ON THE BEHAVIOR OF THE SOLUTION OF THE WAVE EQUATION. 1. Introduction. = u. x 2 j

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1 ON THE BEHAVIO OF THE SOLUTION OF THE WAVE EQUATION HENDA GUNAWAN AND WONO SETYA BUDHI Abstract. We shall here study some properties of the Laplace operator through its imaginary powers, and apply the results to understand the behavior of the solution of the wave equation. 1. Introduction One of the classical partial differential equation studied in mathematics and physics is the wave equation (1) u tt = n j=1 u x j = u where = n j=1, known as the Laplace operator in n. This equation has been x j studied by many authors since the eighteen century. See, for instance, [4]. Often the wave equation is coupled with the initial condition () u(x, 0) = g(x), u t (x, 0) = f(x), or some boundary conditions (which we shall not discuss here). In this note, we shall study the behavior of the solution of the wave equation subject to the Cauchy data (3) u(x, 0) = 0, u t (x, 0) = f(x), where f is, for instance, in the Lebesgue space L p ( n ) the space that consists of all (equivalence classes of) Lebesgue measurable, complex valued functions f on n for which f p := ( n f(x) p dx ) 1/p <. We shall first discuss some properties of the Laplace operator through its imaginary powers, and then apply the results to study the behavior of the solution of the wave equation with respect to the initial velocity f. 000 Mathematics Subject Classification: 4B0, 4B5. Keywords and Phrases: wave equation, Laplace operator. 1

2 . A review on Fourier transform method From (1) (), by taking the Fourier transform in x, we obtain the ordinary differential equation û tt (ξ, t) = πξ û(ξ, t) with initial condition û(ξ, 0) = ĝ(ξ), û t (ξ, 0) = f(ξ). Here we define the Fourier transform of f by f(ξ) = n f(x)e πiξ x dx. Solving the second order differential equation above, we get (4) û(ξ, t) = ĝ(ξ) cos t πξ + sin t πξ f(ξ). πξ In particular, when g is identically 0 [so that we have the Cauchy data (3)], we obtain û(ξ, t) = sin t πξ f(ξ). πξ Divide both sides by t and then take the inverse Fourier transform, we obtain (5) u(x, t) t = f S(x, t), sin t πξ where Ŝ(ξ, t) =. Here * denotes the convolution product, defined by the t πξ formula: f g(x) := f(x y)g(y) dy. n One property of the convolution product is that (f g) (ξ) = f(ξ)ĝ(ξ). instance, [4], 7.1. See, for Nothing much we can say from (5), and so in the next sections we shall study the behavior of the solution of the wave equation subject to the Cauchy data by studying the Laplace operator and its imaginary powers. 3. Laplace operators and their imaginary powers One way to have a better understanding of the behavior of the solution of the wave equation with Cauchy data, especially its dependence on the initial velocity f, is by studying the Laplace operator, as we shall do here. Let S( n ) denote the Schwartz space, consisting of sufficiently smooth functions that are rapidly decreasing at infinity (see [11]). Inspired by the relation ( f) (ξ) = πξ f(ξ), f S( n ),

3 one may define β/ for any (complex) exponent β by ( β/ f) (ξ) = (π ξ ) β f(ξ), f S( n ). In particular, for each 0 < α < n, the operator I α : f ( ) α/ f, S( n ), is known as the iesz potential. Here I α may be expressed as I α f = K α f, f S( n ), with K α (x) = π n/ α Γ ( ) n α Γ ( ) x n+α α being its kernel. 1 This tells us that I α is an integral operator. For further details, see [1], p elated to the solution of the wave equation, we shall consider the operator I iu, u \ {0}, given by which makes sense via I iu f = K iu f, f S( n ), (I iu f) (ξ) = (π ξ ) iu f(ξ), f S( n ), that is, I iu = ( ) iu/, an imaginary power of. This operator was studied by B. Muckenhoupt [10] in 1960 and used by Cowling and Mauceri [1] in 1978 to prove E.M. Stein s theorem on the spherical maximal function [13]. Note that K iu (ξ) = (π ξ ) iu = 1, so that by Plancherel s theorem we have (6) I iu f = f. Hence I iu, which is initially defined on S( n ), extends to an isometry hence a bounded operator on L ( n ), because S( n ) is dense in L ( n ). By using further properties of the kernel K iu, particularly the fact that it is locally integrable away from the origin and satisfies K iu (x) C(1 + u ) n/ x n and K iu (x) C(1 + u ) n/+1 x n 1 1 Here Γ denotes the gamma function. An operator T : L p ( n ) L p ( n ) is said to be bounded when there exists a constant C > 0 such that T f p C f p for every f L p ( n ). The norm of the operator T, denoted by T L p Lp, is defined to be the smallest constant C for which the inequality holds. 3

4 for x 0 (see [6]), one may observe that I iu also extends to a bounded operator from the Hardy space H 1 ( n ) to L 1 ( n ), that is, (7) I iu f 1 C(1 + u ) n/ f H 1 (see [7]). ecall that, for 0 < p 1, the Hardy space H p ( n ) consists of all functions f = j λ ja j where a j s are p-atoms and λ j s are real numbers such that j λ j p < C f p Hp. See [14] for further discussion of Hardy spaces. From (6) and (7), interpolation and duality arguments (see [1, 14]) give (8) I iu f p C p (1 + u ) n/p n/ f p for 1 < p <. This tells us that I iu extends to a bounded operator on L p ( n ). Therefore we have the following estimate for ( ) iu/ : Theorem 1 [8]. For each u \ {0}, we have for 1 < p <. ( ) iu/ L p L p C p(1 + u ) n/p n/, 4. Applications For applications, consider a positive self-adjoint operator L given by n Lf = j a jk k f j,k=1 where a jk C ( n ), a jk = a kj for 1 j, k n. Then L has a spectral resolution L = λ de L (λ) + where E L (λ) s are the spectral projectors (see [3]). For any bounded Borel function F : + C, define the operator F (L) by F (L) = F (λ) de L (λ). + As in [9], we define the maximal operator M F,L by Assuming F (0) = 0, we may write F (λ) = M F,L f = sup F (tl)f. t>0 4 A(u)λ iu du,

5 that is, F (λ) is the Mellin transform of A(u). 3 By the Fourier Inversion Theorem, this holds if and only if A(u) = 1 F (λ)λ 1 iu dλ. π + Lemma. If, for some 1 < p <, we have A(u) L iu L p L pdu C p, then, for any t > 0, we have and hence M F,L is bounded on L p ( n ). Proof. For any t > 0, we have F (tl) = and thus, since t iu = 1, we obtain F (tl) L p L p F (tl) L p L p C p, A(u)t iu L iu du, A(u) L iu L p L pdu C p. Since the estimate is independent of t, we conclude that that is, M F,L is bounded on L p ( n ). M F,L L p L p = sup F (tl) L p L p C p, t>0 elated to the solution of the wave equation, consider, for e(α) > 0, the function F α (λ) = 1 1 (1 x ) α 1 e iλx dx, λ +. Looking back at (4), we see that F 0 (λ) = cos λ and F 1 (λ) = sin λ are connected with λ the solution of the wave equation. Moreover, one may observe that for λ > 0 we have F α (λ) = c α λ α+1/ J α 1/ (λ) for some constant c α, while for λ = 0 we have F α (0) = Γ(α)Γ( 1 ) Γ(α + 1 ), where J ν denotes the Bessel function of order ν and Γ is the gamma function (see [4], Chapter 5). 3 The Mellin transform is just a variant of the Fourier transform, see [4]. 5

6 Note particularly that, for L =, F α (L) is a convolution operator with kernel c α (1 ) α n/ 1/ + for some constant c α depending on α. Here the symbol (1 x ) + means that it is equal to 1 x when 1 x 0, and is 0 elsewhere. To be able to apply the above result (namely, Lemma 1), we put F α(λ) = F α (λ) F α (0)e λ, so that in particular we have Fα(0) = 0. Next write Fα(λ) = A α (u)λ iu du, λ +, where A α (u) = 1 F π α(λ)λ 1 iu dλ, u. + Then we have the following lemma. [ Lemma 3. A α (u) = Γ(α)Γ( iu ) iu. 4π 1/ ] 1 Γ(α+ 1 + iu ) Γ(α+ 1 ) The proof of Lemma 3 uses the definition and basic properties of gamma functions (see [15], pp ) and the fact that the Fourier transform of 1 iu, in the distribution sense, is a multiple of iu (see [1], p. 117). See [1] for a similar result. From Lemma 3, we see that A α (u) is bounded near 0 and behaves like u e(α) 1 at infinity. Hence, we have: Corollary 4. A α (u) = O ( (1 + u ) e(α) 1 ). Using this result, we can reprove the following theorem: Theorem 5 [13]. If L =, then M Fα,L is bounded on L p ( n ) for (a) e(α) > n n + 1, 1 < p ; and p (b) e(α) > n n + 1, p. p p Proof. From the previous discussion about the (imaginary powers) of the Laplace operator, we have Theorem 1 which states that L iu L p L p = ( ) iu/ L p L p C p(1 + u ) n/p n/, for 1 < p <. Hence, by Lemma, M F α,l is bounded on L p ( n ) for 1 < p provided that e(α) > n n + 1, since for these α s we have (by Corollary 4) p A α (u) L iu L p L pdu C p (1 + u ) n/p n/ e(α) 1/ du < C p. 6

7 Now the maximal operator f sup e t L f is known to be bounded on each L p ( n ), t>0 and thus the boundedness of M Fα,L on L p ( n ) for 1 < p and those α s follows immediately from Minkowski s inequality. To prove the result for p, we argue as follows. We recall that F α (L) is actually a convolution operator with kernel c α (1 ) α n/ 1/ + for some constant c α depending on α. This fact implies that M Fα,L is bounded on L ( n ) for e(α) > n 1. Interpolating the L and L estimates, we obtain the boundedness of M Fα,L on L p ( n ) for e(α) > n n p + p 1, p. As mentioned earlier, M F1, is connected with the solution of the wave equation. Indeed, for an appropriate constant c n, one may verify that is the weak solution of the wave equation subject to the Cauchy data u(x, t) = c n tf 1 (t )f(x) u tt = u u(x, 0) = 0, u t (x, 0) = f(x). As a consequence of the above theorem, we have: Corollary 6 [14]. M F1, is bounded for (a) n < p, if n 3; and for n+1 (b) n < p < (n ), if n 4. n+1 n 3 Accordingly, the estimate sup u(, t) t>0 t C p f p p holds for the above values of p s. emark. Theorem 5 also has implications on the boundedness of the Hardy-Littlewood maximal operator M F n+1, and the Stein s spherical maximal operator M F n 1, on L p ( n ) for some values of p s (see [13]). eferences [1] M. Cowling and G. Mauceri, On maximal functions, end. Sem. Mat. Fis. Milano 49 (1979), [].. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), [3] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, London, [4] G.B. Folland, Fourier Analysis and Its Applications, Wadsworth & Boorks/Cole, Pacific Grove,

8 [5] J. García-Cuerva and J.-L. ubio de Francia, Weighted Norm Inequalities and elated Topics, North-Holland (1985). [6] H. Gunawan, On weighted estimates for Stein s maximal function, Bull. Austral. Math. Soc. 54 (1996), [7] H. Gunawan, Some weighted estimates for imaginary powers of the Laplace operator, Bull. Austral. Math. Soc. 65 (00), [8] H. Gunawan and J. Wright, Weighted estimates for some singular integrals, esearch eport (001). [9] H. Gunawan and A. Sikora, On maximal operators associated to Laplace operators, esearch eport (00). [10] B. Muckenhoupt, On certain singular integrals, Pacific J. Math. 10 (1960), [11] W. udin, Functional Analysis, McGraw-Hill Book Co., New York, [1] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (1970). [13] E.M. Stein, Maximal functions: spherical means, Proc. Nat. Acad. Sci. USA 73 (1976), [14] E.M. Stein, Harmonic Analysis: eal-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press (1993). [15] E.C. Titchmarsh, The Theory of Functions, nd ed., Oxford Univ. Press, London, Department of Mathematics, Bandung Institute of Technology, Bandung 4013, Indonesia address: hgunawan, wono@dns.math.itb.ac.id 8