Hardy's Inequality for Laguerre Expansions of Hermite Type

Hardy's inequality for Laguerre expansions of Hermite type with the index $\al\in(\{-1/2\}\cup[1/2,\infty))^d$ is proved in the multi-dimensional setting with the exponent $3d/4$. We also obtain the sharp analogue of Hardy's inequality with $L^1$ norm replacing $H^1$ norm at the expense of increasing the exponent by an arbitrarily small value.


Introduction
The well known Hardy inequality states that , where H 1 (R) and H 1 (R + ) denote the real Hardy spaces on R and R + , respectively. Hardy's inequality in the context of Hermite functions was further intensively studied by many authors. Radha [13] proved a similar inequality in an arbitrary dimension. In [14] an improved version of Hardy's inequality was introduced in the multi-dimensional case, d ≥ 2, by Radha and Thangavelu. The exponent in the denominator was 3d/4. This led to the hypothesis that in the one-dimensional case the exponent should be equal to 3/4. It was indeed proved in [8] by Z. Li, Y. Yu and Y. Shi. A generalization of Kanjin's results, with the spaces H p (R) and H p (R + ), p ∈ (0, 1], instead of H 1 (R) and H 1 (R + ), was also considered in the context of Hermite functions (see [2,14]) and in the context of Laguerre functions (see [14,15]).
In this paper we study multi-dimensional Hardy's inequality in the context of Laguerre functions of Hermite type {ϕ α n } n∈N d . In view of the uniform boundedness of the derivatives of functions ϕ α n and [5,Lemma] we have the one-dimensional inequality ∞ k=0 | f, ϕ α k | (k + 1) 29/36 f H 1 (R + ) .
Our aim is to obtain the analogue of this inequality with the power 3d/4, which does not depend on α, and in dimension d ≥ 1.
The proof of one of the main results, Theorem 4.2, is based on the atomic decomposition of functions from H 1 (R d + ) and relies on a uniform estimate for atoms and an additional argument of the "weak" continuity of certain operators. Without this argument, which was often omitted in papers concerning this topic, the proof would have a gap. We remark that the uniform estimate for atoms does not imply continuity of operators that appear in analysis that involves the atomic decomposition of H 1 (R d ) (see [3]).
The range of the Laguerre type multi-index α that is considered in Theorem 4.2, is the set ({−1/2} ∪ [1/2, ∞)) d . This kind of restraint appeared before (see for example [11]). Note that the one-dimensional Laguerre functions of Hermite type with the Laguerre type multi-index equal to −1/2 or 1/2 are, up to a multiplicative constant, the Hermite functions of even or odd degree, respectively. Therefore, it was fair to assume that this values of α should be included. Technically, the restraint emerges from the range of α's for which the derivatives of the Laguerre functions of Hermite type are uniformly bounded. It may also be related to the fact that the associated heat semi-group is a semi-group of L p contractions precisely for this set of α's (see [12]).
In [6] Kanjin proved that if the exponent in one-dimensional Hardy's inequality in the context of Hermite functions is strictly greater than 3/4, then one can replace H 1 (R) norm by L 1 (R) norm and, moreover, the exponent 3/4 is sharp. In Theorem 5.1 we shall prove that this is also the case in the context of Laguerre functions of Hermite type and extend this result to an arbitrary dimension.
Notation. Throughout this paper we write n = (n 1 , . . . , n d ) ∈ N d for a multi-index and |n| = n 1 + . . . + n d for its length, where N = {0, 1, . . . .} and d ≥ 1. The Laguerre type multi-index α = (α 1 , . . . , α d ), unless stated otherwise, is considered in the full range, i.e. α ∈ (−1, ∞) d . We shall also use the notation R d we denote the inner product by f, g . Sometimes we shall use this notation for functions that are not in L 2 (R d + , dx) but the underlying integral makes sense. We shall use the symbol denoting an inequality with a constant that does not depend on relevant parameters. Also, the symbol ≃ means that and hold simultaneously. Moreover, we will denote asymptotic equality by ≈.
Acknowledgement. The author would like to thank Professor Krzysztof Stempak for insightful comments and his continuous help during the preparation of this paper.

Preliminaries
The Laguerre functions of Hermite type of order α on R d + are the functions where ϕ α i n i (x i ) is the one-dimensional Laguerre function of Hermite type defined by The functions {ϕ α n : n ∈ N d } form an orthonormal basis in Note that ϕ α k (u) = (2u) 1/2 L α k (u 2 ). We shall use the pointwise asymptotic estimates (see [9, p. 435] and [1, p. 699]) where ν = ν(α, k) = max(4k + 2α + 2, 2) and with γ > 0 depending only on α. Hence, There is the known formula for the derivatives of functions ϕ α k , where ϕ α −1 ≡ 0. From (1) it follows that for α ≥ −1/2 we have and also by (2) for α ∈ {−1, 2} ∪ [1/2, ∞), We introduce the family of operators {R α r } r∈(0,1) , defined spectrally for f ∈ L 2 (R d + ), by It is easily seen by means of Parseval's identity that for every r ∈ (0, 1), the operator R α r is a contraction on L 2 (R d + ). The kernel associated with R α r is defined by Note that and there is also the explicit formula (compare [17, p. 102]) where I α i denotes the modified Bessel function of the first kind, which is smooth and positive on (0, ∞).
where |B| denotes the Lebesgue measure of B, and B a(x) dx = 0. Every function f ∈ H 1 (R d ) has an atomic decomposition, namely there exist a sequence of complex where the convergence of the first series is in given below follow from [4, Lemma 7.40] stated in the one-dimensional case therein, however easily generalizable to the case of an arbitrary dimension. Every f ∈ H 1 (R d + ) has an atomic decomposition as in (6) We may assume that every ball associated with an We remark (again see [4,Lemma 7.40

One-dimensional kernel estimates
We shall estimate the kernels R α i r (x i , y i ). For the sake of convenience we will write x, y, α instead of x i , y i , α i .
For the proof in the case α > 1/2 see [10, pp. 6-7]. If α = 1/2, then it suffices to use the explicit formulas (see [7, p. 112]) Lemma 3.3 is of paramount importance in our estimates wherever the cancellations are needed. It has been used before in the context of Laguerre functions (see for example [11]).

Main results
be the center of B. Note that since R α r are contractions on L 2 (R d + ) we have for every . This finishes the proof in case |B| ≥ 1. From now on, let us assume |B| < 1. Minkowski's integral inequality and Corollary 3.5 imply Thus, using the above estimates we obtain and this quantity is bounded by a constant that does not depend on |B|.
Now we can state the main theorem. , . Proof. Firstly we prove that n∈N d | a, ϕ α n | (|n| + 1) 3d/4 1, We shall employ the same argument that is used in [8]. For the Beta function there is the known asymptotic B(k, m) ≈ Γ(m)k −m for large k and fixed m. Let a be an H 1 (R d + ) atom. Using Hölder's inequality and Proposition 4.1 we obtain Our aim is to prove that T : (3) and (7) yield .
Let us take f ∈ H 1 (R d + ) and f = ∞ i=0 λ i a i be an atomic decomposition of f . Denote f m = m i=0 λ i a i and note that T (f m ) is a Cauchy sequence in ℓ 1 ((|n| + 1) −3d/4 ). Indeed, we have for l < m, Hence, T (f m ) converges to a sequence g in ℓ 1 ((|n| + 1) −3d/4 ) and, by (12), also in ℓ 1 ((|n| + 1) −d ). Since T : To obtain the boundedness of T : . This finishes the proof.

L 1 result
In this section we shall prove that the inequality in Theorem 4.2 holds also with L 1 (R d + ) norm replacing H 1 (R d + ) norm provided that the exponent in the denominator is strictly greater than 3d/4. Our reasoning is similar to Kanjin's in [6]. The main tool in the proof of this fact is the asymptotic estimate for functions ϕ α n .
Theorem 5.1. Let ε > 0 and α ∈ [−1/2, ∞) d . Then , Proof. Given ε > 0 and α ∈ [−1/2, ∞) d , for the proof of (13) it suffices to verify that We shall prove this estimate in the one-dimensional case. This is indeed sufficient, since But ϕ α 0 (u) 1 uniformly in u ∈ R + , so given α ∈ [−1/2, ∞) we are reduced to proving Denotek = 4k + 2α + 2 and for u ∈ R + define Note that by (1), if k / ∈ N u , then |ϕ α k (u)| k −1/4 uniformly in u and k, and hence the sum over the complement of N u is bounded uniformly in u ∈ R + . We claim that the same is true for the sum over N u .
Assume N u = ∅ and let uniformly in u. Note that |u 2 −k| increases in k providedk ≥ u 2 . Since u 2 ≤ 3 k 0 /2, we have fork ≥ 3 k 0 /2 or, equivalently, for k ≥ k * * 0 := ⌈3k 0 /2 + α/4 + 1/4⌉, that |u 2 −k| increases in k. Hence, This completes the proof of the claim and hence the justification of (15) and thus finishes the verification of (13). Now we pass to the proof of (14). Assume a contrario that the sum in (14) is finite for every f ∈ L 1 (R d + ). The uniform boundedness principle implies that , . Hence, by an obvious adaptation of [6, Lemma 1] we obtain But, as we shall see, it does not hold. In fact, we shall prove that for any x ∈ R d + we have Notice that using the asymptotic estimate for Laguerre polynomials (see [7, (4.22.19)]) and the known asymptotic for the Gamma function, Γ(k + a)/Γ(k + b) ≈ k a−b , k → ∞, where a, b ≥ 0 are fixed, we obtain for u ∈ R + and β ≥ −1/2 Hence, we reduce verifying (16) to checking that We first prove the one-dimensional case. Fix u ∈ R + and notice that, for d = 1, the corresponding sum in (17) is greater than Thus, (17) holds, since for any t ∈ R {0} each of the two series ∞ k=1 sin cos t √ k k converges. Since we could not find a proof of this fact in the literature, we offer a short argument (for the cosine series and t = 1). Let H(k) = k j=1 1/j denote the k-th harmonic number. Applying summation by parts, for any K ∈ N + we obtain We use the asymptotic H(k) = log k + γ + r(k), where r(k) = O(1/k) and γ is the Euler-Mascheroni constant, and plug it into the both summands on the right hand side of the above formula. The terms resulting from the error parts, namely r(K) cos √ K and are easily seen to converge with K → ∞. This is also true for Thus we are left with The latter integral, after a change of variable, is also easily seen to converge with K → ∞. This finishes the proof of the convergence of the investigated series. Now we continue and prove (17) in the multi-dimensional setting. Given x ∈ R d + and proceeding similarly as before we reduce justifying (17) to verifying that each of the 3 d −1 iterated series converges, where J is any non-empty subset of {1, . . . , d} and t j = 0, j ∈ J. We shall use the induction over the dimension. Suppose that every series of the form as in (18) converges. We will prove that also the analogous series in dimension d + 1 converge. Fix such a series and consider the associated set J ⊂ {1, . . . , d + 1}. We distinguish two cases depending on whether d + 1 ∈ J or not.
If d + 1 / ∈ J, then the investigated series is of the form It now suffices to use the asymptotic and the inductive assumption.
The case d+1 ∈ J is more involved. We simplify matters, without any loss o generality, assuming t j = 1. The considered series is of the form (or with the sine in place of the cosine, but this is not an obstacle), where Λ(J, n) is a product of the sines or the cosines taken at √ n j , j ∈ J, respectively. In fact, we shall prove the slightly stronger result that We remark that the cancellation provided by one trigonometric functions are sufficient in our estimates. Note that we cannot use the triangle inequality in the innermost series, because the resulting series would diverge.
To verify (19) we check the convergence of the innermost series with a control of the decrease of its sum in |n|. We will use the following asymptotic estimate ⌊u⌋ k=1 1 |n| + k = log 1 + u |n| + r u (|n|), where r u (|n|) = O(|n| −1 ) uniformly in u ∈ R + . Summation by parts and the above asymptotic yield The term with the error part r K (|n|) converges to zero with K → ∞, while the integral term of the error part r u (|n|) is absolutely convergent with proper decrease in |n|, namely On the other hand, for the main terms, using integration by parts twice we obtain log 1 + K |n| cos √ K (|n| + K) d − √ u ((2d + 1)u − |n|) √ u(|n| + u) d+2 du.
Thus, combining the above and passing to the limit with K → ∞ we get This finishes the justification of the convergence of the considered series and thus the verification of (17). The validation of (14) is completed and also the proof of the whole theorem is finished.