Sharp Hardy's inequality for orthogonal expansions in $H^p$ spaces

Hardy's inequality on $H^p$ spaces, $p\in(0,1]$, in the context of orthogonal expansions is investigated for general basis on a subset of $\mathbb{R}^d$ with Lebesgue measure. The obtained result is applied to various Hermite, Laguerre, and Jacobi expansions. For that purpose some delicate estimates of the higher order derivatives for the underlying functions and of the associated heat kernels are proved. Moreover, sharpness of studied Hardy's inequalities is justified by a construction of an explicit counterexample, which is adjusted to all considered settings.


Introduction
The classical Hardy inequality (see [14]) for Fourier coefficients states that where Re H 1 is the real Hardy space, where belong the real parts of functions in the Hardy space H 1 (D).Here D denotes the unit disk in the plane.Analogues of (1.1) were considered by Kanjin [15], and f (k) were replaced by the expansion coefficients in two orthonormal bases: the Hermite and standard Laguerre function systems.In general, such inequalities are of the form where ϕ k is an orthonormal basis, •, • denotes the inner product in the associated space L 2 , H 1 is an appropriate Hardy space, and E is a positive number which we refer to as the admissible exponent.The difficulty in establishing versions of (1.2) is twofold.Firstly, given an orthonormal basis one can ask if such inequality holds for a certain E.
Secondly, there is a question about sharpness of the admissible exponent.We say that E is sharp if it is the smallest for which (1.2) holds.Moreover, some generalization of (1.2) are possible, such as replacing H 1 by H p , p ∈ (0, 1], or considering the multi-dimensional situation. In the last two decades many authors were interested in various Hardy's inequalities.As mentioned above, Kanjin initiated the study of version of (1.2) for the Hermite functions (he obtained E = 29/36) and the standard Laguerre functions (E = 1).For the latter system Satake [35] generalized this result for p ∈ (0, 1) with E = 2 − p, and for the former expansions Radha [32] extended investigated the multi-dimensional situation d ≥ 1 with E > (17d + 12)/(24 + 12d).Few years later Radha and Thangavelu [33] proved Hardy's inequality associated with Hermite expansions for d ≥ 2 and p ∈ (0, 1] with the admissible exponent E = 3d(2 − p)/4.The lacking case d = 1 was partially covered by Balasubramanian and Radha [2], but the exponent was strictly larger than the expected value 3(2 − p)/4 (see also Kanjin [16]).The inequality with this admissible exponent was proved ten years later by Z. Li, Y. Yu, and Y. Shi [21].Moreover, the Jacobi trigonometric function expansions were studied by Kanjin and K. Sato [17,18].There are also some other paper concerning various Hardy's inequalities in the context of orthogonal expansions, see for instance [8,20,36,37].
The author have already written a few articles in this topic.In [29] the system of Laguerre functions of Hermite type was studied.Secondly, in [28] a general multidimensional method of proving Hardy's inequalities was introduced.It consists in estimating kernels of certain family of operators closely related to the associated heat semigroup.The method was applied to two Laguerre systems: standard and of convolution type.We stress that in the latter the underlying measure is not Lebesgue measure.Furthermore, in the same paper sharpness of obtained admissible exponents was proved.Up to our knowledge, it was the first explicit construction of such counterexamples known in the associated literature.On the other hand, the long study of Hardy's inequality for Hermite expansions were concluded by the author [30], where he justifies that the know exponent E = 3d/4 (for p = 1) was sharp.Finally, four Jacobi systems were also investigated, see [31].
In this paper we prove Hardy's inequalities in the framework of various orthogonal function systems including generalized Hermite, standard Laguerre, Laguerre of Hermite type, and trigonometric Jacobi expansions in H p spaces, p ∈ (0, 1].We focus on systems associated with Lebesgue measure.The main reason behind this restriction is that the atomic H p spaces are not well defined for all p ∈ (0, 1) when the underlying measure is more arbitrary and only assumed to be doubling.On the other hand, if p = 1, then there is no need for such restraint, see [28,Theorem 2.2].
Although we prove Hardy's inequality for certain orthogonal systems, we are interested in establishing a general method which works in the known settings.Therefore, we enhance the approach from [28] and adjust it for the case p ∈ (0, 1].It requires estimating derivatives of an arbitrary order of the kernels R r (x, y) (see (2.2)).In most cases it turns out to be not so difficult as one could expect once we have analogous asymptotics for the functions composing the considered basis.However, for the Laguerre expansions of Hermite type it is much more involved, see the proof of Proposition 4.6.This result can be viewed in terms of the heat kernel, see Subsection 4.3.Moreover, by some minor modifications we were able to add the parameter s ∈ [p, 2] in Theorem 2. 4.
Another novelty of the paper is the unified approach to sharpness.Instead of constructing separate counterexamples in each setting, we construct one sequence of piecewise constant atoms which, with an additional assumption, justifies that the admissible exponent is sharp.In order to verify the added condition in the specific settings we have to subtly estimate the derivatives of the functions in orthonormal basis, see Lemmas 3.4, 4.3, and 5.2.These results can be interesting on their own.
The main result of the paper is Hardy's inequality for general setting, see Theorem 2.4, and sharpness of the admissible exponent, see Propositions 2.5 and 2.7.This theorem is then applied in several settings, see Theorems 3.5, 4.9, 4.13, and 5.9, which generalize many of already known in the literature results (see [2,15,16,18,21,32,33,35]), but also answer some open questions (for instance sharpness or multi-dimensional inequality on H p for Laguerre expansions).
Organization of the paper is as follows.In Section 2 we prepare the necessary tools to prove Hardy's inequality, like Hardy, BMO, and Lipschitz spaces.Moreover, we enhance the method from [28] so it works for H p spaces with p ∈ (0, 1].Furthermore, we construct a counterexample to justify that the obtained formula for the admissible exponent is sharp.In Section 3 we discuss the standard Laguerre functions and estimate their derivatives near zero.This allows us to apply the general theorem.Section 4 is devoted to Laguerre expansions of Hermite type.Similarly as before we estimate the derivatives of the functions from the basis.However, this time it is not immediate to obtain such bounds for the corresponding kernels R r (x, y).For that purpose we need to use the integral formula for the Bessel function, see (4.9) and Proposition 4.6.We also interpret this estimate in term of the heat kernel.Moreover, we deduce Hardy's inequality for the generalize Hermite framework.Lastly, in Section 5 is analysed the Jacobi trigonometric function system.In the latter case we use the same convention as for n.For any u ∈ R we denote the largest integer not greater than u by ⌊u⌋, and the smallest integers not smaller than u by ⌈u⌉.We write for inequalities with non-negative entries which hold with a multiplicative constant.It may depend on the quantities stated beforehand, but not on the ones quantified afterwards.If X and Y X simultaneously, then we write X ≃ Y .
Acknowledgements.Research supported by the National Science Center of Poland, NCN grant no.2018/29/N/ST1/02424.The author is deeply grateful to Professor Krzysztof Stempak for his constant help during the preparation of this paper, and for his numerous comments and suggestions.

Hardy's inequality
In this section we develop a method of proving Hardy's inequality on H p spaces, 0 < p ≤ 1, associated with orthonormal expansions.This is a generalization of the idea described in [28, Section 2].However, Hardy spaces, even in the sense of Coifman-Weiss [9], are not well defined for all p if the underlying measure is only assumed to be doubling.Hence, we will focus our attention only on orthogonal expansions in L 2 (X), where X is a subset of R d equipped with Lebesgue measure.
2.1.Hardy spaces.Recall that given any Schwartz function Φ such that Φ = 0, one can define the Hardy space H p (R d ), p < 0 ≤ 1, as the space of all distributions satisfying where Φ t (x) = t −d Φ(x/t).The p-th norm of the quantity above can be taken as a "norm" in H p (R d ).We remark that • H p (R d ) is indeed a norm only for p = 1.In fact, where ⌋, and |B| denotes the Lebesgue measure of B. Every f ∈ H p (R d ) admits an atomic decomposition, namely there exist a sequence of (p, q)-atoms {a j } j∈N and a sequence of complex coefficients {λ j } j∈N such that There are several possibilities to define equivalent "norms" in H p (R d ).We choose the atomic one, which is given by where the infimum is taken over all atomic decompositions of f .Throughout this paper let X be a convex Lipschitz domain (by which we mean an open connected set with Lipschitz boundary; obviously the latter refers only to the case d ≥ 2) in R d equipped with Lebesgue measure and the Euclidean metric.There is a number of possible definitions of H p spaces on subsets of R d , see for instance the papers of Stein et al. [5,6] and of Miyachi [23].We choose the following one We remark (see [38, p. 137]) that each f ∈ H p (X) admits an atomic decomposition with all atoms supported in X, since X is a Lipschitz domain.We set f H p (X) similarly as in R d .Observe that for f and F as above we have (2.1) since for F the underlying infimum is taken over a possibly larger set.
2.2.Dual type spaces.We need to give some meaning to the paring f, ϕ n for ϕ n from a given orthonormal basis and f ∈ H p (R d ) or, more generally, for f ∈ H p (X).For this purpose we shall make use of the duality relation between H p (R d ) and BMO(R d ) and Lipschitz spaces.A locally integrable function f is in BMO(R d ) (bounded mean oscillation space) if where the supremum is taken over all balls B ⊂ R d and f B = |B| −1 B f is the mean value of f over B. Observe that the expression above vanishes for constant functions.In fact, it is usual to define BMO(R d ) as the quotient of the above space by the space of constant functions.Then BMO(R d ) with the norm • BM O(R d ) becomes a Banach space.For more details we refer to the literature, see [13,38].
The above-mentioned relation is the following: [22, p. 54]), the functional T g has a unique bounded extension to the whole H p (R d ) with the same bound.Now we will show a property of one-dimensional functions which are in Λ ν (R), and then we will justify that the tensor products of such functions belong to Λ ν (R d ).
Let ν > 1 be such that ν / ∈ N (then ⌈ν⌉ − 1 = ⌊ν⌋).We assume a contrario that there exists Moreover, g (⌊ν⌋) (x) does not change the sign on this interval since it is continuous.Now observe that The obtained contradiction proves that g (⌊ν⌋) ∈ L ∞ (R) and, to be more precise, that g (⌊ν⌋) L ∞ (R) ≤ (2k) ⌊ν⌋ + 2 g Λν (R) .Observe that if ν ∈ N, ν ≥ 2, then the proof, with minor modification, works as well.Hence, it suffices to justify that given ℓ ∈ N and g ∈ C ℓ (R) such that g, g L ∞ (R) .This is an easy exercise but for the reader's convenience we give a short proof.
Fix ℓ ∈ N and j ∈ {1, . . ., ℓ − 1}.Now assume a contrario that and this sum does not change the sign in this interval.But on the other hand, This contradiction finishes the proof of the lemma.
Let n be a multi-index such that |n| = ⌊ν⌋.Then, for h = (h 1 , . . ., h d ) ∈ R d \ {0}, we write the difference ∂ n g(x + h) − ∂ n g(x) as g (n 1 ) 1 Hence, by Lemma 2.1 we get Next, assume that ν ∈ N is such that ν ≥ 2.Then, for |n| = ν − 1, we estimate an expression of the form Observe that if n is a multi-index which has at least two non-zero components, then by Lemma 2.1 the expression above is estimated by a constant times d i=1 g i Λν (R) .Indeed, it easily follows from the mean value theorem, more precisely from the estimate Otherwise, we can assume that n = (ν −1, 0, . . ., 0).Hence, denoting x = (x 2 , . . ., x d ), ḡ(x) = d i=2 g i (x i ) and h = (h 2 , . . ., h d ), we write Again, it suffices to use the mean value theorem and Lemma 2.1 to get the required bound.
Observe that this argument is valid also for ν = 1 provided we assume that g ′ i exist and are bounded.This finishes the proof.Now, we shall define the Lipschitz (and BMO) spaces in X and prove similar duality.We say that a function g defined on X belongs to Λ ν (X), ν ≥ 0, if there exists G ∈ Λ ν (R d ) such that G X = g.Note that this type of definition differs from the one of H p (X), where we assumed that the extension vanishes outside X.In this case this is not possible because of the smoothness requirement.
Moreover, we set where the infimum is taken over all G extending g to R d .With those definitions the following lemma holds.
Consequently, the functional T g has a (unique) bounded extension to the whole , where in the last inequality we used (2.1).By taking the infimum over G we obtain the required bound.Now we drop the assumption that f We choose an extension G of g and define T g on the whole H p (X) by T g (f ) = TG (F ) with the notation as above.Hence, It suffices to take the infimum over G to get the claim.
One comment is in order here.Note that T g defined as in the proof of Lemma 2.3 does not depend on the chosen extension G. Indeed, let G 1 and G 2 be extensions of [38, p. 109]).In fact, one can choose F k to be the partial sums of an atomic decomposition of F , where the atoms are supported in X.Such decomposition exists because F ∈ H p (R d ) and supp F ⊂ X, and therefore F X ∈ H p (X) which, in the light of the remark we made before, has such decomposition.Now fix ε > 0 and choose N ∈ N so that .
Observe that . This justifies that T g (f ) does not depend on the chosen extension G.

Main theorem.
Fix p ∈ (0, 1] and let {ϕ n } n∈N d , where ϕ n ∈ Λ d( 1p −1) (X), be an orthonormal basis in L 2 (X).We define the family of operators {R r } r∈(0,1) via Note that the integral makes sense for f We shall apply these operators to the elements of H p (X).For this purpose we need to give more general meaning to f, ϕ n .Indeed, it can be defined by the means of Lemma 2.3, namely Recall that T ϕn defined as in the proof of Lemma 2.3 is unique (see the comment above).Let R r , r ∈ (0, 1), be integral operators for which the associated kernels, denoted by R r (x, y), belong to C P (X) (as functions of x, for any y ∈ X) for P = ⌊d(p −1 − 1)⌋, which means that all of their partial derivatives ∂ j x i , j = 0, . . ., P , exist and are continuous.Moreover, assume that R r (x, y) satisfy the following condition: there exist a constant γ > 0 and a finite set ∆ composed of positive numbers δ strictly greater than d(p −1 − 1) − P , such that for each k = 0, . . ., P , there holds uniformly in r ∈ (0, 1) and for |n| ≤ P + 1.Indeed, it suffices to use Taylor's theorem.
, form an orthonormal basis in L 2 (X), and the associated kernels R r (x, y) satisfy condition (C) with γ > 0. Then the inequality holds uniformly in f ∈ H p (X), where We remark that the above parameter γ is not the same as γ in [28, Theorem 2.2]; in fact if in the cited theorem µ is Lebesgue measure (and hence N = d), then both γ's are equal up to the multiplicative constant (d + 2).
Proof.Fix p and s as in the claim.Firstly, we will prove the theorem for (p, q)-atoms, q ∈ [2, ∞], and then we shall justify that it holds for all f ∈ H p (X).Let a be a (p, q)atom supported in a ball B with the centre in x ′ .The following computation does not depend on a. Similarly as in [28] and [21] in the first step we use an asymptotic estimate for the Beta function obtaining Note that by the definition of P and ∆ there is uniformly in B such that |B| ≤ 1.This finishes the proof of the theorem for the atoms.
To complete the proof let us now justify that the claim holds for any f ∈ H p (X). Fix f ∈ H p (X) and its atomic decomposition f = j∈N λ j a j .Denote f J = J j=0 λ j a j .We Since ℓ s (|n|+1) −E is complete with this metric we have shown that { f J , ϕ n } n∈N d J∈N is a Cauchy sequence.Moreover, for s ∈ [1,2] we use Minkowski's inequality and get and thus the considered sequence is a Cauchy sequence for this range of the parameter s as well.Therefore, there exists We will justify that a n = f, ϕ n .The above equality yields lim On the other hand, by Lemma 2.3 lim , and the latter limit is equal to zero.Hence, by the uniqueness of the limit we justified that a n = f, ϕ n .Finally, fix ε > 0 and , then we proceed as before using Minkowski's inequality.This finishes the proof of the theorem.
2.4.Sharpness.In this subsection we prove that the admissible exponent in Theorem 2.4 cannot be lowered, provided that we pose some additional assumptions on the basis {ϕ n } n∈N d .In fact, we focus only on the case ϕ n (x) = d i=1 ϕ n i (x i ).Therefore, we state our results in the one-dimensional situation and then make an appropriate remark on the general case d ≥ 1.
We remark that although conditions (2.8) and (2.11) may look hard to meet, they turn out to be very natural in the classical orthonormal basis, such as Laguerre, Hermite, or Jacobi function expansions.
Observe that by the equality the cancellation properties come down to (2.6) This is a system of linear equations on C 1 , . . ., C P +1 and one can solve it using Cramer's rule.A calculation shows that Indeed, inserting this into left hand side of (2.6) we obtain Observe that for each j the inner sum vanishes since k ≤ P and hence (2.6) holds.
Proposition 2.5.Let the one-dimensional version of the assumptions of Theorem 2.4 be satisfied.Moreover, we assume that (0, c) ⊂ X for some c > 0 and that there exists τ > 4γ−2pγ−p 2p such that for some , uniformly in u ∈ (0, cK −2γ ), k ≤ K and K ∈ N + , and ϕ k (u) does not change the sign in this interval.Then the admissible exponent in (2.3) cannot be lowered.
Proof.In order to prove this lemma we will construct an explicit sequence of atoms a K , such that for E defined in (2.4) and any ε > 0 (2.9) Let K ∈ N + and a K be an atom defined in (2.5) with A = K 2γ /c and some sufficiently small δ.We will show that This suffices to prove (2.9).Indeed, since τ > (4γ − 2pγ − p)/(2p), we have Let us now justify (2.10).We have Thus, the absolute value of the quantity above is bounded from below by .
Observe that taking δ sufficiently small, by the restraint for τ and (2.7), we obtain (2.10).
Remark 2.6.Notice that if p −1 is not an integer and ϕ k satisfy (2.8) with τ = 4γ−2pγ−p 2p , then (2.10) is also true.This implies for large K the estimate Hence, (2.3) does not hold.But the case Hardy's inequality may be valid, see Proposition 2.7.
Sometimes condition (2.8) holds with τ ≤ 4γ−2pγ−p 2p and hence Proposition 2.5 cannot be applied in order to prove sharpness.However, estimate (2.8) can be replaced by its analogue for the derivatives of ϕ k .We describe this situation in the following proposition.
Proposition 2.7.Let the one-dimensional version of the assumptions of Theorem 2.4 be satisfied.Moreover, we assume that (0, c) ⊂ X for some c > 0, ϕ k are (P + 1)-times differentiable, where P = ⌊p −1 − 1⌋, and that there exists τ > (4γ − 2pγ − p)/(2p) such that for some 0 < m ≤ M there holds Proof.Fix p ∈ (0, 1] and set P = ⌊p −1 − 1⌋.Let K ∈ N and a K be the same H p (X) atom as in Proposition 2.5.We show (2.9).Observe that denoting A = K 2γ /c we have for some ξ u between u and (P + 1)δ/A the following equality du.
The absolute value of the latter integral can be estimated from below by for δ sufficiently small, since we have (2.7).Hence, we obtained uniformly in k ∈ N + and k ≤ K. Thus, for any ε > 0 since τ is large enough.This finishes the proof of the lemma.
Remark 2.8.In the multi-dimensional case the construction also works if the functions in the considered orthonormal basis are products of one-dimensional functions for which properties (2.8) or (2.11) holds.Indeed, denoting A K (x) = d i=1 a K (x i ) we have by (2.10) the following lower bound uniformly in large K.
Remark 2.9.Notice that (2.9), generalized to the multi-dimensional situation, and the uniform boundedness principle (in a stronger version than usual, see for instance [34, Theorem 2.5]) imply that there exists f ∈ H p (X) such that This is consistent with what was proved in author's articles concerning Hardy's inequality on H 1 , see [28,31,30].

Laguerre standard functions
The standard Laguerre functions {L α k } k∈N of order α > −1 are defined on R + by where L α k (u) are the Laguerre polynomials (see [40]).Moreover, in the multi-dimensional case L α n (x) are defined as the tensor product of the one-dimensional functions, namely The following estimates are known for the one-dimensional standard Laguerre functions (see [24, p. 435] and [1, p. 699]) where k ′ = max(4k + 2α + 2, 2) and γ > 0 depends only on α.These estimates imply for all α ≥ 0 the bound (cf.[39, p. 94]), Moreover, using the formula (see [39, p. 95]) where More generally, for j ∈ N and α ∈ {0, 2, . . ., 2j} ∪ (2j, ∞) there holds (see [35, Lemma 1]) Now we will justify that L α n belong to the spaces Λ ν (R d + ).For that purpose we will indicate an extension Lα Following the idea used in [37, p. 94] in the case d = 1 we define where, for α i which is not an even integer, and, for α i which is an even integer, In the latter case the definition of L α i n i is naturally extended by the initial formula (3.1) to the whole real line, and ψ is a smooth function supported in [−1, ∞) such that ψ ≡ 0 on R + and ψ (j) 1, j ∈ N.For an example of such function see [37].In view of [37, Corollary 2.4] we see that given ν > 0 we have Lα i n i ∈ Λ ν (R) for α i ∈ [2ν, ∞).Secondly, if α i is an even integer, then Lα i n i ∈ Λ ν (R) for all ν > 0. And lastly, for α ∈ [0, ∞) the functions L α i n i are bounded and hence are in BMO(R).Thus, by Lemma 2.2 if p ∈ (0, 1] and α ∈ {0, 2, . . ., 2P }∪[2d(p −1 −1), ∞) d , where . In order to satisfy the additional assumption in Lemma 2.2, we have used the fact that for α ∈ {0} ∪ [2, ∞) the functions (L α k ) ′ exist and are bounded.The family of operators {R α r } associated with {L α n } n∈N d and given by R is composed of integral operators, with the kernels of the form It can be explicitly written as the product of the kernels R α i r (x i , y i ) (cf. [28,40]) where I s (u) denotes the Bessel function of the first kind and order s.It is a real, positive, and smooth function for s > −1.
In fact, we do not need this explicit formula for R α r (x, y) to prove Hardy's inequality.However, for the completeness of the presentation we gave it above.On the other hand, its analogue for Laguerre functions of Hermite type will be of paramount importance.Now we are ready to verify condition (C) for the standard Laguerre functions.
uniformly in r ∈ (0, 1) and x, x ′ ∈ R d + , where If for all i = 1, . . ., d there is α i / ∈ (2k, 2k + 2), then apply Taylor's theorem with the reminder of (k + 1)-th order, and Lemma 3.1 with j ≤ k + 1.On the other hand, if some α i ∈ (2k, 2k + 2), then proceed as before but with k-th order reminder, obtaining where for every i ∈ {1, . . ., d} the number ξ i lies between x i and x ′ i .Now for each difference above we apply Lemma 3.2 if α i ∈ (2n i , 2n i + 2), or the mean value theorem and Lemma 3.1 in the opposite situation.
Although the following lemma will be applied strictly to prove sharpness of Hardy's inequality associated with the standard Laguerre expansions, we stress that this is an interesting result and possibly it could be widely used in other problems concerning the functions L α k .Here and later on we use the convention that A ≃ −B for positive B means that A is negative and (−A) ≃ B. Lemma 3.4.Let α ≥ 0 and j, ℓ ∈ N be given.There exists a constant c > 0 such that if ℓ ≥ j, then there holds Proof.We will apply the induction over j separately in both cases.Note that the claim holds for j = 0 and any ℓ ∈ N (this is a known result, see [24, pp. 435, 453]).We assume that it is valid for some j and we will justify it for j + 1. Observe that by (3.3) we have Notice that if ℓ ≥ j + 1, then the components on the right hand side of (3.5) are of the sizes: (k + 1) α/2 u ℓ−j−1 , (k + 1) α/2 u ℓ−j , and (k + 1) α/2+1 u ℓ−j , respectively, and the first one is the dominating.
It remains to justify the case j ≥ ℓ.Let us assume that for some such j the estimate holds.Then we have similarly as above.The second and the third summand on the right hand side of (3.5) are of the sizes (and signs): (−1) j−ℓ+1 (k + 1) α/2+j−ℓ and (−1) j−ℓ+1 (k + 1) α/2+1+j−ℓ , respectively.On the other hand, the first component we decompose and get Again, the first summand can be decomposed, and the two remaining are of the same size (and sign) as before.Moreover, note that the i-th decomposition of the first resulting component brings the multiplicative constant ℓ−i+1.But this proves that the component vanishes, since j ≥ ℓ.Hence, in this case (3.5) is of the size and sign (−1) j−ℓ+1 (k + 1) α/2+1+j−ℓ .This finishes the proof of the lemma.
We now are ready to prove Hardy's inequality associated with the standard Laguerre functions.
The following result is an analogue of Lemma 3.4.
Lemma 4.3.Let α ≥ −1/2 and j, ℓ ∈ N be given.There exists small constant c > 0 such that there holds Proof.The proof is similar to the one of Lemma 3.4, therefore we will only sketch it.If j = 0, then the estimate is well known (cf.(3.6)).For j ≥ 1 we use the induction and (4.3) Note that if ℓ ≥ j + 1, then the first component on the right hand side of the above identity is of the greatest size, (k + 1) α/2+1/2 u ℓ−j−1 , and the others are strictly smaller.
On the other hand, if j ≥ ℓ, then the second summand on the right hand side of (4.4) is of the size (and sign) , and the third .
We see that the latter is the leading one.Moreover, by the simple identity 2 ⌉, i ∈ N, it can be written in the following form: Furthermore, the first component in (4.4) can be decomposed similarly as in the proof of Lemma 4.3, and it gives the same growth and size as the remaining summands.This finishes the proof of the lemma.
Lemma 4.4.Let j ∈ N.For α ≥ −1/2 there holds Proof.In order to prove (4.5) we use the induction over j to prove an auxiliary result: for every ℓ ∈ N there is For j = 0 we simply apply (4.2).Now assume that the claim holds for some j ∈ N.
Observe that by (4.3) we have for any ℓ ∈ N uniformly in k ∈ N and u ≥ 1/2.This proves the auxiliary claim.Observe that for ℓ = 0 we obtain (4.5).

4.2.
Hardy's inequality.The kernels of the operators R α r (cf.(2.2)) associated with the Laguerre functions of Hermite type, are defined by and, in the one-dimensional case, admit the explicit form (cf. [40]) Unfortunately, it is highly complicated to proceed as in [29] while estimating derivatives of R α r of order higher than 2. The cancellations between the underlying Bessel functions are not well understood yet.Therefore, we choose an approach similar to the one applied in the case of the Jacobi expansions [31].This method relies on the following formula (4.9) where Π α in the case α > −1/2 is a measure with the density given by where by E α r (u, v) we denote (4.10) Now we have the following proposition.
Before we state Hardy's inequality associated with the Laguerre functions of Hermite type we will prove some auxiliary results.The next one complements the estimate from Proposition 4.6.Essentially, it says that the mentioned bound holds also for α ∈ (j − 3/2, j − 1/2), j ∈ N + , but only away from the origin.Lemma 4.7.If j ∈ N and α ∈ (j − 1/2, j + 1/2), then Proof.It suffices to proceed as in the proof of Proposition 4.6 with some minor changes.For r ∈ (0, 1/2] use (4.5) instead of (4.6).If r ∈ (1/2, 1), then we arrive at Then we estimate like in (4.12).This finishes the proof of the lemma.
uniformly in r ∈ (0, 1) and x, x The proof is analogous to the one of Proposition 3.3, thus we will only sketch it.
Observe that if α i / ∈ (k − 1/2, k + 1/2) for all i = 1, . . ., d, then the claim, with ∆ α k = {1}, follows from Taylor's theorem and Proposition 4.6 applied for j = k + 1 .On the other hand, if α i ∈ (k − 1/2, k + 1/2) for some i then we apply Taylor's theorem, Proposition 4.6, and (4.13).Then the set ∆ α k is as in (4.14).We omit the details.Now we are ready to state Hardy's inequality associated with the system of Laguerre functions of Hermite type.Theorem 4.9.Let p ∈ (0, 1), s ∈ [p, 2], and denote P := ⌊d( 1 p − 1)⌋.For there holds , where E = d + ds 4p (2 − 3p), and the exponent is sharp.Proof.Similarly as in Theorem 3.5: the inequality follows from Theorem 2.4 and Proposition 4.8, whereas sharpness is a consequence of Propositions 2.5, 2.7 and Lemma 4.3.The case α i + 1/2 = d(p −1 − 1) is excluded due to Remark 2.6, unless it is an integer.4.3.Heat kernel estimates.In this article we estimated or will estimate the kernels R r (x, y) in various contexts.In case of the standard Laguerre functions it was very easy and for the Jacobi expansions we will use the result known in the literature.On the other hand, here the situation was more involved.In Proposition 4.6 we have obtained a result which can be interesting on its own, especially in the context of the associated heat kernel.
Recall that the heat semigroup {T α t } t≥0 is spectrally defined by Proof.The inequality is a consequence of Proposition 4.12 and Theorem 2.4.On the other hand, sharpness follows immediately from Theorem 4.9.Indeed, it is clear that if the admissible exponent for the generalized Hermite functions could be lowered, so could be the exponent corresponding to the Laguerre expansions of Hermite type.
We remark that for λ = (0, . . ., 0), that is in the case of the Hermite functions, the result agrees with the ones already known in the literature ( [33] for d ≥ 2, [21] for d = 1).
We will frequently make use of the formula 5.1.Lipschitz and BMO properties.Let us firstly give some auxiliary lemmas and justify that the Jacobi functions belong to the Lipschitz spaces Λ ν ((0, π)) for certain ν.
and the exponent is sharp.Proof.Proposition 3.3 ensures that the appropriate version of (C) holds for the standard Laguerre functions, and hence by Theorem 2.4 we obtain associated Hardy's inequality.