Sharp multiplier theorem for multidimensional Bessel operators

Consider the multidimensional Bessel operator $$B f(x) = -\sum_{j=1}^N \left(\partial_j^2 f(x) +\frac{\alpha_j}{x_j} \partial_j f(x)\right), \quad x\in(0,\infty)^N. $$ Let $d = \sum_{j=1}^N \max(1,\alpha_j+1)$ be the homogeneous dimension of the space $(0,\infty)^N$ equipped with the measure $x_1^{\alpha_1}... x_N^{\alpha_N} dx_1...dx_N$. In the general case $\alpha_1,...,\alpha_N>-1$ we prove multiplier theorems for spectral multipliers $m(B)$ on $L^{1,\infty}$ and the Hardy space $H^1$. We assume that $m$ satisfies the classical H\"ormander condition $$\sup_{t>0} \left||\eta(\cdot) m(t\cdot)\right||_{W^{2,\beta}(\mathbb{R})}<\infty$$ with $\beta>d/2$. Furthermore, we investigate imaginary powers $B^{ib}$, $b\in \mathbb{R}$, and prove some lower estimates on $L^{1,\infty}$ and $L^p$, $1

We choose the constant d ("homogeneous dimension") as small as possible. In this case max(1, α j + 1).
Here I τ is the modified Bessel function. It is well known that T t (x, y) satisfies the upper and lower gaussian bounds, i.e. there exist constants c 1 , c 2 ,C 1 ,C 2 > 0, such that Since B is self-adjoint and nonnegative, for a Borel function m : (0, ∞) → C the spectral theorem defines the operator where E B is the spectral resolution of B.

Multiplier theorems for B.
Multiplier theorems for B and other operators are one of the main topics in harmonic analysis. Many authors investigated assumptions on m that guarantee boundedness of m(B) on various function spaces, such as L p (X ), H p (X ), L p,q (X ) and others.
In the multidimensional case N ≥ 1 in [4] the authors prove weak type (1, 1) estimates for m(B), where m is of Laplace transform type, i.e. there exists φ ∈ L ∞ (0, ∞), such that Notice, that if m is of Laplace transform type, then m is radial and (as a function on (0, ∞)) satisfies (S) with any β > 0. Another multidimensional result can be found in [16], where it is proved that m(B) is weak type (1,1) and bounded on the Hardy space H 1 (X ) provided that α j > 1 for j = 1, ..., N and m satisfies (S) with β > d /2. See also e.g. [17,18,32] for other multiplier results for the Bessel operator.
Our first main goal is to obtain multiplier theorem for B in the most general case N ≥ 1 and α j > −1, j = 1, ..., N . Let us notice that many of the results before assumed that α j > 0 and the case α j < 0 is more difficult and less known. One reason for that is the singularity at zero of the measure x α j d x j when α j < 0. Another difference is that the generalized eigenfunctions of B are no longer bounded if α j < 0 for some j . As a consequence, there is no so called "generalized translation" operator and convolution structure related to B.
Also, we are interested in multiplier results that are sharp in the sense that we assume (S) with β as small as possible. In this case this is expected to be β > d /2 (we shall discuss this later on).
To state the multiplier result let us recall that the weak L 1 space is given by the seminorm and the Hardy space H 1 (B) is defined by the norm .
For the properties of H 1 (B) in the case N = 1 we refer the reader to [6]. In the general case N ≥ 1 and α j > −1, j = 1, ..., N the atomic characterization of H 1 (B) can be found in [15] (see also [16]). We shall recall this characterization later on.
Part 1. of Theorem A will be proved by using results of [30]. More precisely, we shall check the assumptions of [30,Th. 3.1]. The proof of 2. will be given in Section 2. In fact, in the proof we shall only use general properties of B, such as e.g. (D), (G), and (P q ) below. Thus, the multiplier result in Section 2 will be formulated in a more general context. This section can be read independently of the rest of the paper and we shall use different notation. As usual, 3. is a consequence of either 1. or 2. by duality and interpolation, see e.g. [3].
1.3. Imaginary powers of B. Another goal of this paper is to study the imaginary powers B i b , b ∈ R, of the Bessel operator and establish lower bounds of these operators on some function spaces. We shall concentrate our attention on the depndence of the lower estimates on b for large b. This is related with sharpness of multiplier theorems and may be of independent interest. To state these estimates let us restrict ourselves to the one- Notice, that the integral in (1.3) is not absolutely convergent, thus we have to explain how the kernel K b (x, y) is related to the operators B i b . Indeed, in Subsection 3.3 we shall prove that for f ∈ L ∞ (X ) with compact support we have One of our goals is to provide lower estimates for B i b .
Theorem B. Assume that α > −1. Then there exist a constant C > 0 and a function f such that f L 1 (X ) = 1 and for |b| large enough we have Theorem C. Assume that α > 0 and p ∈ (1, 2). Then there exist C p > 0 and f such that f L p (X ) = 1 and for |b| large enough we have The proofs of Theorems B and C are presented in Subsection 3.3. To prove Theorem B we shall carefully analyze the kernels K b (x, y). More precisely, we prove the following lemma. Lemma 1.6. Assume that α > −1 and b ∈ R. Then Moreover, there exists C > 0 that does not depend on b, such that Notice that the kernel R b (x, y) is related to an operator that is bounded on every L p (X ), 1 ≤ p ≤ ∞, uniformly in b ∈ R. Thus we may think of R b (x, y) as of some kind of "error term". However, for |b| > 1 the size of the constants are the following: c.f. Lemma 4.1. Thus, c 3 (b) grows exponentially when |b| → ∞, while the constants c 1 (b) and c 2 (b) are much smaller. It appears that the growth of the constant c 3 (b) will lead to a problem in deriving lower estimates for B i b (since our goal is to find the exact dependence on b). However, we can overcome this difficulty when analyzing weak (1, 1) norm as in Theorem B. The same trick seems not to work in other function spaces (such as H 1 (B), L p (X ) and L p,∞ (X ) with p > 1), thus the proof of Theorem C is different and uses the integral representation of the Bessel function I τ instead of Lemma 1.6.
As a corollary of Theorems B and C we obtain that Theorem A is sharp (at least for N = 1) in the sense that d /2 cannot be replaced by a smaller number. The argument is standard, but we shall present it now for the convenience of the reader. One can check that for m b (λ) = λ i b we have Also, Theorem A actually gives that m b (B) f L 1,∞ (X ) ≤ C M b f L 1 (X ) , where C does not depend on b. Combining these estimates with Theorem B for |b| large enough we have Therefore β ≥ d /2. Actually, one expects that β = d /2, but this question is beyond the scope of this paper.
Similarly, the constant d /2 cannot be improved for the Hardy spaces. If α < 0 then d /2 = 1/2 and (S) with β < 1/2 would not even guarantee that m is bounded. On the other hand, for α > 0 if we could prove multiplier theorem on H 1 (B) with a constant lower than d /2, then by interpolation we would have better upper bounds for m b (B) on L p (X ) for 1 < p < 2, which contradicts Theorem C by an argument similar to the one above.
1.4. Organization of the paper and notation. In Section 2 we state and prove a "sharp" multiplier theorem on Hardy spaces for self-adjoint operators on spaces of homogeneous type with certain assumptions (Theorem D). This is a slight generalization of Theorem A 2. in the spirit of [30,Th. 3.1]. In Section 2 we shall use different notation, so that it can be read independently of the rest of the paper. In Section 3 we prove the results stated above. More precisely, first we check that B satisfies assumption (P 2 ) (see Section 2 below) in the full generality N ≥ 1, α j > −1 for j = 1, ..., N . Thus Theorem D can be applied for B. Then we prove Lemma 1.6 and Theorems B and C. We shall use standard notations, i.e. C and c denote positive constants that may change from line to line.

Background and general assumptions.
In this section we consider a space Y with a metric ρ and a nonnegative measure µ. We shall assume that the triple (Y , ρ, µ) is a space of homogeneous type, i.e. there exists C > 0 such that µ(B(x, 2r )) ≤ C µ(B(x, r )), for all x ∈ Y and r > 0, where B(x, r ) = y ∈ Y : ρ(x, y) < r , c.f. [12]. It is well-known that this implies the existence of d ,C d > 0 such that As usual, we choose d as small as possible, even at the cost of enlarging C d .
Let A denotes a self-adjoint positive operator and denote by P t = exp(−t A) the semigroup generated by A. Assume that there exists an integral kernel P t (x, y) such that P t f (x) = Y P t (x, y) f (y) d µ(y) and that satisfies the upper gaussian bounds, i.e. there exist c 2 ,C 2 > 0 such that

Multiplier theorems.
Since A is self-adjoint and positive there exists a spectral mea- . By the spectral theorem, for a Borel function m on (0, ∞), we have the operator In the classical case A = −∆, Y = R D , the Hörmader multiplier theorem states that if m satisfies (S) with β > D/2, then m(−∆) is weak type (1, 1) and bounded on L p (R D ) for 1 < p < ∞. It is well-known that the constant D/2 is sharp in the sense that it cannot be replaced by a smaller constant, see e.g. [28].
At this point let us recall one of many multiplier theorems on spaces of homogeneous type. Suppose Y and A are as in Subsection 2.1. Following [30] we introduce additional assumption. Suppose that there exists C > 0 and q ∈ [2, ∞], such that for R > 0 and every Borel function m on R satisfying suppm ⊆ [R/2, 2R] we have At this point let us make a few comments.

2.
For the Bessel operator we are interested in (S q ) and (P q ) for q = 2 only, (S) = (S 2 ).
However, in Section 2 the results are stated and proved with an arbitrary q ∈ [2, ∞].

3.
The assumption (P q ) in some sense plays a role of Plancherel theorem in the proof of Theorem 2.1. It is a key to obtain the sharp range β > d /2. For example, if we would allow m to satisfy (S q ) with β > d /2 + 1/2, then (P q ) would be superfluous.

4.
The assumption (P q ) is written in [30] for m having support in However, a simple inspection of the proof shows that (P q ) is needed only for m with suppm ⊆ [R/2, 2R]. This makes no difference for many operators. However, it matters e.g. when considering the Bessel operator with negative parameters α j . 5. Assumption (P q ) in [30] is written for m( A), but we use equivalent version with m(A) (therefore we replace B(y, R −1 ) by B(y, R −1/2 )).
One of the main goals of this paper is to establish a multiplier theorem on the Hardy space. Let us start by recalling that the Hardy space H 1 (A) associated with A is defined by To state our result we shall assume additionally that P t (x, y) satisfies also the the lower Gaussian bounds, namely there exist c 1 ,C 1 > 0, such that and that the space (Y , ρ, µ) satisfies the following assumption: Notice that (Y) implies that µ(Y ) = ∞ and that µ is non-atomic. Now we are ready to state the theorem.
Theorem D. Assume that (Y , ρ, µ) is a space of homogeneous type, d is as in (D), and (Y) is satisfied. Suppose that there is a self-adjoint positive operator A such that (UG), (LG), and The history of multiplier theorems for spaces of homogeneous type is long and wide. The interested reader is referred to [2,8,9,11,13,16,17,20,22,26,27,29,30] and references therein. Let us concentrate for a moment on the range of parameters β in Theorem D. Obviously, in general, the range β > d /2 is optimal. However, it may happen that for some particular operators one may obtain multiplier results assuming that β > d /2 with d < d , see e.g. [24,25,27]. On the other hand, there are known families of operators for which the constant d /2 cannot be lower. One of the methods to prove this is to derive lower estimates for A i b in terms of b ∈ R, see [10,25,28,30].
Boundedness of operators on the Hardy space H 1 is a natural counterpart of weak type (1, 1) bound. For example, it is a good end point for the interpolation, see e.g. [3]. However, the Hardy spaces are strictly related to some cancellation conditions and it is usually more involving to study properties of operators on the Hardy space, than on L p or L p,∞ spaces. Let us also mention that boundedness from H 1 to H 1 obviously implies boundedness from H 1 to L 1 , which is usually much easier to prove.

Hardy spaces.
The Hardy spaces on spaces of homogeneous type are studied extensively from the 60's, see e.g. [12]. In particular, now we have many atomic decompositions for H p on various spaces and operators acting on this spaces. We refer the reader to e.g. [6,15,21] and references therein.
In this subsection we recall some results on Hardy spaces related to A, assuming that (D), (UG), (LG) and (Y) are satisfied. For the proofs and more details we refer the reader to [15]. Firstly, there exists the unique (up to a multiplicative constant) A-harmonic function ω : Y → R such that The function ω plays a special role in the analysis of A and P t . In particular we have the following Hölder-type estimate.
Using Theorem 2.2 the authors obtained the following atomic decomposition for the elements of H 1 (A). Let us call a function a : Y → C an (µ, ω)-atom, if there exists a ball B in Y , such that:  ∈ N), such that Let us start by recalling a few consequences of (D) and (UG).
In particular and, for ρ(y, z) < R −1/2 , Let us start by showing the following lemma.
Observe that (2.13) is exactly the estimate we look for, but the Sobolev parameter is higher by 1/2 than we want. To sharpen this estimate, we make use of known interpolation method. Notice, that M t (x, y) = P t (K m(A) (·, y))(x). It is well-known that (UG) implies boundedness on L 2 (Y ) of the maximal operator M f = sup t>0 |P t f |. A second estimate needed for an interpolation is the following (2.14) In the last inequality we have used (P q ). Now, Lemma 2.11 follows by interpolating (2.13) and (2.14), see e.g. proofs of [30,Lem. 4

.3(a)] and [16, Lem. 2.2] for details.
Proof of (2.9). By the Cauchy-Schwarz inequality and Lemmas 2.7 and 2.11, Consider for a moment the operator P t m(A) exp(AR −1 ) and let M t,R (x, y) be its kernel. By almost identical arguments as in the proofs of Lemma 2.11 and (2.9), we can show that for β > d /2 we also have Proof of (2.10). Notice, that M t (x, y) = Y M t,R (x, u)P R −1 (u, y) d µ(u). For ρ(y, z) < R −1/2 , by Corollary 2.3 and (2.15), Therefore, it is enough to prove that Let η ∈ C ∞ c (2 −1 , 2) be a fixed function such that j ∈Z η(2 −j λ) = 1 for all λ ∈ (0, ∞). By using this partition of unity, we decompose m as Denote m j ,t (λ) = exp(−t λ)m j (λ) and let M j ,t (x, y) be the kernel of m j ,t (A) = P t m j (A). Obviously, suppm j ,t ⊆ [2 j −1 , 2 j +1 ] and applying (2.9) we obtain that If y ∈ B and j < N , then ρ(y, y 0 ) < r < 2 −j /2 and we can apply (2.10) for the kernel M j ,t with R = 2 j . Using the cancellation condition of a, This finishes the proof of (2.16) and Theorem D.

THE MULTIDIMENSIONAL BESSEL OPERATOR
In this Section we turn back to the analysis related to B and prove the results stated in Section 1. N ∈ N and α j > −1 for j = 1, ..., N . For

The Hankel transform. Recall that
Here J τ denotes the Bessel function of the first kind. By the asymptotics of J τ one has The Hankel transform is defined by As we have already mentioned, φ j ∈ L ∞ if and only if α j ≥ 0. Nevertheless, it is known that H always extends uniquely to an isometric isomorphism on L 2 (X ), see [4] then m is radial and
Proof. In the proof we consider only the case N ≥ 2. Let q = 2 and suppose that m is supported in [R/2, 2R] for some R > 0. Notice that by (3.3) and (3.2) we have and the kernel associated with m(B) has the form Therefore, by the Plancherel identity for H, (P 2 ) is equivalent to For each i = 1, ..., N we consider four cases: Divide the set x ∈ X : 1/2 < |x| 2 < 2 into several disjoint regions using the cases above. Without loss of generality we may consider the set S of points x ∈ X such that: • x i satisfies C1. for i = 1, ..., k 1 , • x i satisfies C2. for i = k 1 + 1, ..., k 2 , • x i satisfies C3. for i = k 2 + 1, ..., k 3 , • x i satisfies C4. for i = k 3 + 1, ..., N , where 0 ≤ k 1 ≤ k 2 ≤ k 3 < N . The fact that k 3 < N is implied by |x| 2 > 1/2. Notice that it may happen that S is empty. Recall that ν(B(y, r ) j d x j is the one-dimensional measure, and Denote d g l = N + α 1 + ... + α N . Using (3.1) and Lemma 3.4 with k = k 2 , we have

Imaginary powers of B.
In this subsection we prove Lemma 1.6 and Theorems B and C. From now on we consider one-dimensional Bessel operator, i.e. N = 1, X = (0, ∞), α > −1, and d ν(x) = x α d x.
Let us start this section by recalling well-known asymptotics of the Bessel function I τ , i.e.
Proof. Using the reflection formula  Similarly we show boundedness of F for z = 2 + bi , |b| ≥ 1.