Heat-diffusion maximal operators for Laguerre semigroups with negative parameters

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1 Journal of Functional Analsis Heat-diffusion maximal operators for Laguerre semigroups with negative parameters R. Macías a,,c.segovia b,, J.L. Torrea c, a IMAL-FIQ, CONICET - Universidad Nacional del Litoral, Güemes 345, 3 Santa Fe, Argentina b Intituto Argentino de Matemática IAM - CONICET, Saavedra 5, 83 Buenos Aires, Argentina c Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid 849 Madrid, Spain Received October 4; accepted 9 Februar 5 Communicated b G. Pisier Available online 3 March 5 Dedicated to the memor of Miguel de Guzmán Abstract We prove L p boundedness for the maximal operator of the heat semigroup associated to the Laguerre functions, {L α k } k, when the parameter α is greater than. Namel, the maximal operator is of strong tpe p, p if p> and +α <p< α, when < α <. If α there is strong tpe for <p. The behavior at the end points is studied in detail. 5 Elsevier Inc. All rights reserved. MSC: 4A45; 4B5; 4B; 4B5; 4C Kewords: Heat and Poisson semigroups; Laguerre functions Partiall supported b Dirección General de Investigación, Ministerio de Ciencia Tecnología, BFM-43-C- and Proecto IALE UAM-Banco Santander Central-Hispano. Corresponding author. addresses: rmacias@ceride.gov.ar R. Macías, segovia@iamba.edu.ar C. Segovia, joseluis.torrea@uam.es J.L. Torrea. Partiall supported b CONICET and Universidad Nacional del Litoral, Argentina. Partiall supported b CONICET, Argentina. -36/$ - see front matter 5 Elsevier Inc. All rights reserved. doi:.6/j.jfa.5..5

2 . Introduction R. Macías et al. / Journal of Functional Analsis The Laguerre polnomials L α k are given b e α L α k = k! d d k e k+α, where is positive. We assume that α >. The Laguerre polnomials {L α k } k= form a orthogonal sstem with respect to the measure e α d. More precisel, L α k Lα j e α d = thus, the Laguerre functions L α k defined b Γk + α + δ kj, Γk + L α Γk + / k = e / α/ L α k Γk + α +, for a given α, form an orthonormal set with respect to the Lebesgue measure. Standard references for Laguerre functions and polnomials are [,8,9]. We define the heat diffusion kernel W α t,,z for α >, t>, > and z> as W α t,,z= n= e tn+α+/ L α n Lα n z, and the heat diffusion integral W α ft, as W α ft,= W α t,, zf z dz. The heat diffusion integral W α ft, satisfies the semigroup propert W α ft + s, = W α t,,zw α fs,zdz. The maximal operator W α, ft associated to the heat diffusion integral W α ft, is given b W α, f= sup t> W α ft,. We define the fractional maximal function M θ fx for θ < as M θ fx= sup h> h θ fx d. h

3 3 R. Macías et al. / Journal of Functional Analsis If θ =, M fx is the Hard Littlewood maximal function. It is well known that if δ is a weight with < δ <p, then M f is of strong tpe p, p for p> and of weak tpe, if p = with the measure δ d. As references see [4,6]. The purpose of this paper is to stud the action of the maximal operator W α, f just defined on the spaces L p,, d. Forα the positive results we give here were obtained b Stempak [7] using techniques similar to those of Muckenhoupt [3]. The situation for < α < is rather different since, for instance, the functions L α k do not belong to ever Lp,, d. This indicates that the general theor of semigroup, see [5], does not appl.. Statement of the results Let N α denote the interval / + α, / α if < α < N α = and, ] if α. With this notation, we have the following result: Theorem Strong tpe. Let < α <. The maximal operator W α, f satisfies W α, f p dc α,p f p d, with a constant C α,p depending onl on α, provided that p N α. That is to sa a If < α, then / + α <p</ α, and b If α, then <p. The following theorem gives the behaviour of W α, f at the end points of N α. Theorem End points. Let < α. At the end points of N α we have: a If < α <, the upper end point of N α is equal to / α. The operator W α, f is of weak tpe and not of strong tpe / α, / α. b If α, the upper end point of N α is equal to. The operator W α, f is of strong tpe,. c If < α <, the lower end point of N α is equal to / + α. The operator W α, f is of restricted weak tpe and not of weak tpe / + α, / + α.

4 R. Macías et al. / Journal of Functional Analsis d If α, the lower end point of N α is equal to. The operator W α, f is of weak tpe and not of strong tpe,. 3. Lemmas Throughout this paper we shall assume that fx is a non-negative function. The constants will not have the same value in each occurrence. Lemma. Given β, β <, there exists a constant C β such that for ever > β/ M β f zz β/ { C β β/ fzz β/ dz + sup h/ β/ h β fzz β/ dz } + M f 3 and sup h/ β/ h β fzz β/ dz C β β/ Mfzz β/. 4 Proof. If h, wehave h β h h β β fzz β/ dz fzz β/ dz + h β fzz β/ dz + sup h/ h β On the other hand, if h, then h h and we get h β h fzz β/ dz h β/ h β h fzz β/ dz h fzdz fzz β/ dz. β/ h β/ M f β/ β/ M f.

5 34 R. Macías et al. / Journal of Functional Analsis Thus, β/ h β β/ h + sup h fzz β/ dz fzz β/ dz β/ h β fzz β/ dz + β/ M f which proves 3. Now, let h and choose k as the unique integer satisfing k <+h k +, then h β fzz β/ dz h β k k+ k= k fz zβ/ z β dz β k k+ h β βk fzz β/ d k= k β h β k k= βk k+ k+ fzz β/ d β c β βk h β M f zz β/ + h β β c β h β M f zz β/ + h β c β M f zz β/ c β M f zz β/, h thus, we get sup h/ β/ h β fzz β/ dz c β β/ M f zz β/, ending the proof of the lemma.

6 R. Macías et al. / Journal of Functional Analsis Lemma. Let < α. We have the following estimates for the heat diffusion integral W α ft,: a If < α, we denote β = α. Then W α ft,c α {e t/4 M f+ e t β/4 β/ M β z β/ f z}. 5 b If α, then, W α ft,c α e t/4 M f. 6 Let us consider the generating function for the Laguerre polnomials Γn + Γn + α + Lα n Lα n zrn = z+ e r r rz α/ I α rz/, r r where r <, and I α x = e iαπ/ J α ix, the modified Bessel function, see [,, p.89]; [8,9] Then, let Q α,z,r be the function Q α,z,r= Γn + Γn + α + e / α/ L α n e z/ z α/ L α n zrn+α+/ = Lα n Lα n zrn+α+/ = r/ r e z+/ e rz+/ r I α rz/ r Taking into account, this shows that Q α,z,e t = W α t,,z. If e t = s, +s then <s holds if and onl if <t. Thus, if we define R α,z,s= Q α,z, s, + s we get that R α,z, e t/ + e t/ = W α t,,z. Moreover,. R α,z,s= s e 4 s+ s / z / e s+ s z / I α s z /.

7 36 R. Macías et al. / Journal of Functional Analsis Thus, W α ft,= W α t,, zf z dz = R α, z, sf z dz for s = e t/ / + e t/. We shall need in the sequel the following estimations for I α x: Let α >, there exist two constants c α and C α such that If x, then c α x α I α xc α x α. If x, then c α x / e x I α xc α x / e x. 7 See [,,{ p.5] and [, 5, p.86]. } Let D s = x : s x. Then, the function χ Ds z is equal to if and onl if z. B 7, χds zr α,z,s is bounded b constant times s Since s e 4 s+ s z / / s+ s s + s + z / χ Ds z s z / e s s = s, we get that 8 is equal to s e 4 s+ s z / / sz / χ Ds z s /. 8 z /4 / z /4 s e 4s z/ / χ Ds z s /. 9 z /4 Analogousl, b 7, χ D c s zr α,z,s is bounded b constant times We denote s e 4s z/ / s α χ D c s z z /. H α, s, = χ Ds zr α, z, sf z dz

8 R. Macías et al. / Journal of Functional Analsis and H α, s, = χ D c s zr α, z, sf z dz. Let >, z> and s>. For ever integer k we define B k ={z : k s / < z / / k+ s / }, and let k be an integer to be fixed later, then k s H α, s, C α + s +C α k + e k /4 e k /4 B k B k χ Ds z χ Ds z s fzdz / z /4 s fzdz / z /4 = H α, s, + H α, s,. For k and B k having the same meaning as before, b, we have that H α, s, is bounded b a constant times k + s + k + e k /4 s s α χ D c s z z / fzdz B k e k /4 s α χ D c s z z / fzdz B k = H α, s, + H α, s,. Let us fixed k as the unique integer satisfing k + s / < / k +3 s /. Since z / / k+ s /,ifk k and z B k then, we get /4 / k+ s / z / + k+ s /. 3 If k k and z B k, since z / / k+ s /,weget z. k s and. k s. 4 Proof of Lemma. We are going to estimate H α, s,, H α, s,, H α, s,, and H α, s, for α >.

9 38 R. Macías et al. / Journal of Functional Analsis Estimate of H α, s,. B and 3, H α, s, is less than or equal to a constant times the sum for k k of the terms s / / e k /4 / + k+ s / fzdz. 5 / k+ s / Since / + k+ s / / k+ s / = 4 / k+ s /, we have that 5 is bounded b a constant times s / e k /4 k / + k+ s / 4 / k+ s / fzdz. 6 / k+ s / Then, b 3 and 6, we get H α, s, C α s / M f. 7 holds for α >. Estimate of H α, s,. B, we have that H α, s, is bounded b a constant times the sum for k>k of the terms s e k /4 B k χ Ds z s fzdz /. 8 z /4 The condition χ Ds z = is equivalent to z Hence, 8 is bounded b s k s e k /4 χ Ds z s fzdz s / z /4 C s e k /4 k k s k fzdz s C s e k /4 k M f. s, and b 4, k s. k s e k /4 fzdz

10 R. Macías et al. / Journal of Functional Analsis Then, we get that H α, s, is bounded b a constant times b s M f, 9 for α >. Estimate of H α, s,, for the case < α <. Let β = α. B3 we have that H α, s, is bounded b a constant times the sum for k k of the terms s e k /4 β s χ D c s z z / fzdz B k β s e k /4 β/ 4 / k+ s / 4 / k+ s / β / + k+ s / χ D c / k+ s / s zz β/ fdz. If s /4 / >, then for z/4 it turns out that s z / >, thus χ D c s z = and the integral above is equal to zero. Therefore, we can assume that s /4 /, which implies that s. Then, b, expression, is smaller than or equal to a constant times s β e k /4 k β β/ / + k+ s / 4 / k+ s / β z β/ fzdz / k+ s / s β e k /4 k β β/ M β z β/ f z. Hence, for < α <, H α, s, is bounded b a constant times s β β/ M β z β/ f z. Estimate of H α, s, for the case α. B 3, we have that H α, s, bounded b a constant times the sum for k k of the terms s e k /4 s α χ D c s z z / fzdz B k

11 3 R. Macías et al. / Journal of Functional Analsis s +α / C e k /4 α + k+ s / χ D c / k+ s / s zf z. 3 Then, b, we get that 3 is bounded b a constant times s / e k /4 k / + k+ s / 4 / k+ s / fzdz / k+ s / C s / e k /4 k M f. Thus, summing up for k k, we get that H α, s, C α s / M f 4 holds for the case α. Estimate of H α, s,, for the case < α. Let β = α. B3 we have that H α, s, is bounded b a constant times the sum for k>k of the terms s e k /4 χ D c s z B k s z / β fzdz s β e k /4 β/ k β s k s k χ s D c s zz β/ fdz. B 4 the former expression is smaller than or equal to a constant times s β e k /4 k β β/ M β z β/ f z. Hence, for < α, H α, s, is bounded b a constant times s β β/ M β z β/ f z. 5 Estimate of H α, s, for α. B and 4, H α, s, is bounded b a constant times the sum for k>k of the terms s e k /4 s α χ D c s z z / fzdz B k s +α k e k /4 α/ s z α/ fzdz s +α k e k /4 α/ k s α/ s fzdz. 6

12 R. Macías et al. / Journal of Functional Analsis Since k s, we obtain that 6 is bounded b a constant times s +α e k /4 +αk k s k fzdz s s +α e k /4 +αk M f. Thus, H α, s, is bounded b a constant times s +α M f, 7 when α. A simple computation show that s 4e t/. Then estimates 7, 9, and 5 obtained for < α, show that 5 holds, and estimates 7, 9, 4 and 7 obtained for α, show that 6 holds. 4. Proof of the results Proof of Theorem. Let us consider the case < α <, and let β = α. B part a of Lemma 4, see 5, we have W α ft,c α {e t/4 M f+ e t β/4 β/ M β z β/ f z}, and from Lemma 3, 3 and 4, we have β/ M β f zz β/ 8 { C β β/ } fzz β/ dz + β/ M f zz β/ + M f, 9 thus, W α, f is smaller than or equal to a constant times β/ M f zz β/ + β/ M f zz β/ + M f. The hpothesis that / + α <p</ α and p> for < α < is equivalent to < pβ/ <pβ/ <p and p>. This shows that the weights pβ/ and pβ/ belong to the Muckenhoupt class A p and therefore β/ M f zz β/ p d C α,p β/ M f zz β/ p d C α,p f p d, f p d,

13 3 R. Macías et al. / Journal of Functional Analsis and M f p dc p f p d, which proves the theorem for < α <. Let us consider the case α. B Lemma part b, inequalit 6, thus W α, fc α M f, and W α fs,c α e t/4 M f, W α, f p dc α M f p dc α,p f p d, which proves the theorem for α. Remark. A second proof of Theorem can be given b interpolation of the results obtained for the end points of N α in Theorem. For the results on interpolation needed, see [, Theorem 3.4.6, p.59]. Proof of Theorem. a If < α < it follows that / α >, and the upper end point of N α is equal to / α. For a fixed s, <s<, consider the points and z satisfing and z, then, b 7, we obtain s s s R α,z,sc α e +s s α s z / C α,s α/ z α/. Thus, denoting a = s, we get that a W α χ,a s, C α,s α/ z α/ dz = C α,s α/, holds for ever a. Since a α/ / α d = a d =, it follows that the operator W α, f is not of strong tpe / α, / α. However, the operator W α, f is of weak tpe. In fact, b 3 in Lemma and 5 in Lemma,

14 R. Macías et al. / Journal of Functional Analsis it will be enough to show that the three terms of β/ M f zz β/ + sup h β/ h β fzz β/ dz + M f satisf the weak tpe condition. Since < α < implies that < < / α, then the weight belongs to A / α. Thus, the first term satisfies = β/ M f zz β/ / α d M f zz β/ / α d C α f β/ / α d= C α f / α d. This shows that the first term is of strong tpe and therefore of weak tpe obviousl of strong and weak tpe / α, / α. For the second term if we denote / α b p, then p = / + α. B Hölder s inequalit, we obtain h β fzz β/ dz h β f L p,+h z α/ L p,+h. In order to estimate z α/ L p,+h we observe that α// + α = + α/ + α > + α/ + α + α/ = + α and hold. Then z α/ L p,+h C α + h +α. Thus, since h, weget h +α + h +α fzz α/ dzc α f L h p,+h C α f L p,. Multipling b β/ and taking the supremum in h/, we obtain sup h/ β/ h β fzz β/ dz C α β/ f L p,. From this inequalit the weak tpe p, p for p = / α follows readil.

15 34 R. Macías et al. / Journal of Functional Analsis b If α, the upper end point of N α is equal to and, b 6, we have W α, fc α M f. Therefore, the operator W α, f is of strong tpe,. c If the lower end point of N α is greater than, then it coincides with / + α. This implies that α <. If for a given a =, as before, the integral a s fzzα/ dz is finite for ever fz L /+α,a, then since / + α = / α, b uniform boundedness, it follows that z α/ L / α,a. This is a contradiction since z α// α = z. Then, there exists f L /+α,a such that a fzzα/ dz =. This shows that for the given f, a Rs,, zf z dzc α,s α/ a zα/ fzdz=, showing that W α, f= for ever a. This is telling us that the operator W α, cannot be of weak tpe at the lower end point / + α >. Let < α < and β = α. B 5, 3 and 4, we have { W α, fc α M f+ β/ } fzz β/ dz + β/ M f zz β/. 3 It is eas to see that the inequalities < / β and < β// β </ β hold. These inequalities impl that the weight β// β belong to A / β. Therefore, the operators M f and β/ M f zz β/ are of strong tpe / β, / β. We have not considered et the second term of 3. Let E be a measurable set contained in,. Then, χ E zz β/ dz χ E zz β/ dz E z β/ dz = β/ E β/ = C β χ E z dz β/. In consequence, β/ χ E zz β/ dzc β β/ β/ χ E z dz. From this inequalit the restricted weak tpe readil obtained. β, β of the second term 3 is

16 R. Macías et al. / Journal of Functional Analsis d Let us show that if the lower end point of N α is equal to, then the operator W α, f cannot be of strong tpe,. B7, we have χ Ds zr α,z,s s / c α e 4s / z / e 4 s / z / e sz/ χ Ds z z /4. Take <ε. Let us assume that <z + ε, + ε, and s = /4. Then it follows that s /4, s, and s / z /4. Thus χ Ds z =, and since 4s / z / = / z / z + ε, 4ε z + we get that R α,z,s C α holds. Then W α, χ,+ε C α ε χ,+ε z dz = C α, holds for +ε. If the operator W α, were of strong tpe, we would have that W α, χ,+ε d A α χ,+ε d = A α ε 3 holds for a finite constant A α depending on α onl. On the other hand, we have +ε W α, χ,+ε d C α +ε In consequence, from 3 and 3, it follows that ε d = C αε log. 3 ε C α ε log A α ε or also C α,ε log A α. ε ε This is a contradiction since the left-hand side of the last inequalit above tends to when ε tends to, proving that there is no strong tpe, for the operator W α, f. However as we are going to prove, the operator is of weak tpe,.

17 36 R. Macías et al. / Journal of Functional Analsis Since / + α, it follows that α. Then, b Lemma, 6, we have that W α, fc α M f which implies the weak tpe, for the operator W α, f. References [] A. Erdéli, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. II, Based, in part, on notes left b Harr Bateman, McGraw-Hill Book Compan Inc., New York, Toronto, London, 953. [] M. de Guzmán, Real Variable Methods in Fourier Analsis, North-Holland Mathematics Studies, vol. 46, North-Holland Publishing Compan, Amsterdam, 98. [3] B. Muckenhoupt, Mean convergence of Hermite and Laguerre expansions, Trans. Amer. Math. Soc [4] B. Muckenhoupt, Weighted norm inequalities for the Hard maximal function, Trans. Amer. Soc [5] E.M. Stein, Topics in Harmonic Analsis Related to the Littlewood Pale Theor, Annals of Mathematical Studies, vol. 63, Princeton Universit Press, Princeton, NJ, 97. [6] E.M. Stein, Singular Integrals and Differentiabilit Properties of Functions, Princeton Mathematical Series, vol. 3, Princeton Universit Press, Princeton, NJ, 97. [7] K. Stempak, Heat-diffusion and Poisson integrals for Laguerre expansions, Tohoku Math. J [8] G. Szego, Orthogonal Polnomials, Amer. Math. Soc. Colloquium Publication XXIII, American Mathematical Societ, Providence, RI, 939. [9] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Mathematical Notes, vol. 4, Princeton Universit Press, Princeton, NJ, 993.