c (s−r) ≤ V(x, s) V(x, r) ≤ C e c (s−r) , ∀ ≤ s ≤ r, x ∈ S, (1.2) and V(x, r) ≤ CV(x, r), If S has the exponential volume growth, M is not in general weak (1,1) bounded. See, for example the counterexamples in Strömberg [24] and Li [14,15]. For more results about the maximal functions on manifolds with exponential volume growth, we refer the readers to [5,7,8,[13][14][15][16][17]24] and references therein.
Recently, various topics about the maximal functions such as dimensional free estimates, multisublinear maximal functions and applications to function spaces have been extensively studied. See, for example [1,2,6,10,18,20,25,26] and references therein.
One natural way to construct new manifolds is taking product of given manifolds.
For every point X = (x , x , · · · , x k ) ∈ X, denote by B(X, r) the ball centered at X with radius r > and by V(X, r) the volume of B(X, r) with respect to µ. The centered maximal functions on X i and X are denoted by M i and M respectively.
The main purpose of this paper is to study that the mapping properties of the centered maximal functions on the product manifolds will to what extend be in uenced by that on submanifolds.
Note rst that when every X i ( ≤ i ≤ k) satis es (1.1), so does X. Therefore the centered maximal function M is of weak (1,1) type. Thus it is natural to ask about the mapping properties of M when at least one X i does not satisfy the doubling volume condition.
To begin with, we consider the product spaces of two manifolds.
Remark 1.2. Since X has the exponential volume growth, we need rst to estimate the volume of balls on X.
Then inspired by the proof for the strong maximal operators( [23]), we control M by M M . Finally, it turns out that the mapping properties of M are almost determined by that on exponential volume growing submanifold.
Next we will study the product spaces of two weighted hyperbolic spaces. Let us begin with some notations and known results. The real hyperbolic space H n+ for n ≥ can be interpreted as R + ×R n with the Riemannian metric ds = dy +dx y . The induced distance is And the induced measure is dµ(y, x) = y −n− dydx with dx the Lebesgue measure on R n . Let α ∈ R. The weighed hyperbolic space of dimension n + with the parameter α, H (n+ ,α) := (H n+ , d, y α−n− dydx), namely R + × R n endowed with the distance d and the measure y α−n− dydx. As is known, H (n+ ,α) is of exponential growth. See for example [11,14] for more details.
By [14,16], the mapping properties of M on H (n+ ,α) where α ∈ R can be summarized as follows: 1. If α ≤ or α > n, then M is weak (1,1) bounded; 2. If n ≤ α ≤ n, then M is unbounded on L p for ≤ p < ∞; 3. If < α < n , letp = n−α n− α . Then M is unbounded on L p for ≤ p <p and bounded on L p for p >p. Let X = H (n+ ,α) , X = H (n+ ,β) . The mapping properties of the centered maximal function on the product space will be investigated for every α, β ≠ n . Without loss of generality, we assume α ≥ β and the following cases will be investigated.
The results read as follows: (ii) M is unbounded on L p for ≤ p < ∞ in the following cases: Case 5 and Case 2 with α − α + β − β ≤ . (iii) In Cases 3, 7, 9 and Case 2 with α − nα + β − nβ > , there exists p > determined by n, α, β such that when p < , M is weak (1,1) bounded; when ≤ p , M is bounded on L p for p < p ≤ ∞ and unbounded for ≤ p < p .
(iv) In Cases 6 and 8, there exists p > such that M is bounded on L p for p < p ≤ ∞ and unbounded for ≤ p < p . Remark 1.4. Since there is no doubling volume condition in X, it fails to use the covering lemmas to prove the mapping properties of M. Instead, in the spirit of the arguments in [14,24], we use an integral operator to control the maximal functions. Moreover, we have proved some convolution inequalities on X which are of independent interest; see, for example [3,19].
Finally we will study the product spaces of several weighted hyperbolic spaces.
Remark 1.6. It is worth noting that our argument still works without the assumption α i > n i for ≤ i ≤ k. However, in that cases, we can only show that there exist two numbers < p ≤ p such that M is bounded The paper is organized as follows. Section 2 is devoted to the volume estimates on the product manifolds. Among others, we give detailed estimates of the volume on product of weighted hyperbolic spaces which is of independent interest. We will prove Theorem 1.1 in Section 3. The proof of Theorem 1.3 and Theorem 1.5 will be given in Section 4 and Section 5, respectively. Now we introduce some notations. If f and g are two functions, we say f ∼ g if and only if there exists a constant c > such that c − f ≤ g ≤ cf . We say f g if and only if there exists a constant c > such that f ≤ cg. Given a real number l, denote by [l] the integer part of l. The constants C, c > may change from line to line.

Volume Estimates
In this section, we give volume estimates of balls in the product spaces X mentioned above. Set X = X × X where X has the exponential volume growth and X satis es the doubling volume condition. We have the following volume estimates of the balls in X. Proposition 2.1. Suppose that X, X , X are de ned as above, then the following holds: To begin with, we obtain dµ and we have where we have used (1.1) and (1.2). On the other hand, where ν is the doubling constant and c is the constant in (1.2). Thus we have nished the proof.
Next we consider the product space of several weighted hyperbolic spaces.
Before proving the proposition, we need the following lemma. Then we have Proof:. By the Coarea formula, we have where dS = y α dσ and dσ is the standard surface measure on the hyperbolic space. Use the Coarea formula again and we obtain Proof of Proposition 2.2:. We will prove it by induction. When k = , the volume estimates can be found in Then by induction we obtain for < r < ∼ a α · · · a α k k r n +···+n k +k .
Now we prove the result for r ≥ C * where C * > is a constant to be determined later. Note rst that Similarly, we have J a α · · · a α k− k− a α k k e rα * k a α · · · a α k k r k− e r α * +···+α * k .
Hence it is su cient to prove J ∼ a α · · · a α k k r k− e r α * +···+α * k for r ≥ C * . In fact, by Lemma 2.3 we have Set A = α * + · · · + α * k− , B = α * k and denote Then we have proved  a α · · · a α k k r n +···+n k +k , < r < a α · · · a α k k r k− e r α * +···+α * k , r ≥ C * , Note that the argument in proving the volume estimates for < r < still works for < r < C * . Moreover when ≤ r ≤ C * , we have a α · · · a α k k r n +···+n k +k ∼ a α · · · a α k k r  r). For the sake of brevity, in the following proof, we will show S(Y, r) ∼ V(Y, r) for ∀r > , Y ∈ X.
By Proposition 2.2 and Lemma 2.3, we know that Then we have Note that by this expression and Lemma 2.3, we can show inductively that S(Y, r) a α · · · a α k k , ∀Y ∈ X, < r ≤ .
We will prove S(Y, r) ∼ V(Y, r) for r > by induction. When k = , it is just the Lemma 2.3. The proof is similar to the proof of Proposition 2.2, so we omit some details here. First we will prove the result for r > C ** where C ** > is a constant to be determined later. First we have Now we are ready to deal with S . By changing variable, we have where A = α * + · · · + α * k− , B = α * k . Following the method in the proof of Proposition 2.2, we have On the other hand, we have for k ≥ Set C ** = max , + ( A B ) , + ( B A ) . Then we have proved S(Y, r) ∼ V(Y, r) for r > C ** . Note that for < r ≤ C ** , both S(Y, r) and V(Y, r) are positive and nite. Then we have nished the proof.

Proof of Theorem 1.1
The positive part of Theorem 1.1 follows from the following lemma.

Lemma 3.1.
Suppose that X has the exponential volume growth and X is noncompact satisfying the doubling volume condition. Then we have

Proof of Lemma 3.1:.
Without loss of generality, we suppose f ≥ .
To begin with, let And we have According to the volume estimates, we have Now we deal with T . For every r ≥ , we have where ν is the doubling constant. It remains to show the rst term in the last inequality can be controlled by M M f .
In fact, we have Thus we have proved the lemma.
Proof of the positive part of Theorem 1.1:. According to the de nition of Lorentz space (see [22]), we have for < p ≤ q < ∞ When q = ∞ and < p < ∞, we have Since M is bounded from L p (X ) to L p,q (X ), by the Minkowski inequality, we have for < p ≤ q < ∞ Note that we have used the fact M is bounded from L p (X ) to L p (X ) for < p ≤ ∞. When q = ∞, with a slight modi cation, the above argument still works.
Thanks to (1.3), M is bounded from L p (X ) to L p (X ) for p > . Denote by A the operator norm of M on L p (X ). Then we obtain Fix a point P ∈ X and set E = X × B (P , √ R) where R > . Now consider the function f (y , y ) = f (y )χ B (P , √ R) and for (x , x ) ∈ E, the following holds, We have used (1.2) in the second inequality. Letting R → ∞, by the Fatou theorem we have Since the constants in the above inequalities are independent of δ, we have proved the desired results.

Proof of theorem 1.3
We need some preparation before the proof. Given a weighted hyperbolic space H (n+ ,α) , de ne the following group operation for any Y = (a, b), Y = (a , b ) ∈ H n+ , (a, b) · (a , b ) = (aa , b + ab ).
It is easy to verify E = ( , ) is the identity element and (a − , −ba − ) is the inverse element of (a, b). For any In this section, we consider the product space X of two weighted hyperbolic spaces X = H (n+ ,α) , X = H (n+ ,β) . To begin with, we consider the positive parts of Theorem 1.3. Since X has the local doubling properties, the operator M de ned by is weak (1,1) bounded. We only need to prove the results for the following operator Without loss of generality, suppose f ≥ and it is su cient to consider the following operator Set Z = Y − · Y and change the variables : where we have used the fact dµ(Y · Z) = a α a β dµ(Z). Denote by Z = ((y , x ), (y , x )) where (y , x ), (y , x ) ∈ H n+ . By the Minkowski inequality, we obtain The above inequality holds when i α i (α i − n i ) < or i α i (α i − n i ) > , p < p . Then we have proved the results.