On the least values of L p -norms for the Kontorovich Lebedev transform and its convolution

Transcription

1 Journal of Approimation Theory On the least values of L p -norms for the Kontorovich Lebedev transform and its convolution Semyon B. Yakubovich Department of Pure Mathematics, Faculty of Sciences, University of Porto, Campo Alegre St., 687, Porto Portugal Received 6 January 4; received in revised form 31 July 4; accepted in revised form 6 October 4 Communicated by Zeev Ditzian Abstract We establish analogs of the Hausdorff Young and Riesz Kolmogorov inequalities and the norm estimates for the Kontorovich Lebedev transformation and the corresponding convolution. These classical inequalities are related to the norms of the Fourier convolution and the Hilbert transform in L p spaces, 1 p. Boundedness properties of the Kontorovich Lebedev transform and its convolution operator are investigated. In certain cases the least values of the norm constants are evaluated. Finally, it is conjectured that the norm of the Kontorovich Lebedev operator K iτ : L p R + ; d L p R + ; sinh πd, p K iτ [f ]= K iτ f d, τ R + π is equal to 1 p 1. It confirms, for instance, by the known Plancherel-type theorem for this transform when p =. 4 Elsevier Inc. All rights reserved. MSC: 44A15; 44A5; 44A35 Keywords: Kontorovich Lebedev transform; Convolution; Hausdorff Young inequality; Riesz Kolmogorov inequality; Fourier transform; Hilbert transform; Plancherel theory; Modified Bessel function address: syakubov@fc.up.pt 1-945/$ - see front matter 4 Elsevier Inc. All rights reserved. doi:1.116/j.jat.4.1.7

2 3 S.B. Yakubovich / Journal of Approimation Theory Introduction and preliminary results Let f, h be comple-valued measurable functions defined on R + =,. The purpose of this paper is to obtain an analog of the Hausdorff Young inequality [] and the norm estimates for the following convolution operator cf. [6,7,11] f h = 1 e 1 u +y f uhy du dy, >. 1.1 Our aim is also to establish an analog of the Riesz Kolmogorov inequality [3] for the Kontorovich Lebedev transformation [5,7,8,11] K iτ [f ]= K iτ f d, τ >, 1. where K iτ is the modified Bessel function of the second kind [1] with respect to the pure imaginary inde ν = iτ. We will consider these operators in appropriate Lebesgue spaces. In particular, the convolution operator 1.1 is well defined in the Banach ring L α R + L 1 R + ; K α d, α R see [11,7], i.e. the space of all summable functions f : R + C with respect to the measure K α d for which f L α R + = f K α d 1.3 is finite. Generally, the modified Bessel function K ν z satisfies the differential equation z d u dz + z du dz z + ν u = 1.4 for which it is the solution that remains bounded as z tends to infinity on the real line. It has the asymptotic behaviour cf. [1, relations 9.6.8, 9.6.9, 9.7.] π 1/ K ν z = e z [1 + O1/z], z z 1.5 and near the origin K ν z = O z Re ν, z, 1.6 K z = log z + O1, z. 1.7 Moreover, it can be defined by the following integral representations [5, 6-1-, 4, vol. I, relation ] K ν = K ν = 1 e cosh u cosh νudu, >, 1.8 ν t e 4t t ν 1 dt, >. 1.9

3 S.B. Yakubovich / Journal of Approimation Theory Hence, we easily find that K ν is a real-valued positive function when ν R and an even function with respect to the inde ν. Moreover, it satisfies the following inequalities: K ν K Re ν, >, 1.1 Re ν K ν Re ν 1 Γ Re ν, >, Re ν =, 1.11 where Γz is Euler s gamma-function [1]. Applying the Hölder inequality to the integral 1.3 and invoking asymptotic formulas with the latter inequalities 1.1, 1.11 for the weighted function K α it is not difficult to establish the following embeddings: L α R + L α R +, L α R + L β R +, α β, α, β R, 1.1 L α R L p R + ; d, <p, α < 1 p, 1.13 where L p R + ; dis a weighted Banach space with the norm 1/p f Lp R + ;d = f d p, 1 p <, 1.14 f L R + ;d = ess sup R+ f Our goal in this paper is to study the boundedness properties in L p R + ; dof the convolution operator 1.1 when one of the functions, say h, is fied and belongs to the space L α R +, where α depends on p. For this we will generalize on L p -case the following Hausdorff Young-type inequality: f h L R + ;d f L R + ;d h L R +, 1.16 which is proved in [9,1]. Furthermore, for some values of p relations between norms of operators 1.1, 1. are confirmed by the Young-type inequality see [1] f h L1/μ R + ;d C μ,γ,β f L1/β R + ;d h L1/γ R + ;d, 1.17 where γ+β 1 γ β γ+β μ 1 C μ,γ,β = μ μ μ K γ β K d, γ β and < γ, β < 1, γ + β 1, γ+β+ γ β < μ 1. In the final section, we prove an analog of the Riesz Kolmogorov theorem for the Kontorovich Lebedev operator 1. when p. Namely, we will prove the following inequality: τ sinh πτ K iτ [f ] p dτ πp p 1 which is equivalent to K iτ [f ] Lp R + ;τ sinh πτ dτ π 1 1 p f p d, 1.19 f Lp R + ;d. 1.

4 34 S.B. Yakubovich / Journal of Approimation Theory We note, that spaces L p R + ; τ sinh πτ dτ, p 1 and L R + ; τ sinh πτ dτ are normed, respectively, by 1/p f Lp R + ;τ sinh πτ dτ = fτ p τ sinh πτ dτ, 1 p <, 1.1 f L R + ;τ sinh πτ dτ = ess sup τ R+ fτ. 1. We will conjecture that the norm of the Kontorovich Lebedev operator 1. K iτ is equal π to, where we define the norm as usual by 1 1 p K iτ = sup K iτ [f ] Lp R + ;τ sinh πτ dτ. 1.3 f LpR +;d=1 As it is known, the product of the modified Bessel functions of the second kind of different arguments can be represented by the Macdonald formula [4, vol. II, relation ] K ν K ν y = 1 e 1 u +y K ν u du u. 1.4 This is a key formula, which is used to prove the factorization property for the convolution 1.1 in terms of the Kontorovich Lebedev transform 1. in the space L α R + [9,11], namely K iτ [f h] =K iτ [f ]K iτ [h], τ R +, 1.5 where the integral 1. eists as a Lebesgue integral. It is also proved in [7,11] that the Kontorovich Lebedev transform is a bounded operator from L α R + into the space of bounded continuous functions on R + vanishing at infinity. Furthermore, the convolution 1.1 of two functions f, h L α R + eists as a Lebesgue integral and belongs to L α R +. It satisfies the Young-type inequality f h L α R + f L α R + h L α R However, it is not difficult to verify that in the case f L R + ; dintegral 1. in general, does not eist in Lebesgue s sense take, for instance { 1 f= log if < 1, if > 1, and use asymptotic formula 1.6. Thus we define it in the form K iτ [f ]= lim N N 1/N K iτ f d, 1.7 where the limit is taken in the mean-square sense with respect to the norm of the space L R + ; τ sinh πτ dτ. It has been proved see [7] that K iτ : L R + ; d L R + ; τ sinh πτ dτ

5 S.B. Yakubovich / Journal of Approimation Theory is a bounded operator and forms an isometric isomorphism between these Hilbert spaces with the Parseval identity of the form τ sinh πτ K iτ [f ] dτ = π f d. 1.8 The two definitions 1. and 1.7 are equivalent, if f L R + ; d L α R +. The inverse operator in the latter case is given by the formula f= lim N f N, where f N = N π τ sinh πτ K iτ K iτ [f ] dτ, 1.9 and the convergence is in the mean-square sense with respect to the norm 1.14 of L R + ; d. It can be written for almost all R + in the equivalent form f= π d d τ sinh πτk iτ yk iτ [f ] dy dτ Boundedness properties of the convolution operator We generalize inequality 1.16 by proving the following: Theorem 1. Let 1 <p, f L p R + ; dand h L p p 1 R +. Then convolution 1.1 eists as a Lebesgue integral for all > and belongs to the space L p R + ; d. Moreover, it satisfies the following inequality: f h Lp R + ;d f Lp R + ;d h p L p 1 R +..1 Proof. Indeed, from Hölder s inequality we have for convolution 1.1 the estimate q 1 + p 1 = 1 f h p 1 p e 1 u +y p/q hy dudy u q/p u f u p e 1 u +y hy dudy.. Hence we use the formula see in [4, vol. I, relation ] e 1 u +y du u q/p = y 1 q/p/ + y K 1 q/p + y.3 and we prove that for all,y > y 1 q/p/ + y K 1 q/p + y 1 q/p K 1 q/p y..4

6 36 S.B. Yakubovich / Journal of Approimation Theory Indeed, invoking the representation 1.9 we deduce y 1 q/p/ + y K 1 q/p + y = y 1 q/p q/p e t +y 4t t 1 q/p 1 dt y 1 q/p q/p y t e 4t t 1 q/p 1 dt = y 1 q/p q/p 1 y 1 q/p K 1 q/p y = 1 q/p K 1 q/p y. Thus by taking relations.3,.4 we estimate the right-hand side of the inequality. as follows: f h p 1 p/q K 1 q/p + y hy dy u f u p e 1 hy dudy 1 u f u p e 1 u +y p/q K 1 q/p y hy dy u +y hy dudy..5 Hence multiplying both sides of.5 by we integrate with respect to R +. Inverting the order of integration by the Fubini theorem we invoke again. and an elementary inequality K u + y K y to obtain f h p d p/q K 1 q/p y hy dy hy dudy u f u p du uk u + y f u p p/q K 1 q/p y hy dy K y hy dy..6 Hence taking into account that q = p/p 1 we recall norms 1.3, 1.14 and write.6 in the form f h Lp R + ;d f Lp R + ;d h 1/p L R + h 1 1/p p L p 1 R +..7

7 S.B. Yakubovich / Journal of Approimation Theory However, it is easily to verify from representation 1.8 that K y K 1 q/p y. Hence from.7 cf. embedding 1.1 we arrive at the desired inequality.1. Theorem 1 is proved. Remark 1. Putting in.1 p = we arrive at When p = it takes the form f h L R + ;d f L R + ;d h L 1 R +..8 Another version of the Hausdorff Young-type theorem for convolution 1.1 when f, h belong to the conjugate spaces 1.14 is given by Theorem. Let 1 <p<, h L p R + ; d, f L q R + ; d, q = p/p 1. Then convolution 1.1 eists as a Lebesgue integral for all >and belongs to the space L r R + ; r d with 1 r <. Furthermore it satisfies the inequality where pq p q f h Lr R + ; r d C r,p,q f Lq R + ;d h Lp R + ;d,.9 C r,p,q = [ K 1/q 1 q/p K1/p 1 p/q ] r d 1/r..1 Proof. Again with Hölder s inequality and employing.3,.4 we majorize 1.1 as f h 1 e 1 e 1 y = 1 +y u + u u +y u +y hy p y dudy u p/q f u q u dudy y q/p 1/q 1/p 1 p/q/ 1/p K 1 p/q +y hy p ydy 1 q/p/ 1/q K 1 q/p + u f u q udu 1 K 1/q 1 q/p K1/p 1 p/q f L q R + ;d h Lp R + ;d. Hence we easily obtain inequality.9 with a constant given by the integral.1. Due to asymptotic formulas we verify by straightforward calculations that it converges when 1 r <. Theorem is proved. pq p q

8 38 S.B. Yakubovich / Journal of Approimation Theory Letting p = q =, r= 1 in.9 we invoke relation from [4, vol. II] to calculate the corresponding value of the integral.1. As a result we deduce the inequality f h L1 R + ;d π f L R + ;d h L R + ;d,.11 which is a particular case of 1.17 when μ = 1, β = γ = 1. For a fied function h we denote by S h, u = 1 e 1 u +y the kernel of the following convolution operator: S h f = hy dy.1 S h, uf u du, >..13 As a consequence of Theorem 1 we establish boundedness properties of the operator.13 in the space L p R + ; d, 1 <p. Thus we have Theorem 3. Let 1 <p, and h L p p 1 R +. Then integral operator.13 is bounded in the space L p R + ; dand its norm S h h p. If, in turn, h is a positive L p 1 R + function on R + and p 1 + 1,, then S 3 h = h p. L p 1 R + Proof. The first part of the theorem easily follows from inequality.1. Let us show that if h is a positive function on R + and p 1 + 1, 3, then the norm of the convolution operator.13 is equal to h p. To do this we prove that the case S h < L R + h p is impossible when p 1 + 1,. Indeed, assuming that S L R + 3 φ < 1, where h φ = h p L R +, > it follows immediately by virtue of the Banach theorem that the operator A = I S φ I denotes the identity operator has an inverse in L p R + ; d, 1 <p<. Consequently, the adjoint operator A = I Ŝ φ, where Ŝ φ f = S φ u, f u du, >,.14 has an inverse in L q R + ; d,q 1 + p 1 = 1. However, we will show now that the operator A in L q R + ; d cannot have an inverse since integral equation A f = has a nontrivial solution f = K p, which belongs to L q R + ; d when p p 1

9 S.B. Yakubovich / Journal of Approimation Theory ,. Indeed, employing the Macdonald formula 1.4 and the norm definition of the space L α R + we substitute formally in.14 the function K p to obtain p 1 S φ u, K p u du = K p p 1 p 1 for any positive h L p p 1 R +. This solution belongs to L q R + ; d, if the integral K q p d <. p 1 When p = it evidently converges via asymptotic formulas 1.5, 1.7 for the modified Bessel function. For other values of p according to 1.5, 1.6 we see that the latter integral is convergent when p p <. This gives the condition p p 1,. So this fact 3 contradicts our assumption above and we conclude that S φ 1. But from the first part of the theorem it follows that S φ 1. Thus we get S φ = 1or S h = h p. L R + Theorem 3 is proved. Let us consider two eamples of convolution operator.13 cf. [7,1] with the corresponding kernels.1, which are calculated for concrete functions h. If, for instance, we put h 1 then we calculate integral.1 by using representation 1.9 and we arrive at the following integral operator K 1 + u Kf = uf u du, > u It is easily seen from 1.3, that h 1 belongs to the space L p p 1 R + if and only if p 3,. Consequently, appealing to Theorem 3 via relation from [4, vol. II] we find that.15 is a bounded operator in L p R + ; d, p 3, and has a least value of its norm when p 1 + 1,, namely 3 π K = cosh π p p 1, p 1 + 1,. 3 When, in turn, we put h = e then calculating the corresponding integral.1 and taking into account, that the modified Bessel function of the inde 1 reduces to π K 1/ z = e z z see [1, 9.6], we arrive at the Lebedev convolution operator cf. [7,11] π e u Lef = uf u du, > u

10 4 S.B. Yakubovich / Journal of Approimation Theory In this case h = e belongs to the space L p p 1 R + if and only if p 5 3, 3. Moreover, using relation from [4, vol. II, Theorem 3] we obtain that the Lebedev operator.16 is bounded in L p R + ; d, p 5 3, 3 and we have π 1 Le = π, p cosh π p p 1 5 3, Riesz Kolmogorov-type theorem In this final section, we study the Kontorovich Lebedev transformation 1. as an operator, which maps the weighted space L p R + ; d into the weighted space L p R + ; τ sinh πτ dτ for p. We will show that for 1 p < this map in general, does not eist. Finally, we prove an analog of the Riesz Kolmogorov theorem [3], which is known for the Hilbert transform in L p. Namely, we will prove inequality 1.19 and consider least values of the norm constants 1.3 for the Kontorovich Lebedev operator 1.. The case p = has been studied in [7,9,1]. It forms an isometric isomorphism between the corresponding Hilbert spaces with the Parseval equality 1.8. It has also a relationship with convolution 1.1 by means of the following Parseval equality cf. [9,1] τ sinh πτ K iτ [f ]K iτ [h] dτ = π f h d. 3.1 However, when <p and f L p R + ; dit has been shown in [8] that integral 1. is understood as a Lebesgue integral. We will prove that the image of the Kontorovich Lebedev operator belongs to L p R + ; τ sinh πτ dτ. It fails certainly when 1 p <. For instance, we take f= { 1if 1, if>1. Then f clearly belongs to L p R + ; d, 1 p <. Nevertheless, we find that the corresponding value of the Kontorovich Lebedev transform 1. is 1 K iτ d. The latter integral is a continuous function with respect to τ and behaves as Oe πτ/ when τ + cf. [7]. Consequently, integral 1.1 is divergent for this function when 1 p <. In order to establish inequality 1.19 we need the boundedness of the Kontorovich Lebedev transform as an operator see 1.15, 1. K iτ : L R + ; d L R + ; τ sinh πτ dτ. This result is given by Theorem 4. The Kontorovich Lebedev transformation 1. is a bounded operator L R + ; d L R + ; τ sinh πτ dτ and its norm is equal to π.

11 S.B. Yakubovich / Journal of Approimation Theory Proof. Taking into account inequality 1.1 for the modified Bessel function and definitions of norms 1.15, 1. we deduce from 1. the following estimate: K iτ [f ] L R + ;τ sinh πτ dτ f L R + ;d K d = π f L R + ;d, 3. where the latter integral is equal to π due to relation from [4, vol. II]. Consequently, the Kontorovich Lebedev transform is bounded from L R + ; dinto the space L R + ; τ sinh πτ dτ and its norm 1.3 K iτ π. However, it attains its least value by taking f 1. Indeed, by the same relation from [4, vol. II] we get that the transform 1. is gτ = K iτ d = π 1 cosh πτ, and clearly, g L R + ;τ sinh πτ dτ = π. Theorem 4 is proved. It is not difficult to find the norm of the Kontorovich Lebedev operator when p =. In this case the Parseval equality 1.8 gives the value π. When <p<, the norm estimate is established by the Riesz Kolmogorov-type theorem. So, finally we prove Theorem 5. Let p. The Kontorovich Lebedev transformation 1. is a bounded operator L p R + ; d L p R + ; τ sinh πτ dτ. Moreover, inequality 1.19 holds true and the norm 1.3 K iτ π. 1 1 p Proof. In fact, by the Plancherel-type theorem for the Kontorovich Lebedev transformation [7] we conclude that integral operator K iτ is of type,. Meantime, Theorem 4 states that this operator is of type,. Hence by the Riesz Thorin conveity theorem [] the Kontorovich Lebedev operator 1. is of type p, p, i.e. maps the space L p R + ; d into L p R + ; τ sinh πτ dτ, where p 1 = θ, θ 1. This means that p and we find π θ π 1 θ K iτ [f ] Lp R + ;τ sinh πτdτ f Lp R + ;d π /p π 1 /p = f Lp R + ;d = π f Lp R + ;d p Thus we easily get inequality Furthermore, from 3.3 and 1.3 it follows that K iτ π. Theorem 5 is proved. 1 1 p

12 4 S.B. Yakubovich / Journal of Approimation Theory Remark. The problem of the inversion formula for the Kontorovich Lebedev transformation 1. is considered in [8]. When p = it is given by relations 1.9, 1.3. But if <p< the corresponding relation is a particular case of the main theorem in [8] and reciprocally to 1. we obtain f= lim ε + π τ sinhπ ετk iτ K iτ [f ] dτ, >, where the limit is taken with respect to the norm 1.14 or eists almost for all >. Our final conjecture states that the norm 1.3 of the Kontorovich Lebedev operator is π equal to. However, it is still an open question for <p<. 1 1 p Acknowledgments The present investigation was supported, in part, by the Centro de Matemática of the University of Porto. The author sincerely indebted to the referee for the valued suggestions and comments, which rather improved the presentation of the paper. References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 197. [] R.E. Edwards, Fourier Series II, Holt Rinehart and Winston, New York, [3] S.K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math [4] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, vol. I: Elementary Functions, vol. II: Special Functions, Gordon & Breach, New York, London, [5] I.N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 197. [6] S.B. Yakubovich, On the convolution for the Kontorovich Lebedev transform and its applications to integral equations, Dokl. Akad. Nauk BSSR in Russian. [7] S.B. Yakubovich, Inde Transforms, World Scientific Publishing Company, Singapore, New Jersey, London, Hong Kong, [8] S.B. Yakubovich, On the Kontorovich Lebedev transformation, J. Integral Equations Appl [9] S.B. Yakubovich, Integral transforms of the Kontorovich Lebedev convolution type, Collect. Math [1] S.B. Yakubovich, Boundedness and inversion properties of certain convolution transforms, J. Korean Math. Soc [11] S.B. Yakubovich, Yu.F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions Kluwers Series in Mathematics and Applications vol. 87, Dordrecht, Boston, London, 1994.