NEGATIVE POWERS OF LAGUERRE OPERATORS ADAM NOWAK AND KRZYSZTOF STEMPAK 0 Abstract. We study negative powers of Laguerre differential operators in Rd, d 1. 1 ≥ For these operators we prove two-weight Lp Lq estimates, with ranges of q depending 0 − 2 on p. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here n a these results are applied in certain Laguerre settings. The procedure is fairly direct for J Laguerre function expansions of Hermite type, due to some monotonicity properties of 4 the kernels involved. The case of Laguerre function expansions of convolution type is 1 less straightforward. For half-integer type indices α we transfer the desired results from ] the Hermite setting and then apply an interpolation argument based on a device we call A the convexity principle to cover the continuous range of α [ 1/2, )d. Finally, we C ∈ − ∞ investigate negative powers of the Dunkl harmonic oscillator in the context of a finite h. reflectiongroupacting onRd andisomorphicto Zd2. The two weightLp Lq estimates we − t obtaininthissettingareessentiallyconsequencesofthoseforLaguerrefunctionexpansions a m of convolution type. [ 2 v 8 1. Introduction 3 0 Consider the fractional integral operator (also referred to as the Riesz potential) 0 . 2 1 1 Iσf(x) = f(y)dy, x Rd, 9 ZRd x y d−σ ∈ 0 k − k : 0 < σ < d, defined for any function f for which the above integral is convergent x-a.e.; for v i instance, f Lp(Rd) with 1 p < d/σ is good enough. X ∈ ≤ Then, with an appropriate constant c , r σ a ( ∆)−σ/2f = c Iσf, f (Rd), σ − ∈ S where ∆ = d ∂2 is the standard Laplacian in Rd, d 1, and the negative power j=1 j ≥ ( ∆)−σ/2 is dPefined in L2(Rd) by means of the Fourier transform. − A classical result concerning Iσ is the following, see e.g. [6, 20]. Theorem 1.1 (Hardy-Littlewood-Sobolev). Let 0 < σ < d, 1 p < d and 1 = 1 σ. ≤ σ q p − d Then for p > 1 we have the strong type (p,q) estimate Iσf . f , f Lp(Rd), q p k k k k ∈ 2010 Mathematics Subject Classification: Primary 47G40; Secondary 31C15, 26A33. Key words and phrases: potential operator, fractional integral, Riesz potential, negative power, harmonic oscillator, Laguerre operator, Dunkl harmonic oscillator. 1 2 A.NOWAK ANDK. STEMPAK while for p = 1 the weak type (1,q) estimate holds, q f x Rd: Iσf(x) > λ . k k1 , λ > 0, f L1(Rd). |{ ∈ | | }| (cid:18) λ (cid:19) ∈ The Hardy-Littlewood-Sobolev theorem was extended to a two-weight setting in [21]. Theorem 1.2 (E. M. Stein and G. Weiss). Let 0 < σ < d, 1 < p q < , a < d/p′, ≤ ∞ b < d/q, a+b 0 and 1 = 1 σ−a−b. Then ≥ q p − d Iσf . f , f Lp(Rd, x ap). Lq(kxk−bq) Lp(kxkap) ∈ k k (cid:13) (cid:13) (cid:13) (cid:13) Note that the co(cid:13)nditi(cid:13)ons 1 = 1 (cid:13)σ (cid:13)or 1 = 1 σ−a−b appearing in the above theorems q p − d q p − d are in fact necessary and forced by a homogeneity type argument. Numerous analogues of the Euclidean fractional integral operator were investigated in various settings, including spaces of homogeneous type, orthogonal expansions, etc. For in- stance, in the seminal article of Muckenhoupt and E. M. Stein [15] the case of ultraspherical expansions was treated. Gasper and Trebels (and one of the authors of the present article) studied fractional integration for one dimensional Hermite and Laguerre function expan- sions [8, 9]; the Laguerre case was also considered by Kanjin and E. Sato [12]. Recently, Bongioanni and Torrea [2] obtained Lp Lq estimates for negative powers of the harmonic − oscillator. In a more general context Bongioanni, Harboure and Salinas [4] investigated weighted inequalities for negative powers of Schro¨dinger operators with weights satisfying the reverse H¨older inequality. Our present results generalize significantly those of [8, 9, 12]. Inthispaperwefocusonnegativepowersof“Laplacians”associatedtomulti-dimensional Laguerre function expansions. For these operators we prove two-weight Lp Lq estimates − in the spirit of Theorem 1.2. Such estimates are of interest, for instance in the study of higher order Riesz transforms or Sobolev spaces related to Laguerre expansions. In all the cases we discuss, spectra of self-adjoint extensions of the considered operators are discrete, separated from zero, subsets of (0, ). Hence negative powers of them are well defined in ∞ appropriate L2 spaces just by means of the spectral theorem. The relevant extensions to weighted Lp spaces of the negative powers are given by suitable integral representations. The emerging integral operators are called the potential operators (sometimes also referred to as the fractional integral operators). Also, we take an opportunity to slightly enhance the result obtained by Bongioanni and Torrea for the harmonic oscillator, by stating and proving a weighted counterpart (with power weights) to their result. In the Laguerre case we consider two different systems of Laguerre functions, ϕα and { k} ℓα . The first one leads to so-called Laguerre function expansions of Hermite type. It { k} occurs that to some extent in this setting the problem of Lp Lq estimates for the potential − operator almost reduces to the Hermite case. This is due to the fact that the heat kernels correspondingtodifferentmulti-indicesoftypeα [ 1/2, )dpossesscertainmonotonicity ∈ − ∞ property with respect to α. Thus it suffices to consider only the specific multi-index α = o ( 1/2,..., 1/2), which corresponds to Hermite function expansions. − − The second system of Laguerre functions is related to so-called Laguerre expansions of convolution type. In this case our approach is quite different and in fact a more involved analysis is necessary. We first deal with half-integer multi-indices α and transfer the desired NEGATIVE POWERS OF LAGUERRE OPERATORS 3 resultsfromtheHermitesetting. Then, tocover thecontinuousrangeofα [ 1/2, )d, we ∈ − ∞ derive certain interpolation argument, which we call the convexity principle. This method is of independent interest and can be applied in other situations. The organization of the paper is the following. In Section 2 we gather known facts concerningthepotentialkernelandthepotentialoperatorrelatedtotheharmonicoscillator. Then we state and prove a two-weight Lp Lq estimate for the Hermite potential operator − in the spirit of Theorem 1.2. In Section 3 we discuss potential operators associated to Laguerre function expansions of Hermite type. Section 4 is devoted to Laguerre function expansions of convolution type. Section 5 establishes the convexity principle which allows to give proofs of the main results of Section 4. In Section 6 we take an opportunity to study negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on Rd and isomorphic to Zd. The results of this section contain as special cases those 2 of Sections 2 and 4, and are strongly connected with the estimates of Section 4. Finally, in Section 7 we gather various additional observations and remarks. Comments explaining how our present results generalize those of [8, 9, 12] are located throughout the paper. We use a standard notation with essentially all symbols referring to either Rd or Rd = + (0, )d, d 1, depending on the context. Thus ∆ denotes either the Laplacian in Rd, ∞ ≥ or its restriction to Rd, and stands for the Euclidean norm. By f,g we denote + k · k h i f(x)g(x)dx (or the same, but with the integration restricted to Rd) whenever the Rd + Rintegral makes sense. For a nonnegative weight function w on either Rd or Rd, by Lp(Rd,w) + or Lp(Rd,w), 1 p , or simply by Lp(w), we denote the usual Lebesgue spaces related + ≤ ≤ ∞ to the measure dw(x) = w(x)dx (we will often abuse slightly the notation and use the same symbol w to denote the measure induced by a density w). If w 1 we simply write ≡ Lp(Rd) or Lp(Rd). Beginning from Section 4, Lebesgue measure dx on Rd is replaced by + + µ (dx), where α ( 1, )d is a multi-index, hence some symbols previously related to dx α ∈ − ∞ are then related to µ (dx). Similar situation occurs in Section 6 where dx on Rd is replaced α by w (dx). α If k Nd, N = 0,1,... , then k = k + ... + k is the length of k. The notation 1 d ∈ { } | | X . Y will be used to indicate that X CY with a positive constant C independent of ≤ significant quantities. We shall write X Y when simultaneously X . Y and Y . X. ≃ Given 1 p , p′ denotes its adjoint, 1/p+1/p′ = 1. ≤ ≤ ∞ 2. Negative powers of the harmonic oscillator The multi-dimensional Hermite functions h (x), k Nd, are given by tensor products k ∈ d h (x) = h (x ), x = (x ,...,x ) Rd, k ki i 1 d ∈ Yi=1 where hki(xi) = (π1/22kiki!)−1/2Hki(xi)e−x2i/2, and Hn denote the Hermite polynomials of degree n N, cf. [14, p.60]. The system h : k Nd is a complete orthonormal system k ∈ { ∈ } in L2(Rd). It consists of eigenfunctions of the d-dimensional harmonic oscillator = ∆+ x 2, H − k k 4 A.NOWAK ANDK. STEMPAK h = λ h , λ = 2 k + d. We shall denote by the same symbol the natural self-adjoint k k k k H | | extension of , whose spectral resolution is given by the h and λ , see [22]. The integral k k H kernel of the Hermite semigroup e−tH : t > 0 is known explicitly (see [23] for this { } symmetric form of the kernel), ∞ G (x,y) = e−(2n+d)t h (x)h (y) t k k Xn=0 |Xk|=n 1 = 2πsinh(2t) −d/2exp tanh(t) x+y 2 +coth(t) x y 2 . (cid:18)− 4 k k k − k (cid:19) h i (cid:0) (cid:1) Given σ > 0, consider the negative power −σ. In view of the spectral theorem, it is H expressed on L2(Rd) by the spectral series, (2.1) −σf = (2 k +d)−σ f,h h . k k H | | h i kX∈Nd Notice that −σ is a contraction on L2(Rd) for any σ > 0. H Motivated by the formal identity 1 ∞ −σ = e−tHtσ−1dt, H Γ(σ) Z 0 it is natural to introduce the potential kernel 1 ∞ (2.2) σ(x,y) = G (x,y)tσ−1dt. t K Γ(σ) Z 0 It follows from the decay of G (x,y), as t and t 0+, that for σ > d/2 the integral t → ∞ → in (2.2) is convergent for every x,y Rd, while for 0 < σ d/2 the integral converges ∈ ≤ provided that x = y. 6 Define the auxiliary convolution kernel Kσ(x), x Rd 0 , by ∈ \{ } Kσ(x) = exp( x 2/8), x 1, −k k k k ≥ and, for x < 1, k k 1, σ > d/2, Kσ(x) = log(4/ x ), σ = d/2,  k k x 2σ−d, σ < d/2. k k  It is immediately seen that Kσ L1(Rd) for all σ > 0. Moreover, if σ > d/2, then ∈ Kσ Lr(Rd) for each 1 r , if σ = d/2, then Kσ Lr(Rd) for 1 r < , while for ∈ ≤ ≤ ∞ ∈ ≤ ∞ σ < d/2 we have Kσ Lr(Rd) if and only if r < d/(d 2σ). ∈ − It was proved in [2] that σ(x,y) is controlled by Kσ(x y). To make this section K − self-contained we include below a short proof of this result. An estimate of the integral 1 E (T) = ζ−aexp( Tζ−1)dζ, T > 0, a Z − 0 is needed. The statement below is a refinement of [22, Lemma 1.1], see also [16, Lemma 2.3]. NEGATIVE POWERS OF LAGUERRE OPERATORS 5 Lemma 2.1. Let a R be fixed. Then ∈ (2.3) E (T) . exp( T/2), T 1, a − ≥ and for 0 < T < 1 1, a < 1, E (T) log(2/T), a = 1, a ≃   T−a+1, a > 1.   Proof. A change of the variable of the integration yields ∞ (2.4) E (T) = T−a+1 ya−2exp( y)dy. a Z − T Now the estimate for T 1 follows since ≥ ∞ T−a+1 ya−2exp( y)dy . T−a+1exp( 3T/4) . exp( T/2). Z − − − T Notice that (2.3) can be improved; in fact we have E (T) . exp( T/(1+ε)) for any fixed a − ε > 0. The estimates for 0 < T < 1 are verified by splitting the integration in (2.4) onto the intervals (T,1) and (1, ). Then in the first resulting integral the exponential factor can ∞ be neglected, and the second integral is just a positive constant. This easily implies the desired bounds from above and below. (cid:3) Proposition 2.2 ([2, Proposition 2]). For each σ > 0, 0 < σ(x,y) . Kσ(x y). K − Proof. The lower estimate is a consequence of the strict positivity of the kernel G (x,y). t To show the upper estimate we write 1 ∞ Γ(σ) σ(x,y) = G (x,y)tσ−1dt+ G (x,y)tσ−1dt σ(x,y)+ σ(x,y). K Z t Z t ≡ J0 J∞ 0 1 Then ∞ 1 x y 2 σ(x,y) . e−dtexp x y 2 tσ−1dt . exp k − k J∞ Z (cid:18)− 4k − k (cid:19) (cid:18)− 4 (cid:19) 1 and 1 1 x y 2 σ(x,y) . exp k − k tσ−d/2−1dt. J0 Z (cid:18)− 4 t (cid:19) 0 To treat the last integral we use Lemma 2.1 and then combine the obtained bounds of σ(x,y) and σ(x,y). The required estimate of σ(x,y) follows. (cid:3) J0 J∞ K Consider the potential operator σ, I σf(x) = σ(x,y)f(y)dy, I Z K Rd defined on the natural domain Dom σ consisting of those functions f for which the above I integral is convergent x-a.e. (heuristically, σf = 1 ∞e−tHf tσ−1dt). By Proposition I Γ(σ) 0 2.2 and the fact that Kσ L1(Rd) we see that Lp(Rd) RDom σ, 1 p . ∈ ⊂ I ≤ ≤ ∞ 6 A.NOWAK ANDK. STEMPAK The following result has recently been proved by Bongioanni andTorrea. Here we include in addition a discussion of the case σ d/2. ≥ Theorem 2.3 ([2, Theorem 8]). Let σ > 0 and 1 p , 1 q . If σ d/2, then ≤ ≤ ∞ ≤ ≤ ∞ ≥ (2.5) σf . f , f Lp(Rd), q p kI k k k ∈ excluding the cases when σ = d/2 and either p = , q = 1 or p = 1, q = . If 0 < σ < d/2, ∞ ∞ then (2.5) holds if 1 2σ 1 < 1 + 2σ, with exclusion of the cases: p = 1 and q = d p − d ≤ q p d d−2σ (in which σ satisfies the weak type (1,q) estimate), and p = d and q = . Moreover, in I 2σ ∞ each of the cases of strong type (p,q), p < , for any k Nd we have ∞ ∈ (2.6) σf,h = λ−σ f,h , f Lp(Rd). hI ki k h ki ∈ Proof. Consider first the case σ d/2. If 1 p q , since 0 < σ(x,y) . Kσ(x y), ≥ ≤ ≤ ≤ ∞ K − theproofof (2.5) reduces to checking asimilar estimate with σ replaced by theconvolution I operator Tσ: f Kσ f. Recall that classical Young’s inequality has the form 7→ ∗ 1 1 1 g f g f , + = 1+ , 1 p,q,r q r p k ∗ k ≤ k k k k p r q ≤ ≤ ∞ (in particular it follows that if g Lr and f Lp, then g f(x) is well defined x-a.e.). ∈ ∈ ∗ Taking g = Kσ above shows that Tσ: Lp(Rd) Lq(Rd) boundedly provided σ > d/2 and → 1 p q ; for σ = d/2 the case p = 1, q = , is excluded. If q < p, then we argue ≤ ≤ ≤ ∞ ∞ as in the proof of [2, Theorem 8,(iii)] to get σf . f for 1 q < with exclusion q ∞ kI k k k ≤ ∞ of q = 1 when σ = d/2. Similarly, we proceed as in the proof of [2, Theorem 8,(iv)] to get σf . f for p < . Then (2.5) follows for 1 < q < p < by interpolation. 1 p kI k k k ∞ ∞ If 0 < σ < d/2 and 1 > 1 2σ, then using Young’s inequality is limited to 1 1. In the q p − d q ≤ p endpoint case when 1 = 1 2σ the desired conclusion follows from Theorem 1.1. This is q p − d because (see [22, (2.9)]) G (x,y) W (x y), t t ≤ − where W denotes the Gauss-Weierstrass kernel in Rd, which implies σf . I2σf for any t I nonnegative f. The case 1 < 1 < 1 + 2σ is more delicate and requires further arguments p q p d based on the estimate x 2σ σ(x,y)dy C, x Rd, k k Z K ≤ ∈ Rd and an interpolation argument; we refer to [2] for details. Toverify (2.6)observethatforeachfixedk Nd themappingf σf,h isabounded k ∈ 7→ hI i linear functional on Lp(Rd). This is because σf,h σf h . h f . k q k q′ k q′ p |hI i| ≤ kI k k k k k k k Moreover, intheproofof Corollary2.4below it ischecked thatthis functional agrees, onthe denseinLp(Rd)linearspanofHermitefunctions, withthelinearfunctionalf λ−σ f,h , 7→ k h ki which is also bounded on Lp(Rd). Hence both functionals coincide and (2.6) is justified. (cid:3) It is worth mentioning that in the case 0 < σ < d/2 of the above theorem, in some occurrences the constraint between p and q gives optimal ranges of p and q for (2.5) to hold. This happens when p = 1 or p = (then 1 q < d or d < q are ∞ ≤ d−2σ 2σ ≤ ∞ NEGATIVE POWERS OF LAGUERRE OPERATORS 7 optimal, respectively), and q = 1 or q = (then 1 p < d or d < p are sharp, ∞ ≤ d−2σ 2σ ≤ ∞ respectively). The corresponding proofs can be found in [2]. Corollary 2.4. Let σ > 0 and (p,q), 1 p < , 1 q , be such a pair that (2.5) ≤ ∞ ≤ ≤ ∞ holds. Then −σ extends to a bounded operator from Lp(Rd) to Lq(Rd). Moreover, denoting H this extension by −σ, in each of the cases, for any k Nd we have Hpq ∈ (2.7) −σf,h = λ−σ f,h , f Lp(Rd). hHpq ki k h ki ∈ Proof. In view of Theorem 2.3 it suffices to show that −σ = σ as operators on L2(Rd). H I This follows by observing that both operators, being bounded on L2(Rd), coincide on the dense in L2(Rd) linear span of Hermite functions. Indeed, to check that −σh = σh , k k H I k Nd, we write ∈ ∞ Γ(σ) σ(x,y)h (y)dy = G (x,y)tσ−1dth (y)dy k t k ZRd K ZRd Z0 ∞ = tσ−1e−tHh (x)dt k Z 0 ∞ = tσ−1e−tλk dth (x) k Z 0 = Γ(σ) −σh (x). k H Application of Fubini’s theorem in the second identity above was possible since, for any fixed x Rd, ∈ ∞ G (x,y)tσ−1 h (y) dtdy = σ(x,y) h (y) dy < ; t k k ZRd Z0 | | ZRd K | | ∞ this is because σ(x, ) L1(Rd) for any fixed x Rd, and h L∞(Rd). k K · ∈ ∈ ∈ Considering (2.7), given 1 p < , the subspace L2(Rd) Lp(Rd) is dense in Lp(Rd), ≤ ∞ ∩ hence the extension −σ coincides with σ as a bounded operator from Lp(Rd) to Lq(Rd). Hpq I Thus (2.7) follows from (2.6). (cid:3) It is worth to point out that for 1 < q < the assertion of Corollary 2.4 remains ∞ valid if in (2.1), the definition of −σ, the multi-sequence (2 k + d)−σ is replaced by H { | | } another multi-sequence of similar smoothness, for instance by ( k + 1)−σ (it would be { | | } reasonabletorefertotheresultingoperatorastothefractionalintegraloperatorforHermite function expansions; then accordingly λ in (2.7) must be replaced by ( k + 1)). Indeed, k | | this is a simple consequence of a multiplier theorem for multi-dimensional Hermite function expansions, see [25, Theorem 4.2.1] or [7, Theorems 7.10-11], since the multiplier multi- sequence (2|k|+d)σ defines a bounded operator on Lq(Rd) for each 1 < q < . { |k|+1 } ∞ Theorem2.3extends theresult ofGasperandTrebels [9, Theorem3]inseveral directions. First of all, the result is multi-dimensional. Secondly, the restriction σ < 1/2 (in the case d = 1 discussed in [9]) is released. Finally, the constraint 1 = 1 2σ (still in the case d = 1) q p− occurs to be unnecessary, and in addition the case p = 1 is admitted. We take an opportunity to generalize Theorem 2.3, and below we give a two-weight extension of Theorem 2.3 in the spirit of the result by Stein and Weiss stated in Theorem 1.2. It is clear that the range of q that depends on p in Theorem 2.5 below is not optimal. 8 A.NOWAK ANDK. STEMPAK Theorem 2.5. Let σ > 0, 1 < p q < , a < d/p′, b < d/q, a+b 0. ≤ ∞ ≥ (i) If σ d/2, then σ maps boundedly Lp(Rd, x ap) into Lq(Rd, x −bq). ≥ I k k k k (ii) If σ < d/2, then the same boundedness holds under the additional condition 1 1 2σ a b (2.8) − − . q ≥ p − d Moreover, under the assumptions ensuring boundedness of σ from Lp(Rd, x ap) into I k k Lq(Rd, x −bq), k k (2.9) σf,h = λ−σ f,h , f Lp(Rd, x ap). hI ki k h ki ∈ k k Note that implicitly Theorem 2.5 asserts the inclusion Lp(Rd, x ap) Dom σ in all k k ⊂ I the cases when weighted Lp Lq boundedness holds. The proof of Theorem 2.5 requires − suitable weighted inequalities for convolutions. We shall use those obtained by Kerman [13], which we formulate below for an easy reference. Lemma 2.6 ([13, Theorem 3.1]). Assume that the parameters p,q,r,a,b,η satisfy 1 1 1 (2.10) 1 < p,q,r < , + , ∞ q ≤ p r 1 1 a+b 1 η (2.11) 1 = + , q − p −(cid:18) d − (cid:19) r d d d d (2.12) a < , b < , η < , p′ q r′ (2.13) a+b 0, a+η 0, b+η 0. ≥ ≥ ≥ If g Lr( x ηr) and f Lp( x ap), then g f(x) is well defined for a.e. x Rd and ∈ k k ∈ k k ∗ ∈ g f C g f Lq(kxk−bq) Lr(kxkηr) Lp(kxkap) k ∗ k ≤ k k k k with a constant C independent of g and f. Proof of Theorem 2.5. We first deal with case (i), that is when σ d/2. As in the proof ≥ of Theorem 2.3, it is enough to prove the statement with σ replaced by the convolution I operator Tσ: f Kσ f. To get the desired boundedness of Tσ we will apply Lemma 7→ ∗ 2.6 with a suitable choice of r and η, so that, in particular, assumptions (2.10)-(2.13) are satisfied. It is easy to check that the kernel Kσ is in Lr( x ηr) if and only if the right-hand side k k in (2.11) is positive. Notice that (2.10) is satisfied with any 1 < r < , since p q. Also, ∞ ≤ the first two inequalities of (2.12) hold by the assumptions, and together they imply that the left-hand side in (2.11) 1 1 a+b ξ := 1 > 0. q − p −(cid:18) d − (cid:19) On the other hand, since by assumption a+b 0, the first inequality in (2.13) holds and it ≥ follows that the quantity ξ is in the interval (0,1) except for the singular case when p = q and a+b = 0, which will be treated in a moment separately. Finally, the third inequality NEGATIVE POWERS OF LAGUERRE OPERATORS 9 in (2.12) is equivalent to saying that the right-hand side in (2.11) is less than 1. To make use of Lemma 2.6 it remains to show that any admissible value of ξ can be attained by the right-hand side in (2.11), with 1 < r < and η such that a + η 0 and b + η 0. If ∞ ≥ ≥ a,b > 0, then we simply take η = 0 and let r = 1/ξ. When a 0 we take η = a and we ≤ − have 1 a 1 1 b 1 1 1 1 1 > = ξ + = +1 > +1 = > 0, r d q − p − d q − p − q p′ so the appropriate choice of r is again possible. Finally, if b 0, then we take η = b and ≤ − can choose suitable r since now 1 b 1 1 a 1 1 1 1 1 > = ξ + = +1 > +1 = > 0. r d q − p − d q − p − p′ q We see that Lemma 2.6 does the job except for the singular case distinguished above. To cover the case when p = q and a + b = 0 we observe that Tσ is controlled by the (centered) Hardy-Littlewood maximal operator M. This is because the convolution kernel is integrable, radial, and essentially radially decreasing, see [6, Proposition 2.7]. But for 1 < p < , M is bounded on Lp(w) with any weight w in Muckenhoupt’s class A . The p ∞ conclusion follows by the easy to verify fact that a power weight w(x) = x ap belongs to k k A if and only if d/p < a < d/p′. This completes proving case (i). p − − We now treat case (ii), when 0 < σ < d/2. Since σ is dominated by a constant times I I2σ, see the proof of Theorem 2.3, the case of equality in (2.8) is covered by Theorem 1.2. Therefore we may assume that 2σ ξ > 1 . − d But the kernel Kσ belongs to Lr( x ηr) if and only if the right-hand side in (2.11) is greater k k than 1 2σ/d. In this position we repeat the reasoning of case (i). − For the proof of (2.9) we copy the argument leading to the proof of (2.6). One only has to know that h < , but this holds in view of the inequality b > d/q′ k kkLq′(kxkbq′) ∞ − following from the assumptions imposed on d,p,q,a,b. (cid:3) 3. Laguerre function expansions of Hermite type Let k = (k ,...,k ) Nd and α = (α ,...,α ) ( 1, )d be multi-indices. The 1 d 1 d ∈ ∈ − ∞ Laguerre function ϕα on Rd is the tensor product k + ϕα(x) = ϕα1(x ) ... ϕαd(x ), x = (x ,...,x ) Rd, k k1 1 · · kd d 1 d ∈ + where ϕαi are the one-dimensional Laguerre functions ki 1/2 2Γ(k +1) ϕαkii(xi) = (cid:18)Γ(k +iα +1)(cid:19) Lαkii(x2i)xαii+1/2e−x2i/2, xi > 0, i = 1,...,d; i i given α > 1 and k N, Lαi denotes the Laguerre polynomial of degree k and order α , i − i ∈ ki i i see [14, p.76]. Each ϕα is an eigenfunction of the differential operator k d 1 1 LH = ∆+ x 2 + α2 , α − k k x2 (cid:18) i − 4(cid:19) Xi=1 i 10 A.NOWAK ANDK. STEMPAK the corresponding eigenvalue being λα = 4 k + 2 α + 2d, that is LHϕα = λαϕα; here by k | | | | α k k k α we mean α = α +...+α (thus α may be negative). The operator LH is symmetric | | | | 1 d | | α and positive in L2(Rd), and the system ϕα : k Nd is an orthonormal basis in L2(Rd). + { k ∈ } + As defined in [16, p.402], LH has a self-adjoint extension H whose spectral decompo- α Lα sition is given by the ϕαk and λαk. The heat-diffusion semigroup {e−tLHα}t>0 generated by H, Lα ∞ e−tLHαf = e−t(4n+2|α|+2d) f,ϕα ϕα, f L2(Rd), h ki k ∈ + Xn=0 |Xk|=n is a strongly continuous semigroup of contractions on L2(Rd). We have the integral repre- + sentation e−tLHαf(x) = Gα,H(x,y)f(y)dy, x Rd, Z t ∈ + Rd + where ∞ Gα,H(x,y) = e−t(4n+2|α|+2d) ϕα(x)ϕα(y), x,y Rd. t k k ∈ + Xn=0 |Xk|=n It is known, cf. [14, (4.17.6)], that d 1 x y Gα,H(x,y) = (sinh2t)−dexp coth(2t) x 2 + y 2 √x y I i i . t −2 k k k k i i αi sinh2t (cid:16) (cid:0) (cid:1)(cid:17)Yi=1 (cid:16) (cid:17) Here I denotes the modified Bessel function of the first kind and order ν; considered on ν the positive half-line, it is real, positive and smooth for any ν > 1. − It was observed in [16] (see the proof of Proposition 2.1 there) that given α [ 1/2, )d, ∈ − ∞ there exists a constant C such that α (3.1) Gα,H(x,y) C Gαo,H(x,y), t > 0, x,y Rd, t ≤ α t ∈ + with α = ( 1/2,..., 1/2). This was based on the asymptotics, cf. [14, (5.16.4),(5.16.5)], o − − (3.2) I (z) zν, z 0+; I (z) z−1/2ez, z ν ν ≃ → ≃ → ∞ (more information on C can be obtained from properties of the function ν I (x), α ν 7→ x R , see [16]). Moreover, we have (cf. [16, (A.2)]) + ∈ (3.3) Gαo,H(x,y) = G (εx,y), x,y Rd, t t ∈ + X ε∈E where = (ε ,...,ε ) : ε = 1 and εx = (ε x ,...,ε x ). 1 d i 1 1 d d E { ± } Given σ > 0, consider the operator ( H)−σ defined on L2(Rd) by the spectral series Lα + (3.4) ( H)−σf = (λα)−σ f,ϕα ϕα. Lα k h ki k kX∈Nd Observe that ( H)−σ is a contraction on L2(Rd) if α [ 1/2, )d. Lα + ∈ − ∞ We next define the potential kernel 1 ∞ (3.5) α,σ(x,y) = Gα,H(x,y)tσ−1dt, x,y Rd, KH Γ(σ) Z t ∈ + 0