ON THE KONTOROVICH- LEBEDEV TRANSFORMATION

Transcription

1 JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS Volume 5, Number, Spring 3 ON THE KONTOROVICH- LEBEDEV TRANSFORMATION SEMYON B. YAKUBOVICH ABSTRACT. Further developments of the results on the Kontorovich-Lebedev integral transformation are given. In particular, properties of the boundedness, compactness in L ν,p, p, ν <, are established. The Bochner type representation theorem is proved. An example of the Fredholm integral equation associated with the Kontorovich- Lebedev operator is considered.. Introduction and auxiliary results. In this paper we investigate mapping properties of the Kontorovich-Lebedev operator [3], [5], [] (.) K ir [f] K iτ (x)f(x) dx, τ R +, which is associated with the Macdonald function K iτ as the kernel [] in its natural domain of definition f L L (R + ; K (x) dx), i.e., (.) L : { f : } K (x) f(x) dx <. In particular, it contains all spaces L α L (R + ; K (αx) dx), <α andl ν,p (R + ), ν<, p, with the norms (.3) f L α K (αx) f(x) dx <, (.4) ( f ν,p ) x νp f(x) p p dx <, f ν, ess sup x x ν f(x) <. Key words and phrases. Kontorovich-Lebedev transform, Macdonald function, regularization, Fourier integrals, Bochner theorem, Hilbert-Schmi operator, Fredholm equation. AMS Mathematics Subject Classification. 44A5, 45B5. Copyright c 3 Rocky Mountain Mathematics Consortium 95

2 96 S.B. YAKUBOVICH When ν p we obtain the usual norm in L p denoted by p. The Macdonald function can be represented by the Fourier integral [5] (.5) K iτ (x) e x cosh u cos(τu) du. Therefore, for x>, τ R, it is real-valued and even function with respect to the index iτ. Furthermore, via the analytic properties of the integrand in (.5), the latter integral can be extended to the strip δ [,/) in the upper half-plan (cf. in [9]), i.e., iδ+ (.6) K iτ (x) e x cosh β+iτβ dβ. iδ Thisgivesusforeachx> an immediate uniform estimate (.7) K iτ (x) e δ τ K (x cos δ), δ δ <. We note that the Macdonald function is the modified Bessel function of the second kind K µ (z) (in our case µ iτ), which satisfies the differential equation z d u dz + z du dz (z + µ )u. It has the asymptotic behavior [] (.8) K µ (z) and near the origin ( ) e z [ + O(/z)], z, z (.9) (.) z µ K µ (z) µ Γ(µ)+o(), z, K (z) log z + O(), z. By using relation (.6.5.8) in [4, Vol. ], we obtain the useful formula (.) τ sinh(( δ)τ)k iτ (x)k iτ (y) dτ xy sin δ K ((x + y xy cos δ) ), (x + y xy cos δ) x, y >, <δ.

3 KONTOROVICH-LEBEDEV TRANSFORMATION 97 In the next section we establish mapping properties of the Kontorovich- Lebedev operator (.) in the spaces L α and L ν,p (see also [8], []). Mapping properties for distributions were considered in [9].. Kontorovich-Lebedev transform. In order to study boundedness properties of the operator (.) we prove its composition representation in terms of the Fourier and Laplace operators, which are defined respectively by (.) (.) (Ff)(x) (Lf)(x) e ixt f(t), e xt f(t). Lemma. Let ν<, p, q p/(p ) and <α. Then the embedding (.3) L ν,p (R + ) L α L is true and (.4) f L f L α Γ(( (αq)ν ν)/)γ/q (q( ν)) f ν,p, Γ( (ν/)) <p, (.5) f L f L α α ν f ν, sup[k (x)x ν ]. x Proof. Letting in (.5) τ we easily find that K (x) is steadily decreasing function and invoking (.3) and (.4) with the Hölder inequality we obtain ( /q f L f L α K q dx) (αx)x( ν)q f ν,p.

4 98 S.B. YAKUBOVICH Hence, applying the generalized Minkowski inequality we derive ( ) /q K q (αx)x( ν)q dx ( ( q ) /q x ( ν)q e du) αx cosh u dx ( ) /q du x ( ν)q e αqx cosh u dx (αq) ν Γ /q du (q( ν)) cosh ν u Γ(( (αq)ν ν)/)γ/q (q( ν)). Γ( (ν/)) Meanwhile, for f L ν, (R + ) we easily verify that K (αx) f(x) dx α ν f ν, sup[k (x)x ν ] < x and complete the proof of Lemma. Lemma. The Kontorovich-Lebedev transformation (.) is a bounded operator from L α, <α, into the space of bounded continuous functions vanishing at infinity. Besides, the following composition representation in terms of operators (.) and (.) takes place (.6) K iτ [f] (F(Lf)(cosh x))(τ). Proof. Substituting (.6) with δ in (.) we have the estimate K iτ [f] f(x) dx e x cosh β dβ f(x) dx K (αx) f(x) dx e αx cosh β dβ f L α.

5 KONTOROVICH-LEBEDEV TRANSFORMATION 99 Hence, in view of Fubini s theorem, we can invert the order of integration in the corresponding iterated integral and arrive at the composition (.6). It converges uniformly with respect to τ and therefore K iτ [f] is a bounded continuous function. Further, under conditions of Lemmaanhelatterestimatewehavethat(Lf)(cosh x) L (R). Hence K iτ [f] whenτ as an immediate consequence of the Riemann-Lebesgue lemma. Lemma is proved. Corollary. Operator K iτ [f] :L ν,p (R + ) L p (R + ), p, ν< is bounded and (.7) K iτ [f] p ( ) ν [ ( )] q p /q q Γ ( ν) f ν,p, 3/q q p p. In particular, for p, ν, we get (.8) K iτ [f] f. Proof. Indeed, by appealing to the composition (.6) and L p -theorem for the Fourier transform [6, Theorem 74], we find (.9) ( /q K iτ [f] p () ( p ) ( ) (Lf)(cosh x) dx) q, q p p. Hence by the generalized Minkowski and Hölder inequalities with

6 S.B. YAKUBOVICH formula (.6..) from [4, Volume ], we obtain ( ) /q () ( p ) ( ) (Lf)(cosh u) q du ( /q () ( p ) ( ) f(x) e du) qx cosh u dx () ( p ) ( ) f(x) K /q (qx) dx ( /q () ( p ) ( ) K (qx)x dx) ( ν)q f ν,p ( ) ν q p 3/q [ ( )] /q q Γ ( ν) f ν,p. Consequently combining with (.9) we have (.7) and (.8). Corollary is proved. Furthermore, we will establish now that the Kontorovich-Lebedev transform (.) is a completely continuous operator. Lemma 3. The transformation K iτ [f] : L ν,p (R + ) L q (R + ), <p, ν<, q p/(p ) is a completely continuous operator. Proof. Indeed, this can be done by approximation of the Macdonald function K iτ (x) in terms of kernels of finite rank. However, first we verify the following integral condition (.) K q iτ (x) x ( ν)q dτ dx <. We treat double integral (.) by using again Fourier integrals (.5), (.6) and Theorem 74 from [6], see (.9). Thus, we have ( K q ) /q iτ (x) x ( ν)q dτ dx () (/q) ( ) ( K /(p ) (px)x ( ν)q dx ) /q ( ) (/p) /(q) q ν [Γ(q( ν))]/q [Γ((p/)( ν))] /p [ Γ ( p( ν)+) ] <. /p

7 KONTOROVICH-LEBEDEV TRANSFORMATION In particular, by taking the values of integrals (.6.5.6) and (.6..) from [4, Volume ] for the Hilbert-Schmi norm, p q,ν,of K iτ [f], we find (.) K iτ (x) dτ dx 4. Therefore, in the Lebesgue space with the norm, ( ) /q (.) h(τ,x) q x ( ν)q dτ dx there is a sequence {K n (τ,x)} of continuous in R + and finite kernels, which converges to / K iτ (x) in the space (.), namely, (.3) K q iτ (x) K n (τ,x) x ( ν)q dτ dx, n. Denoting by K n integral operators with kernels K n (τ,x) weusethe Hölder inequality to obtain K iτ [f] K n f(τ) q [ ] q K iτ (x) K n (τ,x) f(x) dx q ( K q/p iτ (x) K n (τ,x) x ( ν)q dx f(x) p x dx) νp. Hence, K iτ [f] K n f q q K q iτ [f] K n f(τ) dτ q f q ν,p K iτ (x) K n (τ,x) x ( ν)q dx dτ. Consequently, the norm of difference K iτ K n q q K q iτ (x) K n (τ,x) x ( ν)q dx dτ, n

8 S.B. YAKUBOVICH and the Kontorovich-Lebedev operator (.) is the L q -limit of the sequence {K n } of completely continuous integral operators with continuous finite kernels. Thus K iτ [f] is a completely continuous operator. Lemma 3 is proved. Let us introduce for all x> the following regularization operator (cf. in [], [9]) (.4) (I ε g)(x) x τ sinh(( ε)τ)k iτ (x)g(τ) dτ, where ε (,). We will show that (.4) gives in some sense an inversion of (.) in L ν,p (R + ) space when ε. Lemma 4. If g(τ) K iτ [f], f(y) L ν,p (R + ), ν<, p, then (I ε g)(x) (KL ε g)(x) (.5) sin ε K ((x + y xy cos ε) ) yf(y) dy, (x + y xy cos ε) x>. Proof. Substituting g(τ) in (.4) and inverting the order of integration we use relation (.) and immediately arrive at the representation (.5). The motivation of the change of the order of integration is due to Fubini s theorem and the following estimate (see (.7)) (KL ε f)(x) K (x cos δ ) τ sinh(( ε)τ)e (δ +δ )τ dτ x ( /q K q (y cos δ )y dy) ( ν)q f ν,p <, where δ i [,/), i,, δ + δ + ε>, ν<, <p<. Since in view of the asymptotic behavior of the Macdonald function (.8),

9 KONTOROVICH-LEBEDEV TRANSFORMATION 3 (.9) and (.), we have (cf. (.4) and (.5)) K (y cos δ ) f(y) dy (cos δ ) ν f ν, sup [K (y)y ν ] y <, K (y cos δ ) f(y) dy ν (cos δ ) ν Γ ( ν ) f ν, <, it follows that representation (.5) is also true when p,, respectively. Lemma 4 is proved. We are ready to prove now the Bochner type representation theorem for the regularization operator (.5). Theorem. Let f L ν,p (R + ), <ν<, p<. Then (.6) f(x) lim(kl ε f)(x), ε where the limit is with respect to the norm (.4). Besides, the limit (.6) exists for almost all x>. Proof. Making substitution y x(cosε+t sin ε) in the integral (.5), we deduce (KL ε f)(x) x sin ε K (x sin ε t +) t + (.7) f(x(cos ε + t sin ε))(cos ε + t sin ε). Hence, due to the generalized Minkowski inequality and elementary inequality for the Macdonald function xk (x), x, cf. (.5), (.8) and (.9), we estimate the L ν,p -norm, <ν<, p<, of the integral (.7) as follows KL ε f ν,p f ν,p f ν,p t + f(x(cos ε + t sin ε))(cos ε + t sin ε) ν,p (cos ε + t sin ε) ν t + ( + t ) ν t + 4 ( + ) f ν,p. ν

10 4 S.B. YAKUBOVICH Consequently, by using the identity and denoting by t + ε (.8) R(x, t, ε) x sin ε t +K ( x sin ε t + ) we find that KL ε f f ν,p + ε t + t + I (ε)+i (ε). But, since (see [], []) f(x(cos ε + t sin ε))(cos ε + t sin ε)r(x, t, ε) ( ε ) f(x) [ f(x(cos ε + t sin ε))(cos ε + t sin ε) ν,p ( ε ) ] f(x) R(x, t, ε) t + f(x)[r(x, t, ε) ] ν,p d dx [xk (x)] xk (x), and xk (x), x, we obtain the following representation ν,p x sin ε(t +) R(x, t, ε) yk (y) dy. Hence, appealing again to the generalized Minkowski inequality, we deduce

11 KONTOROVICH-LEBEDEV TRANSFORMATION 5 I (ε) ε ( ε ( ε t + ( x sin ε(t x νp +) t + yk (y) x νp f(x) p dx y/(sin ε(t +) ) ε ( ε + ε ( ε x νp f(x) p dx u sin ε εν/ f ν,p ε ε t + u ν K (u) du + p ) yk (y) dy) f(x) p p dx ) p dy uk (u ( ) t +) x νp f(x) p p dx du u sin ε ) + uk (u t +) ε ) p du uk (u ( ) t +) x νp f(x) p p dx du u sin ε ( uk (u) du (t +) (ν/) ( ε ( ε ν/ Γ( ν) f ν,p + ε ε. x νp f(x) p dx ε ( x νp f(x) p dx ε x νp f(x) p dx ε ) p ) p ), Concerning the integral I we first approximate f L ν,p (R + )bya smooth function ϕ with a compact support on R +. This implies that there exists a function ϕ C (R + ) such that f ϕ ν,p ε. Hence, ) p

12 6 S.B. YAKUBOVICH since the kernel (.8), R(x, t, ε), then in view of the representation, ϕ(x(cos ε + t sin ε))(cos ε + t sin ε) ϕ(x) (.9) cos ε+t sin ε cos ε+t sin ε d [yϕ(xy)] dy dy [ϕ(xy)+xyϕ (xy)] dy. In a similar manner, we have (.) I (ε) + f ϕ ν,p + t [f(x(cos ε + t sin ε)) + ϕ(x(cos ε + t sin ε))](cos ε + t sin ε) ν,p t + t + +( ϕ ν,p + ϕ +ν,p ) [ + 4 ε ϕ(x(cos ε + t sin ε))(cos ε + t sin ε) ( ε ) f(x) ν,p (cos ε + t sin ε) ν t + ( ϕ(x) ε ) f(x) ν,p cos ε+t sin ε t + y ν dy ] f ϕ ν,p + ε f ν,p + ϕ ν,p + ϕ +ν,p ( ν) + 4 ν (cos ε + t sin ε) ν t. + The latter integral in (.9) we treat by making the substitution

13 KONTOROVICH-LEBEDEV TRANSFORMATION 7 e v cosε + t sin ε. Then it takes the form (.) (cos ε + t sin ε) ν t + sinh(( ν)v) sinε cosh v cos ε dv ( ) sinh(( ν)v) sinε + cosh v cos ε dv ( sin ε log(cosh v cos ε) sin(v( ν)) + cosh v ( sin ε log sin ε ) + A ν sin ε, ) dv where sinh(v( ν)) A ν + dv, cosh v <ν<. Thus, combining with (.9), we see that lim ε I (ε). Therefore, by virtue of the above estimates lim ε KL ε f f ν,p and relation (.6) is proved. In order to verify the pointwise convergence we use the fact that any sequence of function {ϕ n } C(R + ) which converges to f in L ν,p (R + )-norm (.4) contains a subsequence {ϕ nk } convergent almost everywhere, i.e., lim k ϕ nk (x) f(x) for almost all x>. Then via (.9), (.) and (.), we obtain (.) (KL ε f)(x) f(x) t + ( f(x(cos ε + t sin ε))(cos ε + t sin ε)r(x, t, ε) ε ) f(x) (cos ε + t sin ε) t + f(x(cos ε + t sin ε) ϕ nk (x(cos ε + t sin ε))

14 8 S.B. YAKUBOVICH + ( ν) sup t ν ϕ nk (xt)+xt +ν ϕ n k (xt) t [ ( sin ε log sin ε ] ( )+A ν + ε ) ϕ nk (x) f(x) + ε f(x). Meantime, by taking p<, q p/(p ), we have (.3) (cos ε + t sin ε) t f(x(cos ε + t sin ε) ϕ nk (x(cos ε + t sin ε)) + ( x ν f ϕ nk ν,p (cos ε + t sin ε) q( ν+( p )) ) /q sin p ε (t +) q ( x ν f ϕ v q( ν) ) /q dv n k ν,p sin ε (v cosεv +) q. However, appealing to the relation (..9.7) from [4, Volume I] we get the value of the latter integral in terms of the Gauss function v q( ν) dv Γ(q( ν))γ(qν) (v cosεv +) q Γ(q) F (q qν, qν ; q + ) ;sin ε Γ(q( ν))γ(qν), ε. Γ(q) Therefore we see that the righthand side of inequality (.) tends to zero for almost all x>whenε, k k. Finally let us consider the pointwise convergence when f L ν, (R + ). In this case for any k k, we get f ϕ nk ν, δ. M aking substitution v cosε + t sin ε the first integral in (.3) becomes (cos ε + t sin ε) t f(x(cos ε + t sin ε) ϕ nk (x(cos ε + t sin ε)) + sin ε v ν (v cos ε) +sin ε vν f(xv) ϕ nk (xv) dv x ν sin ε f ϕ nk ν, sup v v ν (v cos ε) +sin ε.

15 KONTOROVICH-LEBEDEV TRANSFORMATION 9 However, by using the methods of calculus it is not difficult to show that sup v v ν (v cos ε) +sin ε v ν (v cos ε) +sin ε ν sin ε, where v ((cos ε)/ν)+ ((cos ε)/ν ) + (( cos ε)/ν). Thus, sin ε v ν (v cos ε) +sin ε vν f(xv) ϕ nk (xv) dv ν x ν δ sin ε, which tends to by choosing first ε and then δ. The theorem is proved. 3. Fredholm equation. In this final section we consider an example of Fredholm integral equation in the space L (R + ), which is associated with the Kontorovich-Lebedev operator (.). Precisely, let us consider the following integral equation of the second kind (3.) f(x) K ix (t)f(t) g(x), where x >, g L (R + ) and the function f L (R + )istobe determined. According to Corollary and norm inequality (.8) the Kontorovich-Lebedev transformation is a bounded operator in L (R + ) and we have that K ix / <. Consequently, by virtue of the Banach theorem integral operator I K ix has a bounded inverse operator [I K ix ] with the norm [I K ix ] /. Furthermore, the unique L -solution of the equation (3.) is represented through the absolutely and uniformly convergent Neumann series (3.) f(x) Kix[g], n n

16 S.B. YAKUBOVICH where the iterates Kix n [g] are defined by the recurrence relation (3.3) K ix[g] g, K n ix[g] K ix [K n ix n. [g]] K n (x, t)g(t), We show that the kernel K n (x, t) of the compositions (3.3) can be expressed in terms of the Fourier transform (.) and Fourier convolution operator (3.4) (ϕ ψ)(x) ϕ(x t)ψ(t). To do this, we use the fact from [6, Theorem 65] that (ϕ ψ)(x) L (R) and F(ϕ ψ)(x) (Fϕ)(x)(Fψ)(x) in L (R) if ϕ L (R) and ψ L (R). Then from (.6) we easily find that the kernel of Kix [g] is K (x, t) (Fe t cosh u )(x). Further, for n, we obtain the composition (3.5) K ix[g] K ix (y) dy K (x, t)g(t), K iy (t)g(t) where (3.6) K (x, t) K ix (y)k iy (t) dy. Note that the interchange of the order of integration in (3.5) is plainly verified via Fubini s theorem by means of the inequality (.7) and condition g L (R + ). Hence, invoking (.5) and the value of integral (.6.4.) from [4, Volume ] after using the Parseval equality for the Fourier cosine transform (cf. in [6]), we derive (3.7) K (x, t) cosh(x/) cosh(x/) e t cosh u cos ( τ log(u + u +) ) e t cosh sinh u cos(xu) du. du u +

17 KONTOROVICH-LEBEDEV TRANSFORMATION Meantime with the integral ( ) in [4, Volume ] and the factorization property for Fourier convolution (3.4), the latter integral in (3.7) implies K (x, t) e t cosh sinh u cos(xu) du cosh(x/) ( (Fe t cosh sinh u )(x) F ) (x) cosh u ( ) cosh sinh u F e t (x). cosh u Thus for the nth iterate it is not difficult to deduce the corresponding formula for the kernel K n (x, t). It gives ( K n (x, t) F ) cosh u K n (sinh u, t) (x) cosh(x/) F(K n (sinh u, t))(x), n. Acknowledgments. The present investigation was supported, in part, by the Centro de Matemática of the University of Porto. REFERENCES. A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions, Vol. II, McGraw-Hill, New York, H.-J. Glaeske and S.B. Yakubovich, On the stabilization of the inversion of some Kontorovich-Lebedev like integral transforms, J. Anal. Appl. 7 (998), N.N. Lebedev, Sur une formule d inversion, C.R. Acad. Sci. URSS 5 (946), A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and series, Vol. I: Elementary functions, Vol.:Special functions, GordonandBreach,NewYork, I.N. Sneddon, The use of integral transforms, McGraw Hill, New York, E.C. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, S.B. Yakubovich, Index transforms, World Scientific Publ. Co., Singapore, 996.

18 S.B. YAKUBOVICH 8., Index transforms and convolution method, Dr. Sc. Thesis, Minsk, S.B. Yakubovich and B. Fisher, On the Kontorovich-Lebedev transformation on distributions, Proc. Amer. Math. Soc. (994), , A class of index transforms with general kernels, Math.Nachr. (999), S.B. Yakubovich and Yu.F. Luchko, The hypergeometric approach to integral transforms and convolutions, Math. Appl., vol. 87, Kluwer Acad. Publ., Dordrecht, 994. Department of Pure Mathematics, Faculty of Sciences, University of Porto, Campo Alegre St., 687, Porto, Portugal address: syakubov@fc.up.pt