CUBO A Mathematical Journal Vol.10, N o ¯ 02, (83–106). July 2008 The Hilbert Transform on a Smooth Closed Hypersurface F. Brackx and H. De Schepper Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, B–9000 Gent, Belgium email: fb@cage.ugent.be ABSTRACT In this paper a condensed account is given of results connected to the Hilbert transform on the smooth boundary of a bounded domain in Euclidean space and some of its related concepts, such as Hardy spaces and the Cauchy integral, in a Clifford analysis context. Clifford analysis, also known as the theory of monogenic functions, is a multidimensional function theory, which is at the same time a generalization of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. It offers a framework which is particularly suited for the integrated treatment of higher dimensional phenomena, without having to rely on tensorial approaches. RESUMEN En este art́ıculo damos un relato condensado de los resultados conectados con la trans- formada de Hilbert sobre dominios acotados con frontera suave en espacios euclideanos y también damos conceptos relacionados, tales como espacios de Hardy y la integral de Cauchy en el contexto del análisis de Clifford. El análisis de Clifford, también conocido 84 F. Brackx and H. De Schepper CUBO 10, 2 (2008) como la teoria de funciones monogénicas, es una teoŕıa de funciones multidimension- ales, la cual es al mismo tiempo una generalización de la teoŕıa de funciones holomorfas en el plano complejo y un refinamiento del análisis armónico clásico. El art́ıculo ofrece un referencial que es particularmente conveniente para el tratamiento integrado de fenómenos en dimensiones altas, sin tener que recurrir a un abordaje tensorial. Key words and phrases: Hilbert transform, Hardy space, Cauchy integral. Math. Subj. Class.: 30G35, 44A15 1 Introduction In one–dimensional signal processing the Hilbert transform is an indispensable tool for both global and local descriptions of a signal, yielding information on various independent signal properties. The instantaneous amplitude, phase and frequency are estimated by means of so–called quadra- ture filters, which allow for distinguishing different frequency components and in this way locally refine the structure analysis. These filters are essentially based on the notion of analytic signal, which consists of the linear combination of a bandpass filter, selecting a small part of the spectral information, and its Hilbert transform, the latter basically being the result of a phase shift by π 2 on the original filter (see e.g. [18]). Mathematically, if f(x) ∈ L2(R) is a real valued signal of finite energy, and H[f] denotes its Hilbert transform given by the Cauchy Principal Value H[f](x) = 1 π Pv ∫ +∞ −∞ f(y) x − y dy then the corresponding analytic signal is the function 1 2 f + i 2 H[f], which belongs to the Hardy space H2(R) and arises as the L2 non–tangential boundary value (NTBV) for y → 0+ of the holomorphic Cauchy integral of f in the upper half of the complex plane. Though initiated by Hilbert, the concept of a conjugated pair (f,H[f]), nowadays called a Hilbert pair, was developed mainly by Titchmarch and Hardy. The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so–called Riesz transforms in each of the Cartesian variables separately. As opposed to these tensorial approaches Clifford analysis is particularly suited for a treatment of multidimensional phenomena encompasssing all dimensions at the same time as an intrinsic feature. During the last fifty years Clifford analysis has gradually developed to a comprehensive theory offering a direct, elegant and powerful generalization to higher dimension of the theory of holomorphic functions in the complex plane. In its most simple but still useful setting, flat m–dimensional Euclidean space, Clifford analysis focusses on so–called monogenic functions, i.e. null solutions of the Clifford– vector valued Dirac operator ∂ = ∑ m j=1 ej∂xj where (e1, . . . ,em) forms an orthogonal basis for the quadratic space R m underlying the construction of the Clifford algebra R0,m. Monogenic functions CUBO 10, 2 (2008) The Hilbert Transform ... 85 have a special relationship with harmonic functions of several variables in that they are refining their properties. The reason is that, as does the Cauchy–Riemann operator in the complex plane, the rotation–invariant Dirac operator factorizes the m–dimensional Laplace operator. This has, a.o., allowed for a nice study of Hardy spaces of monogenic functions, see [7, 24, 8, 9, 1, 11]. In this context the Hilbert transform, as well as more general singular integral operators have been studied in higher dimensional Euclidean space (see [14, 24, 31, 19, 10, 12]), in particular on Lip- schitz hypersurfaces (see [25, 21, 20, 22]) and also on smooth closed hypersurfaces, in particular the unit sphere (see [11, 3, 6]). The subject of this paper is the Hilbert transform, within the Clifford analysis context, on the smooth boundary of a bounded domain in Euclidean space of dimension at least three. For the two–dimensional case we refer to the inspiring book [2]. We have gathered the relevant results spread over the literature and have moulded them together with some new results and insights into a comprehensive text. 2 Clifford analysis: the basics In this section we briefly present the basic definitions and some results of Clifford analysis which are necessary for our purpose. For an in–depth study of this higher dimensional function theory and its applications we refer to e.g. [4, 13, 14, 15, 16, 17, 26, 27, 28, 29, 30]. Let R 0,m be the real vector space R m , endowed with a non–degenerate quadratic form of signature (0,m), let (e1, . . . ,em) be an orthonormal basis for R 0,m , and let R0,m be the universal Clifford algebra constructed over R 0,m . The non–commutative multiplication in R0,m is governed by the rules eiej + ejei = −2δi,j, i,j ∈ {1, . . . ,m}. For a set A = {i1, . . . , ih} ⊂ {1, . . . ,m} with 1 ≤ i1 < i2 < ... < ih ≤ m, let eA = ei1ei2 . . .eih . Moreover, we put e ∅ = 1, the latter being the identity element. Then (eA : A ⊂ {1, . . . ,m}) is a basis for the Clifford algebra R0,m. Any a ∈ R0,m may thus be written as a = ∑ A aA eA with aA ∈ R or still as a = ∑ m k=0[a]k where [a]k = ∑ |A|=k aA eA is the so–called k–vector part of a (k = 0, 1, . . . ,m). If we denote the space of k–vectors by Rk0,m, then the Clifford algebra R0,m decomposes as ⊕ m k=0 R k 0,m. We will identify an element x = (x1, . . . ,xm) ∈ R m with the one–vector (or vector for short) x = ∑ m j=1 xj ej . The multiplication of any two vectors x and y is given by xy = x ◦ y + x ∧ y 86 F. Brackx and H. De Schepper CUBO 10, 2 (2008) with x ◦ y = − m∑ j=1 xjyj = 1 2 (xy + yx) = −〈x,y〉 x ∧ y = ∑ i ε } . By a classical argument involving Cauchy’s Theorem it is found that lim ε→0+ ∫ ∂Ωε E(ζ − ξ) dσζ = 1 2 leading to lim x→ξ C[u](x) = 1 2 u(ξ) + lim ε→0+ ∫ ∂Ωε E(ζ − ξ) dσζ u(ζ) 88 F. Brackx and H. De Schepper CUBO 10, 2 (2008) or finally lim Ω+∋x→ξ C[u](x) = 1 2 u(ξ) + 1 2 H[u](ξ), ξ ∈ ∂Ω (3.1) where we have put for ξ ∈ ∂Ω: H[u](ξ) = 2 lim ε→0+ ∫ ∂Ωε E(ζ − ξ) dσζ u(ζ) = 2 am Pv ∫ ∂Ω ζ − ξ |ζ − ξ|m dσζ u(ζ) = 2 am Pv ∫ ∂Ω ξ − ζ |ξ − ζ|m ν(ζ) u(ζ) dS(ζ) This integral transform H is mostly called the Hilbert transform; it is sometimes denoted by S∂Ω, a notation we will not use any further in this paper. Note that, in view of the example given above H[1](ξ) = 2 am Pv ∫ ∂Ω ξ − ζ |ξ − ζ|m dσζ = 1, ξ ∈ ∂Ω. Similarly we find for the exterior NTBV of the Cauchy integral: lim Ω−∋ x→ξ C[u](x) = − 1 2 u(ξ) + 1 2 H[u](ξ), ξ ∈ ∂Ω. (3.2) The obtained results (3.1)–(3.2) are the so–called Plemelj–Sokhotzki formulae, leading to the Cauchy transforms defined on C∞(∂Ω) by C + [u] = 1 2 u + 1 2 H[u], C−[u] = − 1 2 u + 1 2 H[u]. It follows that u = C + [u] − C − [u], H[u] = C + [u] + C − [u], expressing the function u ∈ C∞(∂Ω) as the jump of its Cauchy integral over the boundary ∂Ω. In section 5 the operators H and C± will be extended to operators on L2(∂Ω). 4 The double–layer potential There is a nice connection between the Cauchy integral and the related operators H and C± on the one side and the double–layer potential on ∂Ω on the other. Indeed, the splitting of the product of two Clifford vectors into the scalar valued dot product and the bivector valued wedge product allows for rewriting the Cauchy integral of the function u ∈ C∞(∂Ω) as C[u](x) = 1 am ∫ ∂Ω (ζ − x) ◦ ν(ζ) |ζ − x|m u(ζ)dS(ζ) + 1 am ∫ ∂Ω (ζ − x) ∧ ν(ζ) |ζ − x|m u(ζ)dS(ζ) (4.1) Denoting by −→ x the geometric vector associated with the Clifford vector x, we have that (ζ − x) ◦ ν(ζ) = 〈 −→ ζ − −→ x , −→ ν (ζ)〉, where 〈., .〉 here denotes the standard Euclidean scalar product. As −→ ∇−→ ζ 1 | −→ ζ − −→ x |m−2 = − (m − 2) −→ ζ − −→ x | −→ ζ − −→ x |m CUBO 10, 2 (2008) The Hilbert Transform ... 89 we have (ζ − x) ◦ ν(ζ) |ζ − x|m = − 1 m − 2 〈 −→ ∇ζ 1 | −→ ζ − −→ x |m−2 , −→ ν (ζ)〉 = − 1 m − 2 ∂ ∂ −→ ν ( 1 | −→ ζ − −→ x |m−2 ) This first term in the Cauchy integral (4.1) then takes the form − 1 m − 2 1 am ∫ ∂Ω ∂ ∂ −→ ν ( 1 | −→ ζ − −→ x |m−2 ) u(ζ) dS(ζ) in which one recognizes, up to constants, the double–layer potential with density u(ζ) on ∂Ω. Note that for the constant density u = 1 on ∂Ω, one has ∫ ∂Ω (ζ − x) ∧ ν(ζ) |ζ − x|m dS(ζ) = 0, x ∈ Ω + ∪ Ω − (4.2) whence ∫ ∂Ω ∂ ∂ −→ ν ( 1 |ζ − x|m−2 ) dS(ζ) = −(m − 2)am, x ∈ Ω + (4.3) ∫ ∂Ω ∂ ∂ −→ ν ( 1 |ζ − x|m−2 ) dS(ζ) = 0, x ∈ Ω− (4.4) confirming known results about Gauß’s Integral (see e.g. [23, p.360]). It is well–known from classical potential theory that the double–layer potential is harmonic in Ω + ∪ Ω−. Under the assumptions made on the region Ω and the function u, the double–layer potential is even defined for ξ ∈ ∂Ω; the value at ξ ∈ ∂Ω is called its direct value and denoted by W̃(ξ). This function W̃(ξ) is a continuous function on ∂Ω and moreover one has (see e.g. [23, p.360]): lim Ω+∋ x→ξ ∫ ∂Ω ∂ ∂ −→ ν ( 1 |ζ − x|m−2 ) u(ζ)dS(ζ) = − 1 2 (m − 2)amu(ξ) + W̃(ξ) lim Ω−∋ x→ξ ∫ ∂Ω ∂ ∂ −→ ν ( 1 |ζ − x|m−2 ) u(ζ)dS(ζ) = 1 2 (m − 2)amu(ξ) + W̃(ξ) with W̃(ξ) = −(m − 2) ∫ ∂Ω (ξ − ζ) ◦ ν(ζ) |ξ − ζ|m u(ζ) dS(ζ), ξ ∈ ∂Ω. It follows that the Hilbert transform of a scalar valued function contains a scalar and a bivector part: H[u](ξ) = − 1 m − 2 2 am W̃(ξ) + 2 am Pv ∫ ∂Ω (ξ − ζ) ∧ ν(ζ) |ξ − ζ|m u(ζ)dS(ζ), ξ ∈ ∂Ω, 90 F. Brackx and H. De Schepper CUBO 10, 2 (2008) and that the Principal Value has to be taken only of the bivector part. As we know from the above example that H[1] = 1, we obtain the following formulae completing (4.2)–(4.4): Pv ∫ ∂Ω (ζ − ξ) ∧ ν(ζ) |ζ − ξ|m dS(ζ) = 0, ξ ∈ ∂Ω, (4.5) ∫ ∂Ω ∂ ∂ −→ ν ( 1 |ζ − ξ|m−2 ) dS(ζ) = − 1 2 (m − 2)am, ξ ∈ ∂Ω, (4.6) again confirming well–known properties of Gauß’s Integral. 5 The Hardy spaces H± 2 (∂Ω) By M∞(Ω + ) we denote the space of left–monogenic functions in Ω + which are moreover C∞(Ω +). Similarly M∞(Ω − ) denotes the space of left–monogenic functions in Ω − , moreover being C∞(Ω −) ánd vanishing at infinity. The Cauchy integral operator C maps C∞(∂Ω) into M∞(Ω + ) as well as into M∞(Ω − ), while the operators H and C± map C∞(∂Ω) into itself. We call M ± ∞ (∂Ω) the spaces of functions on ∂Ω which are the NTBVs of the functions in M∞(Ω ± ) respectively, and we define the Hardy spaces H ± 2 (∂Ω) as the closure in L2(∂Ω) of M ± ∞ (∂Ω). It should be emphasized that the usual notation for H + 2 (∂Ω) is H 2 (∂Ω), and that H − 2 (∂Ω) is mostly not considered. Our notation however reflects the symmetry in the properties of both Hardy spaces. The operators C,H and C± may be extended, through a density argument, to operators defined on L2(∂Ω). Introducing the Hardy spaces H2(Ω ± ) of left–monogenic functions in Ω ± , which do have NTBVs in L2(∂Ω), and, in the case of Ω − , also vanish at infinity, we have the following properties of those operators. Theorem 5.1. (i) The Cauchy integral operator C maps L2(∂Ω) into H2(Ω ± ) and the NTBVs of C[f],f ∈ L2(∂Ω), are given by C ± [f] = ± 1 2 f + 1 2 H[f]; (ii) The Cauchy transforms C± are bounded linear operators from L2(∂Ω) into H ± 2 (∂Ω); (iii) The Hilbert transform H is a bounded linear operator from L2(∂Ω) onto L2(∂Ω); (iv) H2 = 1 or H−1 = H on L2(∂Ω); (v) The adjoint operator H∗ of H is given by H∗ = νHν and H∗2 = 1 on L2(∂Ω); (vi) H±2 (∂Ω) are eigenspaces of H with respective eigenvalues ±1. It is important to note that a function g ∈ L2(∂Ω) belongs to the Hardy space H + 2 (∂Ω) if and only if C + [g] = g, which is equivalent with C−[g] = 0 and still with H[g] = g. Thus a function g ∈ H + 2 (∂Ω) may be identified with its left–monogenic extension C[g] ∈ H2(Ω + ), which is tacitly done most of the time. On the other hand, due to Cauchy’s Theorem, C[g] = 0 in Ω− for each g ∈ H + 2 (∂Ω). Similarly a function g̃ ∈ L2(∂Ω) belongs to H − 2 (∂Ω) if and only if C − [g̃] = −g̃ or C + [g̃] = 0 or still H[g̃] = −g̃. A function g̃ ∈ H − 2 (∂Ω) may thus be identified with its left– monogenic extension C[−g̃] ∈ H2(Ω − ), while here C[g̃] = 0 in Ω+ for all g̃ ∈ H − 2 (∂Ω). CUBO 10, 2 (2008) The Hilbert Transform ... 91 Clearly the Cauchy transforms ±C ± are (skew) projection operators on L2(∂Ω), sometimes called the Hardy projections, since (±C ± ) 2 [f] = 1 4 (1 ± H)2[f] = 1 2 (1 ± H)[f] = (±C±)[f] and C + (−C − )[f] = 1 4 (1 + H)(1 − H)[f] = 0 = (−C − )C + [f] By means of the Hardy projections a skew direct sum decomposition of L2(∂Ω) is obtained at once: L2(∂Ω) = H + 2 (∂Ω) ⊕ H − 2 (∂Ω) with f = C + [f] + (−C − )[f] = 1 2 (1 + H)[f] + 1 2 (1 − H)[f] and H[f] = C+[f] + C−[f] = 1 2 (1 + H)[f] − 1 2 (1 − H)[f] Naturally we have H + 2 (∂Ω) = im C + = ker C − and H − 2 (∂Ω) = im C − = ker C + expressing the fact that C± are projections parallel to H∓2 (∂Ω) (see also Figure 1). Figure 1 92 F. Brackx and H. De Schepper CUBO 10, 2 (2008) Although this decomposition of L2(∂Ω) is rather immediate, it is an important result. In fact it states that an L2(∂Ω)–function f may be split into a sum of a function C + [f] ∈ L2(∂Ω) with left–monogenic extension to Ω + and a function (−C−)[f] ∈ L2(∂Ω) with left–monogenic extension to Ω − vanishing at infinity. This result is the general counterpart of the classical result in Clifford analysis stating that each spherical harmonic may be split into the sum of an inner and an outer spherical monogenic (see e.g. [13]). We conclude this section by giving two examples. As already mentioned before H[1] = 1, which means that the constant function 1 belongs to H + 2 (∂Ω) with C[1] = 1 ∈ H2(Ω + ), C[1] = 0 in Ω− and 〈1, 1〉 = area(∂Ω). The function x |x| m is left–monogenic in R m \ {0} and vanishes at infinity. Its restriction to ∂Ω, given by x |x|m |∂Ω = ζ |ζ|m , belongs to H − 2 (∂Ω) with C[ ζ |ζ|m ] = 1 am ∫ ∂Ω x − ζ |x − ζ|m dσζ ζ |ζ|m =    x |x|m , x ∈ Ω−, 0 , x ∈ Ω+. and also H[ ζ |ζ|m ] = − ζ |ζ|m , ζ ∈ ∂Ω. 6 The orthogonal decomposition of L2(∂Ω) As the Hardy space H + 2 (∂Ω) is a closed subspace of L2(∂Ω), it is itself a Hilbert space and it induces the following orthogonal direct sum decomposition of L2(∂Ω): L2(∂Ω) = H + 2 (∂Ω) ⊕ H + 2 (∂Ω) ⊥. The orthogonal projections P and P ⊥ on H + 2 (∂Ω) and H + 2 (∂Ω) ⊥ respectively are called the Szegö projections. The Hilbert space H + 2 (∂Ω) possesses a reproducing kernel S(ζ,x), ζ ∈ ∂Ω, x ∈ Ω + , the so–called Szegö kernel, for which 〈S(ζ,x),g(ζ)〉 = C[g](x), x ∈ Ω+ for all g ∈ H + 2 (∂Ω). Stricly speaking the reproducing character is only obtained by identifying the function g ∈ H + 2 (∂Ω) with its left–monogenic extension C[g] to Ω + . Note that the Szegö kernel CUBO 10, 2 (2008) The Hilbert Transform ... 93 S(ζ,x) is only defined for x ∈ Ω+. It is the kernel function of the integral transform expressing the projection P of L2(∂Ω) on H + 2 (∂Ω): 〈S(ζ,x),f(ζ)〉 = P[f](x), f ∈ L2(∂Ω), x ∈ Ω +. There is an intimate relationship between the Cauchy and Szegö kernels as established in the following proposition. Proposition 6.1. The Szegö kernel is the Szegö projection of the Cauchy kernel on the Hardy space H+2 (∂Ω), i.e. for all x ∈ Ω + holds S(ζ,x) = P[C(ζ,x)] = P[ 1 am ν(ζ) ζ − x |ζ − x|m ], ζ ∈ ∂Ω. Proof. Take g ∈ H + 2 (∂Ω) and x ∈ Ω + . Then 〈S(ζ,x),g(ζ)〉 = C[g](x) = 〈C(ζ,x),g(ζ)〉 = 〈P[C(ζ,x)],g(ζ)〉. Proposition 6.2. The Szegö kernel is Hermitean symmetric, i.e. for all x,y ∈ Ω+ it holds that S(x,y) = S(y,x). Proof. Take x,y ∈ Ω+. Then, with ζ ∈ ∂Ω, 〈S(ζ,x),S(ζ,y)〉 = 〈S(ζ,x), P[C(ζ,y)]〉 = 〈S(ζ,x),C(ζ,x)〉 = 〈C(ζ,x),S(ζ,x)〉 = C[S(ζ,x)](y) The result follows in view of the identifications C[S(ζ,y)](x) ≈ S(x,y), C[S(ζ,x)](y) ≈ S(y,x). Proposition 6.3. One has S(x,x) > 0 for all x ∈ Ω+. Proof. First observe that it is impossible that S(ζ,x) = 0 for a.e. ζ ∈ ∂Ω, since for all x ∈ Ω+: ∫ ∂Ω S(ζ,x) dS(ζ) = 〈S(ζ,x), 1ζ〉 = C[1](x) = 1. As a consequence of Proposition 6.2 one has for all x ∈ Ω+: S(x,x) = S(x,x) = 〈S(ζ,x),S(ζ,x)〉 = ∫ ∂Ω S(ζ,x)S(ζ,x)dS(ζ) 94 F. Brackx and H. De Schepper CUBO 10, 2 (2008) or, as the Szegö kernel is parabivector valued (i.e. the sum of a scalar and a bivector): S(x,x) = S(x,x) = ∫ ∂Ω |S(ζ,x)|2dS(ζ) from which the result follows. Using the Szegö kernel the Cauchy integral of a function f ∈ L2(∂Ω) may now be expressed as follows: C[f](x) = 〈C(ζ,x),f(ζ)〉 = 〈P[C(ζ,x)], P[f]〉 + 〈P ⊥ [C(ζ,x)], P ⊥ [f]〉. (6.1) For x ∈ Ω− this reduces to C[f](x) = 〈P⊥[C(ζ,x)], P⊥[f]〉 since C[P[f]] = 0 in Ω−. In particular for a function g ∈ H + 2 (∂Ω), and still with x ∈ Ω − , we obtain C[g] = 〈C(ζ,x),g(ζ)〉 = 0 showing that for x ∈ Ω− the Cauchy kernel C(ζ,x) = ν(ζ)E(ζ −x) belongs to H + 2 (∂Ω) ⊥ and hence coincides with P ⊥ [C(ζ,x)], while P[C(ζ,x)] = 0. This is confirmed by the fact that for x ∈ Ω− the fundamental solution E(ζ − x) ∈ H + 2 (∂Ω), since it may be extended left–monogenically to Ω + by the function E(y − x). For x ∈ Ω+ the expression (6.1) for the Cauchy integral reduces to C[f](x) = 〈S(ζ,x), P[f]〉 + 〈P⊥[C(ζ,x)], P⊥[f]〉 which in general cannot be simplified further. From the previous section we know that for f ∈ L2(∂Ω) the Hardy projection P[f] ∈ H + 2 (∂Ω) possesses a left–monogenic extension C[P[f]] ∈ H2(Ω + ) with C[P[f]] = 0 in Ω−, and also that P[f] = H[P[f]] = C+[P[f]], while C−[P[f]] = 0. Now we search for similar properties of the other Hardy projection P ⊥ [f] ∈ H + 2 (∂Ω) ⊥ . In any case its Cauchy integral C[P⊥[f]], though left–monogenic in Ω+ and in Ω−, is not an extension to Ω − of P ⊥ [f]. Proposition 6.4. For a function h ∈ L2(∂Ω) to belong to H + 2 (∂Ω) ⊥ it is necessary and sufficient that H∗[h] = −h. Proof. If h ∈ H + 2 (∂Ω) ⊥ then 〈g,h〉 = 〈H[g],h〉 = 0 for all g ∈ H + 2 (∂Ω) and conversely. This is equivalent with 〈g,H∗[h]〉 = 0 for all g ∈ H + 2 (∂Ω) and so H ∗ [h] ∈ H + 2 (∂Ω) ⊥ . We also have that for all f ∈ L2(∂Ω): 〈C + [f],h〉 = 1 2 〈f + H[f],h〉 = 0 CUBO 10, 2 (2008) The Hilbert Transform ... 95 or 〈f,h〉 + 〈f,H∗[h]〉 = 0 = 〈f,h + H∗[h]〉 which means that h + H∗[h] = 0. Conversely, if H∗[h] = −h, then for all g ∈ H + 2 (∂Ω): 〈g,h〉 = 〈H[g],h〉 = 〈g,H∗[h]〉 = 〈g,−h〉 and hence 〈g, 2h〉 = 0, which means that h ∈ H + 2 (∂Ω) ⊥ . Note that for a non–zero function h ∈ H + 2 (∂Ω) ⊥ there cannot exist a left–monogenic function in Ω + such that its NTBV is h. However there is a one–to–one correspondence between H + 2 (∂Ω) and H + 2 (∂Ω) ⊥ , which is easily expressed by means of the unit normal vector ν(ξ), ξ ∈ ∂Ω. Proposition 6.5. A function g ∈ L2(∂Ω) belongs to H + 2 (∂Ω) if and only if νg ∈ H + 2 (∂Ω) ⊥, and vice–versa. Proof. If g ∈ H + 2 (∂Ω) then H[g] = g and so H ∗ [νg] = νHν[νg] = −νH[g] = −νg, from which it follows that νg ∈ H + 2 (∂Ω) ⊥ , and conversely. If h ∈ H + 2 (∂Ω) then −ννh ∈ H + 2 (∂Ω) ⊥ and so νh ∈ H + 2 (∂Ω), and conversely. Corollary 6.6. The orthogonal direct sum decomposition of L2(∂Ω) takes the form L2(∂Ω) = H + 2 (∂Ω) ⊕ ν H + 2 (∂Ω) = ν H + 2 (∂Ω) ⊥ ⊕ H + 2 (∂Ω) ⊥. 7 The Kerzman–Stein operator The Hilbert operator H on L2(∂Ω) is not self–adjoint. The so–called Kerzman–Stein operator, defined by A = 1 2 (H − H∗) measures the ”non–selfadjointness” of the Hilbert transform. We will find alternative expressions for this operator at the end of this section. To this end, we first introduce four self–adjoint bounded operators on L2(∂Ω), by means of the unit normal function ν on ∂Ω. Proposition 7.1. The operators Hν,νH,νH∗ and H∗ν are self–adjoint bounded operators on L2(∂Ω) moreover satisfying (i) (νH)2 = (H∗ν)2 = H∗H; (ii) (νH∗)2 = (Hν)2 = HH∗; (iii) (νH)(Hν) = −1 = (Hν)(νH); (iv) (νH∗)(H∗ν) = −1 = (H∗ν)(νH∗); (v) 〈Hνf,Hνg〉 = 〈H∗f,H∗g〉 = 〈νH∗f,νH∗g〉, f,g ∈ L2(∂Ω); (vi) 〈H∗νf,H∗νg〉 = 〈Hf,Hg〉 = 〈νHf,νHg〉, f,g ∈ L2(∂Ω). 96 F. Brackx and H. De Schepper CUBO 10, 2 (2008) Note that the function ν belongs to H + 2 (∂Ω) ⊥ since the constant function 1 ∈ H + 2 (∂Ω). It thus follows that H∗[ν] = −ν. Moreover one has (i) ||ν||2 = 〈ν,ν〉 = 〈1, 1〉 = area(∂Ω); (ii) 〈ν,Hν〉 = 〈H∗ν,ν〉 = −〈ν,ν〉 = −area(∂Ω); (iii) ||Hν||2 = 〈Hν,Hν〉 = 〈H∗1,H∗1〉 = ||H∗1||2. As we have seen in the previous section the unit normal vector function ν allows for an alternative form of the orthogonal decomposition of L2(∂Ω). Take f ∈ L2(∂Ω) then also νf ∈ L2(∂Ω) and we have on the one side f = P[f] + P⊥[f] so that νf = νP⊥[f] + νP[f] and on the other νf = P[νf] + P ⊥ [νf]. Hence P[νf] = νP⊥[f] and P⊥[νf] = νP[f] while also P[f] = −νP⊥[νf] and P⊥[f] = −νP[νf]. This leads to f = P[f] − νP[νf] = −νP⊥[νf] + P⊥[f] νf = P[νf] + νP[f] = νP⊥[f] + P⊥[νf]. Taking the Hilbert transform into account we obtain H[f] = H[P[f]] + H[P⊥[f]] = P[f] + H[P⊥[f]] from which it follows that 1 2 (1 − H)[f] = 1 2 (1 − H)[P⊥[f]]. By a similar argument we find 1 2 (1 + H∗)[f] = 1 2 (1 + H∗)[P[f]]. In the operator 1 2 (1 − H) we clearly recognize the Cauchy transform (−C−) for which indeed( 1 2 (1 − H) )2 = 1 2 (1 −H) and 1 2 (1 −H) 1 2 (1 + H) = 0, with 1 2 (1 + H) = C+. On grounds of analogy we put 1 2 (1 + H∗) = D− and 1 2 (−1 + H∗) = D+ defining in this way two bounded linear operators on L2(∂Ω) which moreover satisfy (i) (D − ) 2 = D − ; (ii) (−D+)2 = (−D+); (iii) D+D− = D−D+ = 0. CUBO 10, 2 (2008) The Hilbert Transform ... 97 In a similar way as the Hardy space H + 2 (∂Ω) is characterized by C ± , we may now characterize its orthogonal complement H + 2 (∂Ω) ⊥ by means of the operators D±: h ∈ H + 2 (∂Ω) ⊥ ⇐⇒ D − [h] = 0 ⇐⇒ D+[h] = −h ⇐⇒ H∗[h] = −h. Notice that these newly introduced operators are the adjoints of the Cauchy transforms, i.e. (C+)∗ = D − and (C − ) ∗ = D + , and for each function f ∈ L2(∂Ω) 〈C + [f],D + [f]〉 = 0 and 〈C − [f],D − [f]〉 = 0 meaning that D+[f] belongs to H + 2 (∂Ω) ⊥ , while D−[f] belongs to H − 2 (∂Ω) ⊥ . To the authors’ knowledge no integral transform, similar to the Cauchy integral, has D± as its NTBV. Figure 2 The four operators C ± and D ± are really fundamental; they are the building blocks of the operators 1, H and H∗ (see Figure 2): 1 = C+ − C− = D− − D+, while H = C+ + C− and H∗ = D+ + D−. 98 F. Brackx and H. De Schepper CUBO 10, 2 (2008) Figure 3 Moreover coming back now to the Kerzman–Stein operator, we observe that A = 1 2 (H − H ∗ ) = 1 2 (1 + H) − 1 2 (1 + H ∗ ) = C + − (C + ) ∗ = C + − D − as well as A = 1 2 (H − H ∗ ) = 1 2 (−1 + H) − 1 2 (−1 + H ∗ ) = C − − (C − ) ∗ = C − − D + . In a similar way we define the operator B by C + + D + = C − + D − = 1 2 (H + H ∗ ) = B which clearly is a self–adjoint bounded operator on L2(∂Ω) as well. Next 1 + A = 1 2 (1 + H) + 1 2 (1 − H ∗ ) = C + − D + , −1 + A = − 1 2 (1 − H) − 1 2 (1 + H ∗ ) = C − − D − , and 1 + B = 1 2 (1 + H) + 1 2 (1 + H ∗ ) = C + + D − , −1 + B = − 1 2 (1 − H) − 1 2 (1 − H ∗ ) = C − + D + . CUBO 10, 2 (2008) The Hilbert Transform ... 99 It follows that P(1 + A) = PC + = C + = PB, P⊥(1 + A) = −P⊥D+ = −D+ = −P⊥B since C+[f] ∈ H + 2 (∂Ω) and D + [f] ∈ H + 2 (∂Ω) ⊥ for all f ∈ L2(∂Ω), which means that (1 + A)[f] and B[f] lie ”symmetric” w.r.t H + 2 (∂Ω) (see Figure 3). It should be noted that one of the above formulae relating the Hardy and Szegö projections to each other is the famous Kerzman–Stein Formula P(1 + A) = C + which in fact allows for proving the boundedness of the Cauchy operator C+ on L2(Ω), since the Kerzman–Stein operator A may be expressed as an integral operator which is no longer singular: A[f](ξ) = 1 2 (H − H∗)[f](ξ) = 1 am Pv ∫ ∂Ω ξ − ζ |ξ − ζ|m ν(ζ)f(ζ)dS(ζ) − 1 am Pv ∫ ∂Ω ν(ξ) ξ − ζ |ξ − ζ|m ν(ζ)ν(ζ)f(ζ)dS(ζ) = 1 am ∫ ∂Ω (ξ − ζ)ν(ζ) + ν(ξ)(ξ − ζ) |ξ − ζ|m f(ζ)dS(ζ) = 1 am ∫ ∂Ω (ξ − ζ) ◦ (ν(ζ) + ν(ξ)) |ξ − ζ|m f(ζ)dS(ζ) + 1 am ∫ ∂Ω (ξ − ζ) ∧ (ν(ζ) − ν(ξ)) |ξ − ζ|m f(ζ)dS(ζ). Moreover we have AP = C − P − D + P = −D + P BP = C − P + D − P = D − P implying that for each function g ∈ H + 2 (∂Ω), A[g] belongs to H + 2 (∂Ω) ⊥ and B[g] belongs to H − 2 (∂Ω) ⊥ , and similarly AP ⊥ = C + P ⊥ − D − P ⊥ = C + P ⊥ = (1 + B)P ⊥ BP ⊥ = C − P ⊥ + D − P ⊥ = C − P ⊥ = (−1 + A)P ⊥ implying that for each function h ∈ H + 2 (∂Ω) ⊥ , A[h] = (1 + B)[h] belongs to H + 2 (∂Ω) and B[h] = (−1 + A)[h] belongs to H − 2 (∂Ω). 100 F. Brackx and H. De Schepper CUBO 10, 2 (2008) 8 A second orthogonal decomposition of L2(∂Ω) Making use of the Hardy space H − 2 (∂Ω), introduced in section 5, a second orthogonal direct sum decomposition of L2(∂Ω) is at hand: L2(∂Ω) = H − 2 (∂Ω) ⊕ H − 2 (∂Ω) ⊥. Both components may be characterized in a similar way as was done for H + 2 (∂Ω) and H + 2 (∂Ω) ⊥ . Proposition 8.1. (i) A function g̃ belongs to H−2 (∂Ω) if and only if H[g̃] = −g̃ or C + [g̃] = 0, or still C−[g̃] = −g̃. (ii) A function h̃ belongs to H−2 (∂Ω) ⊥ if and only if H∗[h̃] = h̃ or D+[h̃] = 0, or still D−[h̃] = h̃. Note that for a non–zero function h̃ ∈ H − 2 (∂Ω) ⊥ there cannot exist a left–monogenic function in Ω − , vanishing at infinity, such that its NTBV is h̃. However there again is a one–to–one correspondence, now between H − 2 (∂Ω) and H − 2 (∂Ω) ⊥ , established by means of the unit normal vector ν(ξ), ξ ∈ ∂Ω. Proposition 8.2. A function g̃ ∈ L2(∂Ω) belongs to H − 2 (∂Ω) if and only if νg̃ ∈ H − 2 (∂Ω) ⊥, and vice–versa. Proof. Similar to the proof of Proposition 6.5. Corollary 8.3. The second orthogonal direct sum decomposition of L2(∂Ω) takes the form L2(∂Ω) = H − 2 (∂Ω) ⊕ ν H − 2 (∂Ω) = ν H − 2 (∂Ω) ⊥ ⊕ H − 2 (∂Ω) ⊥ . We denote the orthogonal projections on H − 2 (∂Ω) and H − 2 (∂Ω) ⊥ by Q and Q ⊥ respectively, and we put for x ∈ Ω− T (ζ,x) = −Q[C(ζ,x)], ζ ∈ ∂Ω clearly the counterpart of the Szegö kernel for the Hilbert space H − 2 (∂Ω). The function T (ζ,x), x ∈ Ω − , possesses the reproducing property since for each g̃ ∈ H − 2 (∂Ω) 〈T (ζ,x), g̃(ζ)〉 = 〈−Q[C(ζ,x)], g̃(ζ)〉 = 〈C(ζ,x),−g̃(ζ)〉 = C[−g̃](x) where at the utmost right hand side the functions g̃ ∈ H − 2 (∂Ω) and C[−g̃] are identified. In the same order of ideas it is also the kernel function of the integral transform expressing the projection Q of L2(∂Ω) on H − 2 (∂Ω): 〈T (ζ,x),f(ζ)〉 = Q[f](x), f ∈ L2(∂Ω), x ∈ Ω −. On the other hand for x ∈ Ω+ we obtain for the Cauchy integral of an arbitrary function f ∈ L2(∂Ω), C[f] = 〈C(ζ,x),f(ζ)〉 = 〈Q[C(ζ,x)], Q[f(ζ)]〉 + 〈Q⊥[C(ζ,x)], Q⊥[f(ζ)]〉 = C[Q[f]] + 〈Q⊥[C(ζ,x)], Q⊥[f(ζ)]〉 = 〈Q⊥[C(ζ,x)], Q⊥[f(ζ)]〉 CUBO 10, 2 (2008) The Hilbert Transform ... 101 since the Cauchy integral of Q[f] ∈ H − 2 (∂Ω) vanishes in Ω + . For a function g̃ ∈ H − 2 (∂Ω) this leads in particular to 0 = C[g̃] = 〈C(ζ,x), g̃(ζ)〉, x ∈ Ω + which means that for x ∈ Ω+ the Cauchy kernel C(ζ,x) = ν(ζ)E(ζ − x) belongs to H − 2 (∂Ω) ⊥ , which is confirmed by the fact that for x ∈ Ω+ the fundamental solution E(ζ −x) ∈ H − 2 (∂Ω), since it may be extended left–monogenically to Ω − by the function E(y − x). Returning to the Kerzman–Stein operator A and the related operator B, it also follows that Q(−1 + A) = QC − = C − = QB, Q⊥(−1 + A) = −Q⊥D− = −D− = −Q⊥B since C−[f] ∈ H − 2 (∂Ω) and D − [f] ∈ H − 2 (∂Ω) ⊥ for all f ∈ L2(∂Ω), which means that (−1 + A)[f] and B[f] lie ”symmetric” w.r.t H − 2 (∂Ω) (see again Figure 3). Moreover AQ = C + Q − D − Q = −D − Q BQ = C + Q + D + Q = D + Q implying that for each function g̃ ∈ H − 2 (∂Ω), A[g̃] belongs to H − 2 (∂Ω) ⊥ and B[g̃] belongs to H + 2 (∂Ω) ⊥ , and similarly AQ ⊥ = C − Q ⊥ − D + Q ⊥ = C − Q ⊥ = −(1 − B)Q ⊥ BQ ⊥ = C + Q ⊥ + D + Q ⊥ = C + Q ⊥ = (1 + A)Q ⊥ implying that for each function h̃ ∈ H − 2 (∂Ω) ⊥ , A[h̃] = (−1 + B)[h̃] belongs to H − 2 (∂Ω) and B[h̃] = (1 + A)[h̃] belongs to H + 2 (∂Ω). 9 Extension of the unit normal function ν As the boundary ∂Ω is assumed to be C∞–smooth, it is always possible to introduce the vector function N(x) in an open neighbourhood ∂̃Ω of ∂Ω such that (i) N(x) is a smooth function (ii) |N(x)| = 1 for all x ∈ ∂̃Ω (iii) the restriction of N(x) to ∂Ω is precisely ν(ξ), ξ ∈ ∂Ω. If the closed surface ∂Ω has a defining C∞–function ρ(x), i.e. ∂Ω = {x : ρ(x) = 0}, while Ω + = {x : ρ(x) < 0} and Ω− = {x : ρ(x) > 0}, then ν(ξ) = ∂ρ(ξ) |∂ρ(ξ)| 102 F. Brackx and H. De Schepper CUBO 10, 2 (2008) for all ξ ∈ ∂Ω and the function N(x) = ∂ρ(x) |∂ρ(x)| , x ∈ ∂̃Ω satisfies all above requirements. Note that certainly |∂ρ(ξ)| 6= 0 for all ξ ∈ ∂Ω due to the supposed smoothness of ∂Ω, so that |∂ρ(x)| 6= 0 in an appropriate open neighbourhood ∂̃Ω of ∂Ω. For a given function f ∈ L2(∂Ω) we consider in ∂̃Ω the function F(x) = C[P[f]](x) − N(x)C[P[νf]](x). The first term F1(x) = C[P[f]](x) is left–monogenic in Ω + while vanishing in Ω − and moreover for ξ ∈ ∂Ω it holds that lim Ω+∋x→ξ F1(x) = P[f](ξ) lim Ω−∋x→ξ F1(x) = 0. The function F2(x) = N(x) C[P[νf]](x) apparently is not left–monogenic in ∂̃Ω + = ∂̃Ω ∩ Ω+, but still vanishes in ∂̃Ω − = ∂̃Ω ∩ Ω−, and for ξ ∈ ∂Ω it holds that lim ∂̃Ω + ∋x→ξ F2(x) = ν(ξ)P[νf](ξ) = −P ⊥ [f](ξ) lim ∂̃Ω − ∋x→ξ F2(x) = 0. It follows that F(x) is not left–monogenic in ∂̃Ω, but for ξ ∈ ∂Ω it holds that lim ∂̃Ω + ∋x→ξ F(x) = P[f](ξ) − ν(ξ)P[νf](ξ) = f(ξ) lim ∂̃Ω − ∋x→ξ F(x) = 0. We will now show that F(x) is harmonic in ∂̃Ω\∂Ω. To that end consider the operator ∂ ∗ = N ∂ N for which also holds ∂ ∗ N = −N∂ and N∂ ∗ = −∂N. For this operator, the following lemmata are easily proved. Lemma 9.1. If F is sufficiently smooth then ∂F = 0 if and only if ∂∗(Nf) = 0. Lemma 9.2. The operator ∂∗ factorizes the Laplace operator: (∂ ∗ ) 2 = −∆. We then arrive at the desired result. Proposition 9.3. In ∂̃Ω \ ∂Ω one has: CUBO 10, 2 (2008) The Hilbert Transform ... 103 (i) F1(x) = C[P[f]](x) is left–monogenic with F1 = 0 in ∂̃Ω − ; (ii) F2(x) = N(x) C[P[νf]](x) is a null solution of ∂ ∗ with F2 = 0 in ∂̃Ω − ; (iii) F(x) = F1(x) − F2(x) is harmonic with F = 0 in ∂̃Ω − . Proof. (i) This is a property of the Cauchy integral in R m \ ∂Ω. (ii) Follows from Lemma 9.1 since a Cauchy integral is left–monogenic in R m \ ∂Ω. (iii) ∆F = ∆F1 − ∆F2 = ∂ 2 F1 − ∂ ∗2 F2 = 0. We thus have proved that, given a function f ∈ L2(∂Ω), there exists a function F in ∂̃Ω \ ∂Ω, namely F(x) = C[P[f]](x) − N(x)C[P[νf]](x), which is harmonic in ∂̃Ω \ ∂Ω, vanishes in ∂̃Ω − and for which one has lim ∂̃Ω + ∋x→ξ F(x) = f(ξ). Remark 9.4. Unfortunately the function N(x) is only defined in an open neighbourhood of ∂Ω. Solving the Dirichlet problem and constructing the associated Poisson kernel by means of the Szegö projections and the Cauchy integral, remains an open problem. This problem can be reformulated as follows. Let H2(Ω + ) be the Hardy space of harmonic functions in Ω+ with a NTBV in L2(∂Ω). Clearly H2(Ω + ) is a closed subspace of H2(Ω + ) leading to the direct sum decomposition H2(Ω + ) = H2(Ω + ) ⊕ H2(Ω + ) ⊥ the orthogonal complement being taken in H2(Ω + ). The question now is: what is H2(Ω + ) ⊥? In the specific case where Ω is the unit ball the answer is known (see e.g. [11]), in general it is not. Finally the unit vector function N may also be used in the construction of a reproducing kernel for the Hilbert space H + 2 (∂Ω) ⊥ . Indeed, take h ∈ H + 2 (∂Ω) ⊥ , then νh ∈ H + 2 (∂Ω) and by means of the Szegö kernel we obtain for x ∈ Ω+: C[νh](x) = 〈S(ζ,x),ν(ζ)h(ζ)〉 and hence for x ∈ ∂̃Ω + : N(x) C[νh](x) = −N(x) 〈ν(ζ)S(ζ,x),h(ζ)〉 or −N(x) C[νh](x) = N(x) 〈L(ζ,x),h(ζ)〉 where we have introduced to so–called Garabedian kernel for H + 2 (∂Ω) ⊥ : L(ζ,x) = ν(ζ)S(ζ,x), ζ ∈ ∂Ω, x ∈ Ω+. This Garabedian kernel is reproducing for H + 2 (∂Ω) ⊥ in the sense that for h ∈ H + 2 (∂Ω) ⊥ and for x ∈ ∂̃Ω + , the function N(x) 〈L(ζ,x),h(ζ)〉 equals −N(x) C[νh](x) which in ∂̃Ω + is a null solution 104 F. Brackx and H. De Schepper CUBO 10, 2 (2008) of the operator ∂ ∗ and has NTBV h(ξ) for x ∈ ∂̃Ω + tending to ξ ∈ ∂Ω. As S(ζ,x) = P[C(ζ,x)] we also have for x ∈ Ω+ and ζ ∈ ∂Ω that L(ζ,x) = ν(ζ)P[C(ζ,x)] = P⊥[ν(ζ)C(ζ,x)] = P⊥[E(x − ζ)]. Note that the translated fundamental solution E(x−ζ) = ν(ζ)C(ζ,x) of the Dirac operator belongs to H − 2 (∂Ω). 10 Conclusions The central notion in this paper is the Hilbert transform on the smooth boundary ∂Ω of a bounded domain Ω in Euclidean space, which has been defined quite naturally as a part of the inner and outer NTBVs of the Cauchy integral of an L2–function on ∂Ω, the success of this approach being entirely due to the powerful concept of monogenic function in Clifford analysis. At the same time we have devoted some attention to the concept of Hardy space, to which the Hilbert transform is closely related. In this we have treated the inner and the outer region determined by the considered closed hypersurface ∂Ω on equal footing, enabling us to obtain new bounded linear operators on ∂Ω, similar to the Cauchy transforms, as well as to derive new rela- tions in between those operators, similar to the traditional Kerzman–Stein formula. 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