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Communications in Partial Differential Equations, 34: 233 256, 29 Copyright Taylor & Francis Group, LLC ISSN 36-532 print/532-433 online DOI:.8/36539282384 Local Smoothing for the Backscattering Transform INGRID BELTIŢĂ AND ANDERS MELIN 2 Institute of Mathematics Simion Stoilow of the Romanian Academy, Bucharest, Romania 2 Center for Mathematical Sciences, Lund University, Lund, Sweden. Introduction An analysis of the backscattering data for the Schrödinger operator in odd dimensions n 3 motivates the introduction of the backscattering transform B C n C n. This is an entire analytic mapping and we write Bv = B N v where B N v is the N th order term in the power series expansion at v =. In this paper we study estimates for B N v in H s spaces, and prove that Bv is entire analytic in v H s E when s n 3 /2. Keywords Backscattering; Scattering matrix; Ultra-hyperbolic operator; Wave equation. Mathematics Subject Classification Primary 35R3; Secondary 8U4. The present paper is devoted to proving continuity and smoothing properties of the backscattering transform for the Schrödinger operator in odd dimensions n>. In order to state the main result a brief description of the mathematical objects involved is necessary. (The reader is referred to Chapter 4 in [4] and [7 9] for details.) Consider the Schrödinger operator H v = + v in n, where v L 2 cpt n is real. Assume that H v with domain H 2 n is self-adjoint. Then the wave operators W ± = lim t ± eith v e ith exist and are complete. Let S = W + W be the scattering operator and denote A = S I. The distribution kernel of the operator F AF, where F denotes the Fourier transform, is of the form 2i 2 2 Received December 28, 27; Accepted January 4, 29 Address correspondence to Ingrid Beltiţă, Institute of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box -764, Bucharest, Romania; E-mail: Ingrid.Beltita@imar.ro 233
234 Beltiţă and Melin where is a smooth function of with values in L 2 loc n. In turn, when k>,, n, k k is a constant times ka k, where a k is the classical far field pattern associated to H v. Hence is given by the classical backscattering amplitude. It was proved in [9] that when v C n the inverse Fourier transform of the distribution 2 n /2 /2 is equal to 2 n v x y W + x y x + y dy (.) where W + x y is the distribution kernel of W +. The real part of the expression in (.) is equal to v x = 2 n v x y W x y x + y dy where the operator W = W + + W /2 has a real-valued distribution kernel. Hence v is the real part of the inverse Fourier transform of the distribution 2 n /2 /2. The backscattering transform Bv of v C n is a slight modification of v. Let K v t be the wave group associated to the operator v = 2 t x + v i.e., u x t = K v t f x is, for every f C n, the unique solution in C L 2 n to the Cauchy problem v u x t = u x = t u x = f x Then K v t is a strongly continuous function of t with values in the space of bounded linear operators on L 2 n. (See [8] for details.) We have that x y t in the support of the distribution kernel K v x y t of K v and x y =t in the support of K x y t. This ensures that the operator G = K v t v K t dt is well defined and continuous on L 2 cpt n, where the dot denotes derivative in the variable t. Theorem 7. in [9] gives the relation between G and W above: there exists an orthonormal basis f j j of real eigenfunctions corresponding to the negative part of the spectrum of H v and a set g j j of smooth real-valued functions such that W = I + G + f j g j It turns out (see below) that G = G v, considered as function of v with values in the space of continuous linear operators in L 2 cpt n, extends to an entire analytic function of v C n, i.e., to the space of complex-valued v in C. Also, if v is sufficiently small (in a sense that we do not make precise here), there are no bound
Local Smoothing 235 states and W = I + G then. For these reasons it is natural to modify the definition of v by subtracting the contribution from f j g j. Definition. Assume v C n, not necessarily real-valued. The backscattering transform Bv of v is defined by Bv x = v x + 2 n v x y G x y x + y dy Here the integral is taken in distribution sense and v x G x y is the distribution kernel of the operator vg. It was proved in [8] that G = G v extends to an entire analytic function of v L q cpt n when q>n. For such v we can define Bv again as in the previous definition and Bv will be entire analytic in v with values in n. We write Bv = B N v (.2) where B N v is the N th order term in the power series expansion at v =. There are other spaces of v (containing C as a dense subset) to which Bv can be extended analytically. Such extensions can be studied by deriving a priori estimates for the B N v when v C. In this paper we shall study estimates for B N v in H s spaces, and prove that Bv is entire analytic in v H s E when s n 3 /2. We remark that the discussion before the definition above shows when v C n that v, i.e., the real part of the inverse Fourier transform of 2 n /2 /2, equals Bv + 2 n v y f j y g j 2x y dy Hence our backscattering transformation is, modulo a correction term associated to the negative spectrum of H v, the inverse Fourier transformed version of the restricted backscattering mapping introduced in []. The latter mapping is real analytic in a dense open set only of potentials, while removing the correction term gives rise to a mapping that can be continued to an entire analytic mapping in appropriate spaces of complex potentials. We recall some basic ingredients in the construction of Bv when v C n. We recall from [8], or Section in [9], that K v t = N N K N t (.3) where K N are inductively defined by K t = sin t D D K N t = K N vk t = t K N s vk t s ds N (.4)
236 Beltiţă and Melin One has the estimate K N t L 2 L 2 v N L t2n + / 2N +! Since the distribution kernel K N x y t of K N t is supported in the set where x y t, it makes sense to consider G N = N K N t v K t dt (.5) This is a continuous linear operator in L 2 cpt n, and the estimates for the K N show that G = G N is an entire analytic function of v. We see that B N v x = 2 n v y G N y 2x y dy N 2 (.6) The following theorem (Theorem 8, [8]) reveals the smoothing properties of B N for large N. Theorem.. Let q>n and k be a nonnegative integer. Then there is a positive integer N = N n q k such that k B N v L 2 loc n when v L q n has compact support and N N. Moreover, if R R 2 >, there is a constant C, depending on n, k, R, R 2 and q only such that k B N v L 2 B R C N v N Lq/N! N whenever v L q n has support in the ball B R 2. N The aim of this paper is to give formulas for B N and study their (local) continuity properties in H s spaces. Let s denote the norm in the Sobolev space H s n. Also H s, s, is the space of functions which are restrictions to of functions from the Sobolev space H s n, when is an open set with smooth boundary. The norm in H s, s, is equivalent to the quotient norm f H s = inf { F s F H s n F = f in } Our main result here is the next theorem. Theorem.2. Assume a is nonnegative, s n 3 /2, and let N a s be the smallest integer N such that a<n and a N s n 3 /2. Then there is a constant C, which depends on n, s and a only, such that B N v H s+a B R C N R N /2 N N/2 v N s when N N a s, R> and v C B R.
Local Smoothing 237 A more refined form of this result is given in Theorem 4., where the estimates are given for the N -linear operators associated to B N. A first corollary is the above-mentioned analyticity of the backscattering transformation. Corollary.3. The mapping C n v Bv C n extends to an entire analytic mapping from H s n E n to H s loc n whenever s n 3 /2. Summarizing, we have defined a backscattering transformation, which appears as a natural generalization to higher dimensions of a transform arising in the Gelfand Levitan theory (see Section 4 of [9]). Apart from a contribution from the bound states, it is the real part of the inverse Fourier transform of the antidiagonal part of the far field pattern viewed as a function of the momentum variable. This transformation is entire analytic between certain spaces of functions. We show that the N -linear operator associated to the N th order term B N appearing in the power series expansion (.2) is a singular integral operator, and its distribution kernel is computed. We give local smoothing estimates for each of these operators. Some other consequences of the main result, concerning the regularity of the difference between v and its backscattering transform and local uniqueness properties of B, are given in the last section of the paper. The outline of this paper is as follows. In the next section we derive a formula that generalizes to arbitrary N>2 the formula ( B 2 v x = E 2 y y 2 v n 2 x y 2 y 2 ) ( v x y ) + y 2 dy 2 dy 2 which appears in Corollary.7 of [9]. Here E 2 is the unique fundamental solution of the ultra-hyperbolic operator x y such that E 2 x y = E 2 y x and E 2 is rotation invariant separately in x and y. When N>2 we have to replace E 2 by a distribution E N n N which is a fundamental solution of the operator P N = xn x xn x2 xn xn. The distribution E N is discussed in more detail in Section 3. Once these formulas have been obtained, the proof of the theorem becomes elementary. The third section contains estimates of the Fourier transforms of (cutoffs of) E N. These are in turn used in the fourth section when the estimates in Theorem.2 are obtained by Fourier transforming the formula for B N v. We close this presentation with a few words on the existing literature on backscattering problems for the potential scattering in odd dimensions. The classical complex backscattering map was studied in [] for dimension 3 and in [2] for arbitrary dimensions. The main result in [2] shows that the mapping that associates to the Fourier transform of v is a continuously Fréchet differentiable, hence analytic, map in a open dense subset of a weighted Hölder space (with weight of the form + N with N>n 2) with values in the same space, and its Fréchet derivative at ˆv is an isomorphism when ˆv belongs to open dense subset 2 of. No power series expansion is given for the classical backscattering map. For the same map, generic uniqueness was proved in [3] for compactly supported bounded potentials in dimension 3. The problem of recovering the singularities of v from the classical backscattering data was considered in [3, 5,, 2]. Let us mention the results in [2], where the cases of dimensions n = 2, 3 are considered. In the case
238 Beltiţă and Melin n = 3, these results imply that, for L 2 compactly supported potentials, the difference between the potential and the Born series approximation of the Fourier transform of the backscattering data belongs, modulo a smooth function, to the Sobolev space H with </2, and thus the regularity improves independently of the regularity of the potential. From the point of view of regularity our results are better, in the case n = 3, only for potentials in H s with s /2 and compact support. However, no continuity estimates are given. We also mention [, 4] for an approach using Lax Phillips scattering. Melrose and Uhlmann consider a generalized backscattering transform for compactly supported potentials in H n+ /2 and prove that this is entire analytic and globally Fredholm of index zero. As a result they obtain generic local injectivity for the backscattering operator. The actual backscattering transformation defined as above was considered in [6] for dimension 3. Then it was proved to be analytic when defined on small potentials v such that v L and with values in the same space, and consequently uniqueness for the inverse backscattering problem was obtained for small potentials in this space. Finally, let us fix some notation we use throughout the paper. If N 2we use the notation x = x x N m N where x x N m, for m a positive integer. If x m we shall set x = + x 2 /2. The Fourier transform of a distribution u will be denoted either by û or by F u. 2. A Formula for B N In this section we are going to write B N v as the value at v v of a N -linear operator defined from C n C n to C n, following the procedure in [8]. The key point here is the fact that K t obeys Huygens principle, more specifically, that its convolution kernel k x t is supported in the set where x =t. When N = 2 we define Q N n N + inductively by Q N x x N t = Then the mapping t Q x t = k x t (2.) Q N x x N t s Q x N s ds when N 2 (2.2) + t Q N x x N t ( n N ) is smooth when N. It is easily seen that Q N is symmetric in x x N, rotation invariant separately in these variables and, since x =t in the support of k x t, it follows that Next we define E N n N, N 2, by x + + x N =t in supp Q N (2.3) E N x x N = N Q N x x N t k x N t dt (2.4) It follows from (2.3) that x + + x N = x N in supp E N (2.5)
Local Smoothing 239 E N is rotation invariant separately in all variables, and symmetric in x x N. We recall here that E 2 x y = 4 i n n 2 x 2 y 2 in n n is the unique fundamental solution of the ultra-hyperbolic operator x y such that E 2 x y = E 2 y x and E 2 is rotation invariant separately in x and y. (See Theorem.4 and Corollary.2 in [9].) The next lemma follows easily from (.4) and (.5) by induction and some simple computations. Lemma 2.. Assume v C n. Then K N x y t = v x v x N Q N + x x x x 2 x N x N x N y t d x (2.6) G N x y = v x v x N E N + x x x x 2 x N x N x N y d x n N (2.7) for every N. Proposition 2.2. For N 2 ( B N v x = E N y y N v x y ) N n N 2 Y ( v x y ) ( N 2 Y v x y ) N 2 Y N d y when v C n, where Proof. Y = 2 N y j j= and Y k = Y k y j j= k N We use (2.7) to express G N y 2x y in (.6) and get thus B N v x = 2 n n n N v y v x v x N E N y x x x 2 x N 2 x N x N + y 2x dy d x The proposition follows by changing variables y x = y, x x 2 = y 2 x N 2 x N = y N, x N + y 2x = y N, hence y = x 2 N j= y j = x y N 2 Y x = y + y = x y N 2 Y x N = x N 2 + y N = x y N 2 Y N
24 Beltiţă and Melin Here we have made use of the invariance properties of E N, which in particular ensure that E N y y N is even in each y j. 3. The Distribution E N We need some further information on the distribution E N defined in (2.4). The first result is a characterization of E N. We denote P N = N N N where j in the Laplacian in the variables x j. Lemma 3.. The distribution E N is a fundamental solution of P N. It has the following properties: (i) E N x x N is rotation invariant in each x j ; (ii) x + + x N = x N in the support of E N ; (iii) E N is homogeneous of degree 2 N nn. If E is a fundamental solution of P N that satisfies (i) (iii), then E = E N. Proof. We first prove that P N E N = x x N, and when doing this we may assume that N 3. Since 2 t k x t = x k x t, it follows easily from (2.2) with N replaced by N that 2 t Q N x x N t = N Q N x x N t + Q N 2 x x N 2 t x N It follows from (2.4) then that N E N x x N = N Q N x x N t 2 k t x N t dt = N 2 t Q N x x N t k x N t dt = N N Q N x x N t k x N t dt + N Q N 2 x x N 2 t x N k x N t dt = N E N x x N E N x x N 2 x N x N We have proved therefore that N N E N x x N = E N x x N 2 x N x N (3.) Assuming, as we may, that the assertion has been proved for lower values of N and letting N N 2 N act on both sides of (3.) we may conclude that P N E N x x N = x x N. The conditions (i) and (ii) are simple consequences of the definitions, together with the fact that k x t is rotation invariant in x and supported in the set where
Local Smoothing 24 x =t. Since k is homogeneous when considered as a distribution in x and t, it follows that E N is a homogeneous distribution. Its degree of homogeneity must be equal to the degree of P N minus the dimension of n N. This proves (iii). It remains to prove that = if = x x N is a distribution satisfying the conditions in (i) (iii) and P N =. Define x x N = N N 2 N x x N (with the interpretation = E 2 degree 2 nn and if N = 2). This a homogeneous distribution of N N = Since is rotation invariant in each x j, it follows from Theorem. of [9] that is symmetric in x N x N. Since x + + x N = x N in the support of this implies that x = =x N 2 = in its support. Hence x x N = x x N 2 u x N x N where the u x y n n are solutions to the ultra-hyperbolic equation. The rotation invariance of in the x j implies that the summation takes place over even only and that the u x y are rotation invariant separately in x and y. Also, u x y = u y x and u is homogeneous of degree, where = 2 nn + N 2 n + =2 + 2n is even. Since > 2n the proof is completed if we prove that u vanishes outside the origin in n n. In this set we may view u as a function f s t in s = x, t = y. Since it is supported in the set where s = t we may write f s t = c j j s t s + t j+ j J where = + is odd, and the summation takes place over even j only, since f s t = f t s. We assume that f and shall see that this leads to a contradiction. Assume now that c J. Expressing the Laplacian in polar coordinates, we get the equation = ( 2 s 2 t + n s s t t ) f s t The right-hand side here is a linear combination of j s t s + t j+ 2 with j J +, and a simple computation shows that the coefficient in front of J+ s + t J+ is equal to 4c J, where = J + + n This gives us a contradiction, since we know that = while the right-hand side above is an odd integer. We have proved therefore that u vanishes outside the origin.
242 Beltiţă and Melin We need to establish estimates for the Fourier transforms of certain cut-offs of E N. Namely, we shall consider distributions of the form N Q N x x N t k x N t t dt N = 2 3 (3.2) where C. We notice that x + + x N = x N <R in the support of this distribution whenever the support of is contained in the interval R. Also, if t = when t R, then the restrictions to n N B R of the distribution in (3.2) and of E N coincide. We start with some preparatory computations. When a define a t = Y + t sin ta a t where Y + is the Heaviside s function. Lemma 3.2. Assume N 2 and a a N are real numbers such that a 2 j a2 k when j k. Then we have the identity Proof. a an t = N j= k j a 2 k aj t (3.3) a2 j Let > and define j t = e t aj t. A simple computation shows that If = N it follows that ˆ j = + i 2 + a 2 j ( ) N N = = + i 2 + a 2 j j= k j a 2 k a2 j + i 2 + a 2 j ( ) N = a 2 k ˆ j a2 j j= k j Hence t = ( N j= k j a 2 k a2 j ) j t The lemma then follows when tends to. Lemma 3.3. When N 2, a a N,, Re >, define F a a N = a an t cos ta N e t dt
Then F a a N = 2 ( j N Local Smoothing 243 a 2 j a N i 2 + j N a 2 j a N + i 2 ) (3.4) Proof. Since both sides of (3.4) depend continuously on a a N it is no restriction to assume that a 2 j a2 k when j k. First notice that when a, b and, Re >, one has a t cos tb e t dt = When N = 2 then (3.4) follows directly from this formula. Assume N 3. The previous lemma and (3.5) give a 2 b 2 + 2 (3.5) a 2 b 2 + 2 2 + 4b 2 2 F a a N = = N j= k j N j= k j We can simplify this expression by writing Then F a a N = a 2 k aj t cos ta N e t dt a2 j ( ) a 2 j a2 N a 2 k ( + 2 a2 j a 2 j a 2 N + 2) 2 + 4a 2 N 2 t j = a 2 j a2 N + 2 j N and b = 2a N = 2 = 2 N j= ( N j= k j ( t k t j k j j N t k t j ) tj t 2 j + b 2 t j ib + 2 ) t j ib + 2 j N N j= ( t j + ib k j k N t k t j ) t j + ib This finishes the proof of the lemma, after noticing that t j ± ib = a 2 j a N i 2. The next lemma is a direct consequence of Theorem.4.2 in [4]. Lemma 3.4. There is a sequence N in C such that N t = when t, N t = when t > 2 and k N t C k N k k 2N + 2 Here C> is independent of N.
244 Beltiţă and Melin In what follows R is an arbitrary positive number. We set N R t = N t/r, so that N = N. We define and E N R = N Q N x x N t k x N t N R t dt N = 2 3 (3.6) We notice that x + + x N = x N 2R in supp E N R (3.7) E N R x x N = E N x x N when x N R (3.8) We shall derive estimates for the Fourier transform F E N R N of E N R. We notice here that, due to the homogeneity of Q N t and of k t and to the definition of N R, we have It follows that E N R Rx Rx N = R 2N 2 R Nn E N x x N F E N R N = R 2N 2 F E N R R N (3.9) Therefore it is enough to establish estimates for F E N. The distribution E N x x N is rotation invariant in the variables x x N and compactly supported. The Fourier transform F E N N of E N is smooth and rotation invariant in each variable j. We define F N r r N when r j by F E N N = F N r r N when r j = j (3.) Hence we need estimates of F N. Consider >. Let us define the functions h r s through h r s = + r s + r + s Lemma 3.5. When s, t, one has + s t + s + t Consequently h s r + t 2 + t 2 h s r when s, t, r. Proof. The lemma follows from the inequalities + s t + s t + t + s t = + s + t + t
Local Smoothing 245 The estimate of F N that we need is contained in the next lemma. Lemma 3.6. There is a constant C, which does not depend on N and, such that F N r r N C N N 2N + 2N + e 2 h r j r N (3.) j N Proof. It follows from (3.6) and (2.2) that F N r r N = N N r r N t N t dt (3.2) where N r r N t = r rn t cos tr N As a function of t, N r r N t is supported in and of polynomial growth at infinity. Define Then N r r N t = e t N r r N t N t = e t N t F N r r N = N r r N t N t dt = 2 F N r r N F N d (3.3) where the Fourier transform is taken in the variable t. We notice that F N r r N = N r r N t e t dt = F r r N = + i with F as in Lemma 3.3. Then an application of that lemma gives the estimate F N r r N 2 = 2 + 2 2 N 2 r 2 j r N + i 2 + 2 j N r 2 j r N i 2 j N r j r N i r j + r N + i j N r j r N + i r j + r N + + i j N j N + r j r N + r j + r N + 2 N 2 + r j r N + + r j + r N + j N = 2 N 2 h r j r N + 2 N 2 j N j N h r j r N +
246 Beltiţă and Melin Next we see that F N = N t e t +i dt = + i 2N +2 N 2N + 2 t e t +i dt From this and Lemma 3.4 we deduce that there is a constant C, which is independent of N and, such that F N C N N 2N +2 e 2 + 2N 2 Then (3.3), (3.4) and the above inequality, together with Lemma 3.5, give ( ) F N r r N C N N 2N +2 e 2 + 2N 2 h r j r N d This finishes the proof. j N ( )( C N N 2N +2 2 N e 2 + 4 d C N 2N + N 2N +2 e 2 2C N 2N + N 2N + e 2 j N j N h r j r N h r j r N j N ) h r j r N The following theorem gives the estimate we need for the Fourier transform of E N R. Theorem 3.7. There is a constant C>, which depends on n only, such that F E N R N C N N/ R 2N + e 2R j N h j N N n for every N 2, R> and >. Proof. Let R>. The identity (3.9) and previous lemma show that there is a constant C>, which depends on n only, such that F E N R N C N N/ 2N + R 2N 2 e 2 j N h R j R N when N n, for every N 2, R> and >. This in turn shows that, with the same C, one has F E N R N C N N/ 2N + e 2 j N h /R j N The theorem follows by replacing /R by.
Local Smoothing 247 4. Proof of the Main Result and Some Consequences We introduce an N -linear version of B N, N 2. Namely, for v = v v N, v j C n, define B N v x = E N y y N n N ( (4.) v x y ( N 2 Y )v 2 x y ) ( N 2 Y v N x y ) N 2 Y N d y Here the Y k s are defined as in Proposition 2.2, that is, Y = 2 N y j k Y k = Y y j k = N j= Then B N v is a smooth compactly supported function in n and B N v = B N v v for every v C n. Therefore the result in Theorem.2 is contained in the next theorem. Here and in the rest of the section we use the notation m = n 3 2 Theorem 4.. Assume that < <, s j m and a j = min s j m, j = N, where N 2. Set = min s j a j + N a j Then there is a constant C which is independent of N and the s j, but may depend on and n, such that j= B N v 2 H B R CN N 2 min s j a j m R/N N N v j 2 s j (4.2) for every R>, v v N C B R. The present section is devoted to the proof of the above result and a presentation of some of its consequences. We start with some preparations. Let R> and recall that the distributions E N R E n N were defined in (3.6). When v = v v N, v j C n, we consider B N R v x = E N R y y N n ( N (4.3) v x y ( N 2 Y )v 2 x y ) ( N 2 Y v N x y ) N 2 Y N d y It is easy to see that B N R v is a smooth compactly supported function in n. The following lemma gives the connection between B N R v and B N v.
248 Beltiţă and Melin Lemma 4.2. Assume v v N C B R. Then B N 4R v x = B N v x in a neighborhood of B R and B N 2 N R v = B N v. Proof. Choose > such that the v j are supported in B R and define V x y = v x y N /2 Y v N x y N /2 Y N (4.4) Since Y + Y N =, it follows that 2x y N = x y N /2 Y + x y N /2 Y N 2R 2 when y supp V x. When x <R+ /2 we see that y N < 4R when y is in the support of V x and, since E N 4R = E N when y N < 4R, it follows from (4.3) that B N 4R v x = B N v x when x <R+ /2. This proves the first assertion. When proving the second assertion we notice that y j = x y N /2 Y j x y N /2 Y j < 2R when j N and y supp V x, hence N y j < 2 N R. This shows that the support of V x does not intersect the support of E N 2 N R E N, hence B N 2 N R v = B N v. Let s = s s N be a sequence of nonnegative real numbers and let, N 2, R>. Define where A N R s = sup N + 4 N 2 F E N R N 2 M s N 2 d d N (4.5) M s N = + N s 2 s2 N N s N Then A N R s. Lemma 4.3. We have that B N R v 2 2 n N A N R s s N N v j 2 s j (4.6) for every v j C n, j N. Proof. Let V x be defined as in (4.4). In order to compute the Fourier transform of V x we introduce the linear map L in n N through L z = y, where y j = z j z j+ j N y N = z + z N
Local Smoothing 249 It is easily seen that det L = 2 n and that y N /2 + Y j = z j when j N. Therefore we may write Hence V x y = v v N L y y N y N 2x F V x N = 2 n e 2i x N ˆv ˆv N L Here L denotes the transpose of L. It is easy to see that L =, where = + N j = j j 2 j N It follows that F V x N = 2 n e 2i x N ˆv + N ˆv 2 2 ˆv N N N Write w j = F D s j vj and W = w + N w 2 2 w N N N It follows from (4.3) and the computations above that B N R v x = 2 nn F E N R F V x d = 2 nn 2 n e 2i x N N d N = 2 nn e i x N N /2 d N where N = = F E N R ˆv + N ˆv 2 2 ˆv N N N d d N F E N R M s W d d N This shows that B N R v 2 = 2 2n N /2 N 2 N /2 2 d N = 2 n 2 2n N /2 + 4 N 2 N 2 d N { 2 n 2 2n N /2 + 4 n 2 (4.7) ( ) F E N R 2 M s 2 d d N ( W 2 d d N )}d N 2 n 2 2n N /2 A N R s W 2 d
25 Beltiţă and Melin The proof is then completed by the observation that W = w w 2 w N L It follows that W 2 d = 2 n w w N 2 d N = 2 n w j 2 = 2 n 2 nn N D s j v j 2 = 2 n 2 nn N v j 2 s j The lemma follows if this is inserted into (4.7). We shall arrive at estimates for B N R v by combining the inequality (4.6) with estimates for the expression A N R s in (4.5). The following lemma will be needed. Lemma 4.4. Assume < <. Then there is a constant C = C n such that h 2 2s d C 2m 2s (4.8) when n,, >, m s m +. Proof. Assume r> and n \. Set f s r = r 2s d n If u = / then a simple computation shows that r 2 + r 2 u If f is a continuous function, and c n 2 is the area of the n 2-dimensional unit sphere, then n f / d = c n 2 f t t 2 m dt This shows that f s r c n 2 + r 2 t s t 2 m dt 2 m+ c n 2 + r 2 t s t m dt 2 m+ c n 2 + r 2 t s t m dt 2 m+ c n 2 r 2s t m s dt This gives the estimate f s r C r 2s (4.9) where C = 2 m+ c n 2 /, for n \. This inequality clearly holds for = as well.
Local Smoothing 25 Using (4.9) and introducing polar coordinates in the integration one gets h 2 2s d C h 2 r r2m+2 r 2s dr (4.) Assume first that. Then h 2 r r2m+2 r 2s dr = Assume next that <. Then r 2m+2 + r 2 + r + 2 r 2 + dr s r 2m+2 dr (4.) 2 s m + r 2 r 2 + m+ 2 2 s m + 2 s m + r dr 2 23 2 s m h 2 r r2m+2 r 2s dr r 2m + r 2 r 2 + dr dr (4.2) s + r 2 2 2 s m + 2 + s m 2 3 2 s m Combining (4.) (4.2) we see that the lemma holds with C = 2 3 C. Now we are going to estimate A N R s. Recall that Theorem 3.7 gives that F E N R N C N N/ R 2N + e 2R j N h j N (4.3) where >, N 2, R> and the constant C is independent of these parameters. We notice that M s N 2 s 2 + N s M s2 s N 2 N + 2 s 2 s 2 M s s 3 s N 2 N (4.4) In fact, since 2 + N 2 + + N either 2 2 + N /2 or + N 2 + N /2. In the first case M s N 2 s 2 + N s M s2 s N 2 N and in the second case M s N 2 s 2 s 2 M s s 3 s N 2 N
252 Beltiţă and Melin When N 2 we define T N s = ( j N ) 2 h j M 2 s N d d 2 d N Let < < and assume that m s j m + when j N. It follows from (4.3) and (4.5) that A N R s C N N/ R 4N +2 e 4R sup 2 2 T N s (4.5) Here, and in what follows, C denotes constants that are independent of N, R, s,, (but may depend on and dimension n). Assume N 3. From (4.4) follows that T N s 2 n ( h 2 + 2s d ) ( 2 j N ) h 2 j M 2 s 2 s N 2 N d 2 d N + 2 n ( h 2 2 2s 2 d ) ( 2 j N From Lemma 4.4 we get the estimate ) h 2 j M 2 s s 3 s N 2 N d 2 d N T N s C/2 2m 2s T N s 2 s N + C/2 2m 2s 2 T N s s 3 s N (4.6) Another application of Lemma 4.4 gives T 2 s s 2 = h 2 + 2s 2s 2 d 2s h 2 2s 2 d + 2s 2 h 2 + 2s d C 2m 2s 2s 2 where we may assume that C is the same constant as in (4.6). From this we deduce that the inequality T N s C N N 2 N m s s N (4.7)
Local Smoothing 253 holds when N = 2. Applying (4.6) together with an induction argument we obtain that (4.7) holds for every N 2. Since m s j <m+ we get (with another C) T N s C N N 2 2 N m s s N (4.8) Assume now that s j m, j = N, but not necessarily s j <m+, and let < <. Set a j = min s j m, j = N. We notice that and therefore It follows that + N + 2 + + N N 2 N max + N 2 N N 2 N /N + N 2 N N + 4 N 2 /N 2 /2 N 2 N Then we may write M s N min s j a j +m 2 N min s j a j +m M m+a m+a N This implies that T N s N 2 min s j a j +m 2 2 min s j a j +m T N m + a m+ a N Then (4.8) gives T N s C N N 2 min s j a j +m N 2 2 min s j a j 2 N a j (4.9) Combining (4.5) with (4.9) we get the following lemma. Lemma 4.5. Assume that < <, s j m and a j = min s j m, j = N. Set = min s j a j + N a j Then there is a constant C which is independent of N and the s j, but may depend on and n, such that for every > and R>. j= A N R s C N N 2 min s j a j m N/ R 4N +2 N e 4R (4.2) Next we recall (4.6) which, together with the previous lemma, gives the next proposition.
254 Beltiţă and Melin Proposition 4.6. Assume that < <, s j m and a j = min s j m, j = N, where N 2. Set = min s j a j + N a j Then there is a constant C which is independent of N and the s j, but may depend on and n, such that B N R v 2 CN N 2 min s j a j m N/ R 4N +2 N e 4R N for every v v N C n, R> and >. j= v j 2 s j (4.2) Theorem 4. follows from the previous proposition and Lemma 4.2, by replacing R by 4R and taking = N/ 4R. When replacing R by 2 N R and taking = /R, we obtain the following corollary, where we use Lemma 4.2. Corollary 4.7. Assume that < <, s j m and a j = min s j m, j = N. Set = min s j a j + N a j Then there is a constant C, which depends on n, and the s j only, such that B N v 2 CN R N N for every N 2, R> and v v N C B R. j= v j 2 s j (4.22) We conclude this section by giving some consequences of our main result. The first one, the analyticity of the backscattering transformation, has already been stated in Corollary.3. A second consequence gives the difference of regularity between v and its backscattering transform. Corollary 4.8. Assume s n 3 /2 and a< satisfy a s n 3 /2. Ifv H s n is compactly supported, then v Bv H s+a loc n (4.23) Let now R> be arbitrary fixed. When s we denote by Ḣ s B R the closure of C B R in the H s B R -norm. This is a closed subspace of the Hilbert space H s B R, and let s R H s B R Ḣ s B R be the associated orthogonal projection. The next consequence gives generic local uniqueness for the backscattering transformation. The arguments are quite similar to the arguments in [, 4], used in the proofs of the corresponding results for the backscattering mapping defined in terms of the Lax Phillips scattering.
Local Smoothing 255 When s> n 3 /2 define B R v Ḣ s B R Ḣ s B R B R v = s R Bv B R v Ḣ s B R (4.24) Corollary 4.9. There is a closed subset G s R of Ḣ s B R such that each complex line through the origin of Ḣ s B R intersects G s R in a discrete set only and B R is a local isomorphism at every v not in G s R. In particular, if v Ḣ s B R \ G s R, there is a neighborhood U of v in Ḣ s B R such that v = v 2 if v, v 2 U and Bv = Bv 2 on B R. Proof. The mapping v B R v is entire analytic on Ḣ s B R, according to Theorem 4.. The same theorem shows that the derivative of this mapping at a point v Ḣ s B R has the form I + T v, where T v is a compact operator on Ḣ s B R. Let G s R be the set of those v such that I + T v is not invertible; this is a closed set. By the inverse mapping theorem (see Theorem.2.3, [5]), B R is a local isomorphism at every v not in G s R. The estimates in Theorem 4. show that operator I + T zv is invertible for z small enough. Since z T zv is entire analytic, the analytic Fredholm alternative (see Theorem.8.2 in [6]) ensures that the inversibility can fail only for z in a discrete set. This finishes the proof. Acknowledgments Ingrid Beltiţă was partially supported by SPECT Short Visit Grant 6 and the Grant 2-CEx5--23/25. References [] Eskin, G., Ralston, J. (989). The inverse backscattering problem in three dimensions. Comm. Math. Phys. 24:69 25. [2] Eskin, G., Ralston, J. (992). Inverse backscattering. J. D analyse Mathématique 58:77 9. [3] Greenleaf, A., Uhlmann, G. (993). Recovery of singularities of a potential from the singularities of the scattering data. Comm. Math. Phys. 57:549 572. [4] Hörmander, L. (983, 985). The Analysis of Linear Partial Differential Operators I IV. Grandlehren der Mathematischen Wissenschaften 256, 257, 274, 275. Berlin: Springer- Verlag. [5] Joshi, M. S. (998). Recovering the total singularity of a conormal potential from backscattering data. Ann. Inst. Fourier, Grenoble 48:53 532. [6] Lagergren, R. (2). The Back-Scattering Problem in Three Dimensions. PhD Thesis, Lund University, Centre for Mathematical Studies. [7] Melin, A. (999). Back-scattering and nonlinear Radon transform. Séminaire sur les Équations aux Derivées Partielles, 998 999, Exp. no. XIV. Palaiseau, France: École Polytechnique. [8] Melin, A. (23). Smoothness of higher order terms in backscattering. In: Wave Phenomena and Asymptotic Analysis. RIMS Kokyuroku 35, Kyoto, Japan: RIMS, pp. 43 5. [9] Melin, A. (24). Some transforms in potential scattering in odd dimension. Inverse problems and spectral theory. Contemp. Math. 348:3 34.
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