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Singular integral operators on closed curves
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<h2 id="operators-on-the-unit-circle">Operators on the unit circle</h2> <p class="note">Main article: <a href="/facts/Singular_integral_operators_of_convolution_type/W47sCDdo">Singular integral operators of convolution type</a></p> <p>If <i>f</i> is in L2(T), then it has a Fourier series expansion<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> f ( θ ) = ∑ n ∈ Z a n e i n θ . {\displaystyle \displaystyle {f(\theta )=\sum _{n\in {\mathbf {Z} }}a_{n}e^{in\theta }.}} <p><a href="/facts/Hardy_space/aN45dgx7">Hardy space</a> H2(T) consists of the functions for which the negative coefficients vanish, <i>a</i><i>n</i> = 0 for <i>n</i> < 0. These are precisely the square-integrable functions that arise as boundary values of holomorphic functions in the unit disk |<i>z</i>| < 1. Indeed, <i>f</i> is the boundary value of the function </p> F ( z ) = ∑ n ≥ 0 a n z n , {\displaystyle \displaystyle {F(z)=\sum _{n\geq 0}a_{n}z^{n},}} <p>in the sense that the functions </p> f r ( θ ) = F ( r e i θ ) , {\displaystyle \displaystyle {f_{r}(\theta )=F(re^{i\theta })},} <p>defined by the restriction of <i>F</i> to the concentric circles |<i>z</i>| = <i>r</i>, satisfy </p> ‖ f r − f ‖ 2 → 0 {\displaystyle \displaystyle {\|f_{r}-f\|_{2}\rightarrow 0}} as r → 1 {\displaystyle \displaystyle {r\rightarrow 1}} . <p>The orthogonal projection <i>P</i> of L2(T) onto H2(T) is called the Szegő projection. It is a bounded operator on L2(T) with <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a> 1. </p><p>By Cauchy's theorem </p> F ( z ) = 1 2 π i ∫ | ζ | = 1 f ( ζ ) ζ − z d ζ = 1 2 π ∫ − π π f ( θ ) 1 − e − i θ z d θ . {\displaystyle \displaystyle {F(z)={1 \over 2\pi i}\int _{|\zeta |=1}{f(\zeta ) \over \zeta -z}\,d\zeta ={1 \over 2\pi }\int _{-\pi }^{\pi }{f(\theta ) \over 1-e^{-i\theta }z}\,d\theta .}} <p>Thus </p> F ( r e i φ ) = 1 2 π ∫ − π π f ( φ − θ ) 1 − r e i θ d θ . {\displaystyle \displaystyle {F(re^{i\varphi })={1 \over 2\pi }\int _{-\pi }^{\pi }{f(\varphi -\theta ) \over 1-re^{i\theta }}\,d\theta .}} <p>When <i>r</i> equals 1, the integrand on the right hand side has a singularity at θ = 0. The truncated Hilbert transform is defined by </p> H ε f ( φ ) = i π ∫ ε ≤ | θ | ≤ π f ( φ − θ ) 1 − e i θ d θ = 1 π ∫ | ζ − e i φ | ≥ δ f ( ζ ) ζ − e i φ d ζ , {\displaystyle \displaystyle {H^{\varepsilon }f(\varphi )={i \over \pi }\int _{\varepsilon \leq |\theta |\leq \pi }{f(\varphi -\theta ) \over 1-e^{i\theta }}\,d\theta ={1 \over \pi }\int _{|\zeta -e^{i\varphi }|\geq \delta }{f(\zeta ) \over \zeta -e^{i\varphi }}\,d\zeta ,}} <p>where δ = |1 – <i>e</i><i>i</i>ε|. Since it is defined as convolution with a bounded function, it is a bounded operator on L2(T). Now </p> H ε 1 = i π ∫ ε π 2 ℜ ( 1 − e i θ ) − 1 d θ = i π ∫ ε π 1 d θ = i − i ε π . {\displaystyle \displaystyle {H^{\varepsilon }{1}={i \over \pi }\int _{\varepsilon }^{\pi }2\Re (1-e^{i\theta })^{-1}\,d\theta ={i \over \pi }\int _{\varepsilon }^{\pi }1\,d\theta =i-{i\varepsilon \over \pi }.}} <p>If <i>f</i> is a polynomial in <i>z</i> then </p> H ε f ( z ) − i ( 1 − ε ) π f ( z ) = 1 π i ∫ | ζ − z | ≥ δ f ( ζ ) − f ( z ) ζ − z d ζ . {\displaystyle \displaystyle {H^{\varepsilon }f(z)-{i(1-\varepsilon ) \over \pi }f(z)={1 \over \pi i}\int _{|\zeta -z|\geq \delta }{f(\zeta )-f(z) \over \zeta -z}\,d\zeta .}} <p>By Cauchy's theorem the right hand side tends to 0 uniformly as ε, and hence δ, tends to 0. So </p> H ε f → i f {\displaystyle \displaystyle {H^{\varepsilon }f\rightarrow if}} <p>uniformly for polynomials. On the other hand, if <i>u</i>(<i>z</i>) = <i>z</i> it is immediate that </p> H ε f ¯ = − u − 1 H ε ( u f ¯ ) . {\displaystyle \displaystyle {{\overline {H^{\varepsilon }f}}=-u^{-1}H^{\varepsilon }(u{\overline {f}}).}} <p>Thus if <i>f</i> is a polynomial in <i>z</i>−1 without constant term </p> H ε f → − i f {\displaystyle \displaystyle {H^{\varepsilon }f\rightarrow -if}} uniformly. <p>Define the Hilbert transform on the circle by </p> H = i ( 2 P − I ) . {\displaystyle \displaystyle {H=i(2P-I).}} <p>Thus if <i>f</i> is a trigonometric polynomial </p> H ε f → H f {\displaystyle \displaystyle {H^{\varepsilon }f\rightarrow Hf}} uniformly. <p>It follows that if <i>f</i> is any L2 function </p> H ε f → H f {\displaystyle \displaystyle {H^{\varepsilon }f\rightarrow Hf}} in the L2 norm. <p>This is a consequence of the result for trigonometric polynomials since the <i>H</i>ε are uniformly bounded in <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a>: indeed their Fourier coefficients are uniformly bounded. </p><p>It also follows that, for a continuous function <i>f</i> on the circle, <i>H</i>ε<i>f</i> converges uniformly to <i>Hf</i>, so in particular pointwise. The pointwise limit is a <a href="/facts/Cauchy_principal_value/Gm33Qevf">Cauchy principal value</a>, written </p> H f = P . V . 1 π ∫ f ( ζ ) ζ − e i φ d ζ . {\displaystyle \displaystyle {Hf=\mathrm {P.V.} \,{1 \over \pi }\int {f(\zeta ) \over \zeta -e^{i\varphi }}\,d\zeta .}} <p>The Hilbert transform has a natural compatibility with orientation-preserving diffeomorphisms of the circle.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> Thus if <i>H</i> is a diffeomorphism of the circle with </p> H ( e i θ ) = e i h ( θ ) , h ( θ + 2 π ) = h ( θ ) + 2 π , {\displaystyle \displaystyle {H(e^{i\theta })=e^{ih(\theta )},\,\,\,h(\theta +2\pi )=h(\theta )+2\pi ,}} <p>then the operators </p> H h ε f ( e i φ ) = 1 π ∫ | e i h ( θ ) − e i h ( φ ) | ≥ ε f ( e i θ ) e i θ − e i φ e i θ d θ , {\displaystyle \displaystyle {H_{h}^{\varepsilon }f(e^{i\varphi })={1 \over \pi }\int _{|e^{ih(\theta )}-e^{ih(\varphi )}|\geq \varepsilon }{f(e^{i\theta }) \over e^{i\theta }-e^{i\varphi }}\,e^{i\theta }\,d\theta ,}} <p>are uniformly bounded and tend in the strong operator topology to <i>H</i>. Moreover, if <i>Vf</i>(<i>z</i>) = <i>f</i>(<i>H</i>(<i>z</i>)), then <i>VHV</i>−1 – <i>H</i> is an operator with smooth kernel, so a <a href="/facts/Hilbert%25E2%2580%2593Schmidt_operator/Xhy17yn6">Hilbert–Schmidt operator</a>. </p> <h2 id="hardy-spaces">Hardy spaces</h2> <p class="note">See also: <a href="/facts/Hardy_space/aN45dgx7">Hardy space</a></p> <p>The Hardy space on the unit circle can be generalized to any multiply connected bounded domain Ω with smooth boundary ∂Ω. The Hardy space H2(∂Ω) can be defined in a number of equivalent ways. The simplest way to define it is as the closure in L2(∂Ω) of the space of holomorphic functions on Ω which extend continuously to smooth functions on the closure of Ω. As <a href="/facts/Joseph_Leonard_Walsh/eGYN1jk5">Walsh</a> proved, in a result that was a precursor of <a href="/facts/Mergelyan%2527s_theorem/V9ErA7q8">Mergelyan's theorem</a>, any holomorphic function on Ω that extends continuously to the closure can be approximated in the uniform norm by a rational function with poles in the complementary region Ω<i>c</i>. If Ω is simply connected, then the rational function can be taken to be a polynomial. There is a counterpart of this theorem on the boundary, the <a href="/facts/Hartogs%25E2%2580%2593Rosenthal_theorem/jEd4hR56">Hartogs–Rosenthal theorem</a>, which states that any continuous function ∂Ω can be approximated in the uniform norm by rational functions with poles in the complement of ∂Ω. It follows that for a simply connected domain when ∂Ω is a simple closed curve, H2(∂Ω) is just the closure of the polynomials; in general it is the closure of the space of rational functions with poles lying off ∂Ω.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>On the unit circle an L2 function <i>f</i> with Fourier series expansion </p> f ( e i θ ) = ∑ a n e i n θ {\displaystyle \displaystyle {f(e^{i\theta })=\sum a_{n}e^{in\theta }}} <p>has a unique extension to a harmonic function in the unit disk given by the Poisson integral </p> f ( r e i θ ) = P r f ( e i θ ) = ∑ a n r | n | e i n θ . {\displaystyle \displaystyle {f(re^{i\theta })=P_{r}f(e^{i\theta })=\sum a_{n}r^{|n|}e^{in\theta }.}} <p>In particular </p> ‖ P r f ‖ 2 = ∑ | a n | 2 r 2 | n | , {\displaystyle \displaystyle {\|P_{r}f\|^{2}=\sum |a_{n}|^{2}r^{2|n|},}} <p>so that the norms increase to the value at <i>r</i> = 1, the norm of <i>f</i>. A similar in the complement of the unit disk where the harmonic extension is given by </p> F R ( e i θ ) = F ( R e i θ ) = ∑ a n R − | n | a n e i n θ . {\displaystyle \displaystyle {F_{R}(e^{i\theta })=F(Re^{i\theta })=\sum a_{n}R^{-|n|}a_{n}e^{in\theta }.}} <p>In this case the norms increase from the value at <i>R</i> = ∞ to the norm of <i>f</i>, the value at <i>R</i> = 1. </p><p>A similar result holds for a harmonic function <i>f</i> on a simply connected region with smooth boundary provided the L2 norms are taken over the level curves in a tubular neighbourhood of the boundary.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Using vector notation v(<i>t</i>) = (<i>x</i>(<i>t</i>), <i>y</i>(<i>t</i>)) to parametrize the boundary curve by arc length, the following classical formulas hold: </p> v ˙ ⋅ v ˙ = 1 , v ¨ ⋅ v ˙ = 0. {\displaystyle \displaystyle {{\dot {\mathbf {v} }}\cdot {\dot {\mathbf {v} }}=1,\,\,\,{\ddot {\mathbf {v} }}\cdot {\dot {\mathbf {v} }}=0.}} <p>Thus the unit tangent vector t(<i>t</i>) at <i>t</i> and oriented normal vector n(<i>t</i>) are given by </p> t = v ˙ , n = ( − y ˙ , x ˙ ) . {\displaystyle \displaystyle {\mathbf {t} ={\dot {\mathbf {v} }},\,\,\,\,\,\,\mathbf {n} =(-{\dot {y}},{\dot {x}}).}} <p>The constant relating the acceleration vector to the normal vector is the <a href="/facts/Curvature/RFCAEsj8">curvature</a> of the curve: </p> v ¨ = κ ( t ) n ( t ) , κ ( t ) = v ¨ ⋅ n = y ¨ x ˙ − x ¨ y ˙ . {\displaystyle \displaystyle {{\ddot {\mathbf {v} }}=\kappa (t)\,\mathbf {n} (t),\,\,\,\,\,\kappa (t)={\ddot {\mathbf {v} }}\cdot \mathbf {n} ={\ddot {y}}{\dot {x}}-{\ddot {x}}{\dot {y}}.}} <p>There are two further formulas of <a href="/facts/Jean_Fr%25C3%25A9d%25C3%25A9ric_Frenet/NoFpGMB7">Frenet</a>: </p> n ˙ = − κ t , n ¨ = κ ˙ n − κ 2 t . {\displaystyle \displaystyle {{\dot {\mathbf {n} }}=-\kappa \mathbf {t} ,\,\,\,{\ddot {\mathbf {n} }}={\dot {\kappa }}\mathbf {n} -\kappa ^{2}\mathbf {t} .}} <p>A tubular neighbourhood of the boundary is given by </p> v s ( t ) = v ( t ) + s n ( t ) , {\displaystyle \displaystyle {\mathbf {v} _{s}(t)=\mathbf {v} (t)+s\mathbf {n} (t),}} <p>so that the level curves ∂Ω<i>s</i> with <i>s</i> constant bound domains Ω<i>s</i>. Moreover<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> v ˙ s ( t ) = ( 1 − s κ ) t , ∂ s | v ˙ s | = − κ . {\displaystyle \displaystyle {{\dot {\mathbf {v} }}_{s}(t)=(1-s\kappa )\mathbf {t} ,\,\,\,\,\partial _{s}|{\dot {\mathbf {v} }}_{s}|=-\kappa .}} <p>Hence differentiating the integral means with respect to <i>s</i>, the derivative in the direction of the <i>inward</i> pointing normal, gives </p> ∂ s ∫ ∂ Ω s | f | 2 = − ∫ ∂ Ω s ( ∂ n f f ¯ + f ∂ n f ¯ ) − ∫ ∂ Ω s κ ( 1 − κ s ) − 1 | f | 2 = − 2 ∬ Ω s | ∇ f | 2 − ∫ ∂ Ω s κ ( 1 − κ s ) − 1 | f | 2 , {\displaystyle \displaystyle {\partial _{s}\int _{\partial \Omega _{s}}|f|^{2}=-\int _{\partial \Omega _{s}}(\partial _{n}f{\overline {f}}+f{\overline {\partial _{n}f}})-\int _{\partial \Omega _{s}}\kappa (1-\kappa s)^{-1}|f|^{2}=-2\iint _{\Omega _{s}}|\nabla f|^{2}-\int _{\partial \Omega _{s}}\kappa (1-\kappa s)^{-1}|f|^{2},}} <p>using <a href="/facts/Green%2527s_theorem/WBDk654A">Green's theorem</a>. Thus for <i>s</i> small </p> ∂ s ‖ f | ∂ Ω s ‖ 2 ≤ M ‖ f | ∂ Ω s ‖ 2 , {\displaystyle \displaystyle {\partial _{s}\|f|_{\partial \Omega _{s}}\|^{2}\leq M\|f|_{\partial \Omega _{s}}\|^{2},}} <p>for some constant <i>M</i> independent of <i>f</i>. This implies that </p> ∂ s e − M s ‖ f | ∂ Ω s ‖ 2 ≤ 0 , {\displaystyle \displaystyle {\partial _{s}e^{-Ms}\|f|_{\partial \Omega _{s}}\|^{2}\leq 0,}} <p>so that, on integrating this inequality, the norms are bounded near the boundary: </p> ‖ f | ∂ Ω s ‖ ≤ e M s / 2 ‖ f | ∂ Ω ‖ . {\displaystyle \displaystyle {\|f|_{\partial \Omega _{s}}\|\leq e^{Ms/2}\|f|_{\partial \Omega }\|.}} <p>This inequality shows that a function in the L2 Hardy space H2(Ω) leads, via the Cauchy integral operator <i>C</i>, to a holomorphic function on Ω satisfying the classical condition that the integral means </p> ∫ ∂ Ω s | f | 2 {\displaystyle \displaystyle {\int _{\partial \Omega _{s}}|f|^{2}}} <p>are bounded. Furthermore, the restrictions <i>f</i>s of <i>f</i> to ∂Ω<i>s</i>, which can be naturally identified with ∂Ω, tend in L2 to the original function in Hardy space.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> In fact H2(Ω) has been defined as the closure in L2(Ω) of rational functions (which can be taken to be polynomials if Ω is simply connected). Any rational function with poles only in Ω<i>c</i> can be recovered inside Ω from its boundary value <i>g</i> by Cauchy's integral formula </p> C g ( a ) = 1 2 π i ∫ ∂ Ω g ( z ) z − a d z . {\displaystyle \displaystyle {Cg(a)={1 \over 2\pi i}\int _{\partial \Omega }{g(z) \over z-a}\,dz.}} <p>The estimates above show that the functions <i>Cg</i>|∂Ω<i>s</i> depend continuously on <i>Cg</i>|∂Ω. Moreover, in this case the functions tend uniformly to the boundary value and hence also in L2, using the natural identification of the spaces L2(∂Ω<i>s</i>) with L2(∂Ω). Since <i>Ch</i> can be defined for any L2 function as a holomorphic function on Ω since <i>h</i> is integrable on ∂Ω. Since <i>h</i> is a limit in L2 of rational functions <i>g</i>, the same results hold for <i>h</i> and <i>Ch</i>, with the same inequalities for the integral means. Equally well <i>h</i> is the limit in L2(∂Ω) of the functions <i>Ch</i>|∂Ω<i>s</i>. </p><p>The estimates above for the integral means near the boundary show that <i>Cf</i> lies in L2(Ω) and that its L2 norm can be bounded in terms of that of <i>f</i>. Since <i>Cf</i> is also holomorphic, it lies in the <a href="/facts/Bergman_space/r5moDRvw">Bergman space</a> A2(Ω) of Ω. Thus the Cauchy integral operator <i>C</i> defines a natural mapping from the Hardy space of the boundary into the Bergman space of the interior.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p><p>The Hardy space H2(Ω) has a natural partner, namely the closure in L2(∂Ω) of boundary values of rational functions <i>vanishing at</i> ∞ with poles only in Ω. Denoting this subspace by H2+(∂Ω) to distinguish it from the original Hardy space, which will also denoted by H2−(∂Ω), the same reasoning as above can be applied. When applied to a function <i>h</i> in H2+(∂Ω), the Cauchy integral operator defines a holomorphic function <i>F</i> in Ω<i>c</i> vanishing at ∞ such that near the boundary the restriction of <i>F</i> to the level curves, each identified with the boundary, tend in L2 to <i>h</i>. Unlike the case of the circle, H2−(∂Ω) and H2+(∂Ω) are not orthogonal spaces. By the Hartogs−Rosenthal theorem, their sum is dense in L2(∂Ω). As shown below, these are the ±i eigenspaces of the Hilbert transform on ∂Ω, so their sum is in fact direct and the whole of L2(∂Ω). </p> <h2 id="hilbert-transform-on-a-closed-curve">Hilbert transform on a closed curve</h2> <p>For a bounded simply connected domain Ω in the complex plane with smooth boundary ∂Ω, the theory of the Hilbert transform can be deduced by direct comparison with the Hilbert transform for the unit circle.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>To define the Hilbert transform <i>H</i>∂Ω on L2(∂Ω), take ∂Ω to be parametrized by arclength and thus a function <i>z</i>(<i>t</i>). The Hilbert transform is defined to be the limit in the <a href="/facts/Strong_operator_topology/awhSW5op">strong operator topology</a> of the truncated operators <i>H</i>∂Ωε defined by </p> H ∂ Ω ε f ( s ) = 1 π i ∫ | s − t | ≥ ε f ( t ) z ( t ) − z ( s ) z ˙ ( t ) d t . {\displaystyle \displaystyle {H_{\partial \Omega }^{\varepsilon }f(s)={1 \over \pi i}\int _{|s-t|\geq \varepsilon }\,\,\,\,{f(t) \over z(t)-z(s)}\,{\dot {z}}(t)\,dt.}} <p>To make the comparison it will be convenient to apply a scaling transformation in C so that the length of ∂Ω is 2π. (This only changes the operators above by a fixed positive factor.) There is then a canonical unitary isomorphism of L2(∂Ω) onto L2(T), so the two spaces can be identified. The truncated operators <i>H</i>∂Ωε can be compared directly with the truncated Hilbert transform <i>H</i>ε: </p> H ∂ Ω ε g ( s ) − H ε g ( s ) = 1 π i ∫ | t − s | ≥ ε K ( s , t ) ⋅ g ( t ) d t , {\displaystyle \displaystyle {H_{\partial \Omega }^{\varepsilon }g(s)-H^{\varepsilon }g(s)={1 \over \pi i}\int _{|t-s|\geq \varepsilon }K(s,t)\cdot g(t)\,dt,}} <p>where </p> K ( u , v ) = z ˙ ( t ) z ( t ) − z ( s ) − i e i t e i t − e i s = ∂ t log ( z ( t ) − z ( s ) e i t − e i s ) . {\displaystyle \displaystyle {K(u,v)={{\dot {z}}(t) \over z(t)-z(s)}-{ie^{it} \over e^{it}-e^{is}}=\partial _{t}\log \left({z(t)-z(s) \over e^{it}-e^{is}}\right).}} <p>The kernel <i>K</i> is thus smooth on T × T, so the difference above tends in the strong topology to the Hilbert–Schmidt operator defined by the kernel. It follows that the truncated operators <i>H</i>∂Ωε are uniformly bounded in norm and have a limit in the strong operator topology denoted <i>H</i>∂Ω and called the Hilbert transform on ∂Ω. </p> H ∂ Ω g = lim ε → 0 H ∂ Ω ε g . {\displaystyle \displaystyle {H_{\partial \Omega }g=\lim _{\varepsilon \rightarrow 0}H_{\partial \Omega }^{\varepsilon }g.}} <p>Letting ε tend to 0 above yields </p> H ∂ Ω g ( s ) − H g ( s ) = 1 π i ∫ 0 2 π K ( s , t ) ⋅ g ( t ) d t . {\displaystyle \displaystyle {H_{\partial \Omega }g(s)-Hg(s)={1 \over \pi i}\int _{0}^{2\pi }K(s,t)\cdot g(t)\,dt.}} <p>Since <i>H</i> is skew-adjoint and <i>H</i>∂Ω differs from <i>H</i> by a Hilbert–Schmidt operator with smooth kernel, it follows that <i>H</i>∂Ω + <i>H</i>∂Ω* is a Hilbert-Schmidt operator with smooth kernel. The kernel can also be computed explicitly using the truncated Hilbert transforms for ∂Ω: </p> C ( s , t ) = 1 π i ( z ˙ ( t ) z ( t ) − z ( s ) − z ˙ ( s ) ¯ z ( t ) ¯ − z ( s ) ¯ ) , {\displaystyle \displaystyle {C(s,t)={1 \over \pi i}\left({{\dot {z}}(t) \over z(t)-z(s)}-{{\overline {{\dot {z}}(s)}} \over {\overline {z(t)}}-{\overline {z(s)}}}\right),}} <p>and it can be verified directly that this is a smooth function on T × T.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="plemeljsokhotski-relation">Plemelj–Sokhotski relation</h2> <p class="note">See also: <a href="/facts/Sokhotski%25E2%2580%2593Plemelj_theorem/xoxGrXR6">Sokhotski–Plemelj theorem</a></p> <p>Let <i>C</i>− and <i>C</i>+ be the Cauchy integral operators for Ω and Ω<i>c</i>. Then </p> C ± ∘ H = ± i C ± . {\displaystyle \displaystyle {C_{\pm }\circ H=\pm iC_{\pm }.}} <p>Since the operators <i>C</i>−, <i>C</i>+ and <i>H</i> are bounded, it suffices to check this on rational functions <i>F</i> with poles off ∂Ω and vanishing at ∞ by the Hartogs–Rosenthal theorem. The rational function can be written as a sum of functions <i>F</i> = <i>F</i>− + <i>F</i>+ where <i>F</i>− has poles only in Ω<i>c</i> and <i>F</i>+ has poles only in Let <i>f</i>, <i>f</i>± be the restrictions of <i>f</i>, <i>f</i>± to ∂Ω. By <a href="/facts/Cauchy%2527s_integral_formula/4B35nkvV">Cauchy's integral formula</a> </p> C ± f ± = F ± , C ± f ∓ = 0. {\displaystyle \displaystyle {C_{\pm }f_{\pm }=F_{\pm },\,\,\,C_{\pm }f_{\mp }=0.}} <p>On the other hand, it is straightforward to check that<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> H f ± = ± i f ± . {\displaystyle \displaystyle {Hf_{\pm }=\pm if_{\pm }.}} <p>Indeed, by Cauchy's theorem, since <i>F</i>− is holomorphic in Ω, </p> ∫ ∂ Ω , | z − w | ≥ ε f − ( z ) z − w d z = − ∫ | z − w | = ε , z ∈ Ω F − ( z ) z − w d z . {\displaystyle \displaystyle {\int _{\partial \Omega ,\,\,|z-w|\geq \varepsilon }{f_{-}(z) \over z-w}\,dz=-\int _{|z-w|=\varepsilon ,\,\,z\in \Omega }{F_{-}(z) \over z-w}\,dz.}} <p>As ε tends to 0, the latter integral tends to π<i>i</i> <i>f</i>−(<i>w</i>) by the <a href="/facts/Residue_calculus/4ILprcRr">residue calculus</a>. A similar argument applies to <i>f</i>+, taking the circular contour on the right inside Ω<i>c</i>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p><p>By continuity it follows that <i>H</i> acts as multiplication by <i>i</i> on H2− and as multiplication by −<i>i</i> on H2+. Since these spaces are closed and their sum dense, it follows that </p> H 2 = − I . {\displaystyle \displaystyle {H^{2}=-I.}} <p>Moreover, H2− and H2+ must be the ±<i>i</i> eigenspaces of <i>H</i>, so their sum is the whole of L2(∂Ω). The Plemelj–Sokhotski relation for <i>f</i> in L2(∂Ω) is the relation </p> − i H f = C − f | ∂ Ω − C + f | ∂ Ω . {\displaystyle \displaystyle {-iHf=C_{-}f|_{\partial \Omega }-C_{+}f|_{\partial \Omega }.}} <p>It has been verified for <i>f</i> in the Hardy spaces H2±(∂Ω), so is true also for their sum. The Cauchy idempotent <i>E</i> is defined by </p> H = i ( 2 E − I ) . {\displaystyle \displaystyle {H=i(2E-I).}} <p>The range of <i>E</i> is thus H2−(∂Ω) and that of <i>I</i> − <i>E</i> is H2+(∂Ω). From the above<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> E f = C − f | ∂ Ω , ( I − E ) f = C + f | ∂ Ω . {\displaystyle \displaystyle {Ef=C_{-}f|_{\partial \Omega },\,\,\,\,(I-E)f=C_{+}f|_{\partial \Omega }.}} <h2 id="operators-on-a-closed-curve">Operators on a closed curve</h2> <p class="note">See also: <a href="/facts/Szeg%25C5%2591_kernel/519k3t4d">Szegő kernel</a> and <a href="/facts/Neumann%25E2%2580%2593Poincar%25C3%25A9_operator/3oDjNnFG">Neumann–Poincaré operator</a></p> <p>Two other operators defined on a closed curve ∂Ω can be expressed in terms of the Hilbert and Cauchy transforms <i>H</i> and <i>E</i>. <a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>The Szegő projection <i>P</i> is defined to be the orthogonal projection onto Hardy space H2(∂Ω). Since <i>E</i> is an idempotent with range H2(∂Ω), <i>P</i> is given by the Kerzman–Stein formula: </p> P = E ( I + E − E ∗ ) − 1 . {\displaystyle \displaystyle {P=E(I+E-E^{*})^{-1}.}} <p>Indeed, since <i>E</i> − <i>E</i>* is skew-adjoint its spectrum is purely imaginary, so the operator <i>I</i> + <i>E</i> − <i>E</i>* is invertible.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> It is immediate that </p> E P = P , P E = E . {\displaystyle \displaystyle {EP=P,\,\,\,PE=E.}} <p>Hence <i>PE</i>* = <i>P</i>. So </p> P ( I + E − E ∗ ) = P + E − P = E . {\displaystyle \displaystyle {P(I+E-E^{*})=P+E-P=E.}} <p>Since the operator <i>H</i> + <i>H</i>* is a <a href="/facts/Hilbert%25E2%2580%2593Schmidt_operator/Xhy17yn6">Hilbert–Schmidt operator</a> with smooth kernel, the same is true for <i>E</i> − <i>E</i>*.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p><p>Moreover, if <i>J</i> is the conjugate-linear operator of complex conjugation and <i>U</i> the operator of multiplication by the unit tangent vector: </p> J f ( t ) = f ( t ) ¯ , U f ( t ) = z ˙ ( t ) ⋅ f ( t ) . {\displaystyle \displaystyle {Jf(t)={\overline {f(t)}},\,\,\,Uf(t)={\dot {z}}(t)\cdot f(t).}} <p>then the formula for the truncated Hilbert transform on ∂Ω immediately yields the following identity for adjoints </p> ( H ε ) ∗ = J U H ε U ∗ J . {\displaystyle \displaystyle {(H^{\varepsilon })^{*}=JUH^{\varepsilon }U^{*}J.}} <p>Letting ε tend to 0, it follows that </p> H ∗ = J U H U ∗ J {\displaystyle \displaystyle {H^{*}=JUHU^{*}J}} <p>and hence </p> I − E ∗ = J U E U ∗ J . {\displaystyle \displaystyle {I-E^{*}=JUEU^{*}J.}} <p>The comparison with the Hilbert transform for the circle shows that commutators of <i>H</i> and <i>E</i> with diffeomorphisms of the circle are Hilbert–Schmidt operators. Similar their commutators with the multiplication operator corresponding to a smooth function <i>f</i> on the circle is also Hilbert–Schmidt operators. Up to a constant the kernel of the commutator with <i>H</i> is given by the smooth function </p> A ( s , t ) = f ( s ) − f ( t ) z ( s ) − z ( t ) . {\displaystyle \displaystyle {A(s,t)={f(s)-f(t) \over z(s)-z(t)}.}} <p>The Neumann–Poincaré operator <i>T</i> is defined on real functions <i>f</i> as </p> T f ( w ) = 1 2 π ∫ ∂ Ω ∂ n ( log | z − w | ) f ( z ) = 1 2 ℜ ( H f ) ( w ) . {\displaystyle \displaystyle {Tf(w)={1 \over 2\pi }\int _{\partial \Omega }\partial _{n}(\log |z-w|)f(z)={1 \over 2}\Re (Hf)(w).}} <p>Writing <i>h</i> = <i>f</i> + <i>ig</i>,<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> 2 T h = ℜ ( H f ) + i ℜ ( H g ) = 1 2 ( H f + J H f + i H g + i J H g ) = 1 2 ( H + J H J ) h {\displaystyle \displaystyle {2Th=\Re (Hf)+i\Re (Hg)={1 \over 2}(Hf+JHf+iHg+iJHg)={1 \over 2}(H+JHJ)h}} <p>so that </p> T = 1 4 ( H + J H J ) = 1 4 ( H + U H ∗ U ∗ ) , {\displaystyle \displaystyle {T={1 \over 4}(H+JHJ)={1 \over 4}(H+UH^{*}U^{*}),}} <p>a Hilbert–Schmidt operator. </p> <h2 id="classical-definition-of-hardy-space">Classical definition of Hardy space</h2> <p>The classical definition of Hardy space is as the space of holomorphic functions <i>F</i> on Ω for which the functions <i>F</i><i>s</i> = <i>F</i>|∂Ω<i>s</i> have bounded norm in L2(∂Ω). An argument based on the <a href="/facts/Carath%25C3%25A9odory_kernel_theorem/6Sfk9kTr">Carathéodory kernel theorem</a> shows that this condition is satisfied whenever there is a family of Jordan curves in Ω, eventually containing any compact subset in their interior, on which the integral means of <i>F</i> are bounded.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p><p>To prove that the classical definition of Hardy space gives the space H2(∂Ω), take <i>F</i> as above. Some subsequence <i>h</i><i>n</i> = <i>F</i><i>s</i><i>n</i> converges weakly in L2(∂Ω) to <i>h</i> say. It follows that <i>Ch</i> = <i>F</i> in Ω. In fact, if <i>C</i>n is the Cauchy integral operator corresponding to Ω<i>s</i><i>n</i>, then<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> C h ( a ) − F ( a ) = C h ( a ) − C n h n ( a ) = C ( h − h n ) ( a ) + [ ( C − C n ) h n ] ( a ) . {\displaystyle \displaystyle {Ch(a)-F(a)=Ch(a)-C_{n}h_{n}(a)=C(h-h_{n})(a)+[(C-C_{n})h_{n}](a).}} <p>Since the first term on the right hand side is defined by pairing <i>h</i> − <i>h</i><i>n</i> with a fixed L2 function, it tends to zero. If <i>z</i><i>n</i>(<i>t</i>) is the complex number corresponding to v<i>s</i><i>n</i>, then </p> [ ( C − C n ) h n ] ( a ) = 1 2 π i ∫ h n ( t ) ( z ˙ ( t ) z ( t ) − a − z ˙ n ( t ) z n ( t ) − a ) d t . {\displaystyle \displaystyle {[(C-C_{n})h_{n}](a)={1 \over 2\pi i}\int h_{n}(t)\left({{\dot {z}}(t) \over z(t)-a}-{{\dot {z}}_{n}(t) \over z_{n}(t)-a}\right)\,dt.}} <p>This integral tends to zero because the L2 norms of <i>h</i><i>n</i> are uniformly bounded while the bracketed expression in the integrand tends to 0 uniformly and hence in L2. </p><p>Thus <i>F</i> = <i>Ch</i>. On the other hand, if <i>E</i> is the Cauchy idempotent with range H2(∂Ω), then <i>C</i> ∘ <i>E</i> = <i>C</i>. Hence <i>F</i> =<i>Ch</i> = <i>C</i> (<i>Eh</i>). As already shown <i>F</i><i>s</i> tends to <i>Ch</i> in L2(∂Ω). But a subsequence tends weakly to <i>h</i>. Hence <i>Ch</i> = <i>h</i> and therefore the two definitions are equivalent.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p> <h2 id="generalizations">Generalizations</h2> <p>The theory for multiply connected bounded domains with smooth boundary follows easily from the simply connected case.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> There are analogues of the operators <i>H</i>, <i>E</i> and <i>P</i>. On a given component of the boundary, the singular contributions to <i>H</i> and <i>E</i> come from the singular integral on that boundary component, so the technical parts of the theory are direct consequences of the simply connected case. </p><p>Singular integral operators on spaces of <a href="/facts/H%25C3%25B6lder_continuous/5VUpHT2q">Hölder continuous</a> functions are discussed in Gakhov (1990). Their action on L<i>p</i> and Sobolev spaces is discussed in Mikhlin & Prössdorf (1986). </p> <h2 id="notes">Notes</h2> <ul><li>Bell, S. R. (1992), <i>The Cauchy transform, potential theory, and conformal mapping</i>, Studies in Advanced Mathematics, CRC Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8493-8270-X</li> <li>Bell, S. R. (2016), <i>The Cauchy transform, potential theory, and conformal mapping</i>, Studies in Advanced Mathematics (2nd ed.), CRC Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781498727211</li> <li>Conway, John B. (1995), <i>Functions of one complex variable II</i>, Graduate texts in mathematics, vol. 159, Springer, p. 197, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0387944605</li> <li>Conway, John B. (2000), <i>A course in operator theory</i>, <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>, vol. 21, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, pp. 175–176, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0821820656</li> <li>David, Guy (1984), "Opérateurs intégraux singuliers sur certaines courbes du plan complexe", <i>Ann. Sci. Éc. Norm. Supér.</i>, 17: 157–189, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.24033%2Fasens.1469">10.24033/asens.1469</a></li> <li>Duren, Peter L. (1970), <i>Theory of Hp spaces</i>, Pure and Applied Mathematics, vol. 38, Academic Press</li> <li>Gakhov, F. D. (1990), <i>Boundary value problems. Reprint of the 1966 translation</i>, Dover Publications, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-66275-6</li> <li>Gamelin, Theodore W. (2005), <i>Uniform algebras</i> (2nd ed.), <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, pp. 46–47, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0821840495</li> <li>Garnett, J. B. (2007), <i>Bounded analytic functions</i>, Graduate Texts in Mathematics, vol. 236, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-33621-3</li> <li>Gohberg, Israel; Krupnik, Naum (1992), <i>One-dimensional linear singular integral equations. I. Introduction</i>, Operator Theory: Advances and Applications, vol. 53, Birkhäuser, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-7643-2584-4</li> <li>Goluzin, G. M. (1969), <i>Geometric theory of functions of a complex variable</i>, Translations of Mathematical Monographs, vol. 26, American Mathematical Society</li> <li><a href="/facts/Yitzhak_Katznelson/hDVBzEYv">Katznelson, Yitzhak</a> (2004), <i>An Introduction to Harmonic Analysis</i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-54359-0</li> <li>Kerzman, N.; Stein, E. M. (1978), "The Cauchy kernel, the Szegö kernel, and the Riemann mapping function", <i>Math. Ann.</i>, 236: 85–93, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01420257">10.1007/bf01420257</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121336615">121336615</a></li> <li>Muskhelishvili, N. I. (1992), <i>Singular integral equations. Boundary problems of function theory and their application to mathematical physics</i>, Dover, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-66893-2</li> <li><a href="/facts/Solomon_Mikhlin/Y9wL0IKk">Mikhlin, Solomon G.</a>; Prössdorf, Siegfried (1986), <i>Singular integral operators</i>, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-15967-3</li> <li>Pressley, Andrew; Segal, Graeme (1986), <i>Loop groups</i>, Oxford University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-853535-X</li> <li>Segal, Graeme (1981), <a href="http://projecteuclid.org/euclid.cmp/1103919978">"Unitary representations of some infinite-dimensional groups"</a>, <i>Comm. Math. Phys.</i>, 80 (3): 301–342, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1981CMaPh..80..301S">1981CMaPh..80..301S</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01208274">10.1007/bf01208274</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121367853">121367853</a></li> <li><a href="/facts/Harold_S._Shapiro/C29b4zx5">Shapiro, H. S.</a> (1992), <i>The Schwarz function and its generalization to higher dimensions</i>, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9, Wiley-Interscience, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-57127-X</li> <li><a href="/facts/E.C._Titchmarsh/607FjJwB">Titchmarsh, E. C.</a> (1939), <i>The Theory of Functions</i> (2nd ed.), Oxford University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-853349-7 {{citation}}: ISBN / Date incompatibility (help)</li> <li>Torchinsky, Alberto (2004), <i>Real-Variable Methods in Harmonic Analysis</i>, Dover, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-43508-3</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Torchinsky 2004, pp. 65–66 - Torchinsky, Alberto (2004), Real-Variable Methods in Harmonic Analysis, Dover, ISBN 0-486-43508-3 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Bell 1992, pp. 14–15 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>See: Mikhlin & Prössdorf 1986 Segal 1981 Pressley & Segal 1986 - Mikhlin, Solomon G.; Prössdorf, Siegfried (1986), Singular integral operators, Springer-Verlag, ISBN 3-540-15967-3 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>See: Bell 1992 Gamelin 2005 Duren 1970 Conway 1995 Conway 2000 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Bell 1992, pp. 19–20 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Bell 1992, pp. 19–22 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Bell 1992, pp. 16–21 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Bell 1992, p. 22 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>See: Gohberg & Krupnik 1992 Gakhov 1990 - Gohberg, Israel; Krupnik, Naum (1992), One-dimensional linear singular integral equations. I. Introduction, Operator Theory: Advances and Applications, vol. 53, Birkhäuser, ISBN 3-7643-2584-4 <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Bell 1992, pp. 15–16 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>See: Gakhov 1990 Bell 1992 Goluzin 1969 Gohberg & Krupnik 1992 - Gakhov, F. D. (1990), Boundary value problems. Reprint of the 1966 translation, Dover Publications, ISBN 0-486-66275-6 <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Titchmarsh 1939 - Titchmarsh, E. C. (1939), The Theory of Functions (2nd ed.), Oxford University Press, ISBN 0-19-853349-7 <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Bell 1992 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>See: Kerzman & Stein 1978 Bell 1992 Shapiro 1992 - Kerzman, N.; Stein, E. M. (1978), "The Cauchy kernel, the Szegö kernel, and the Riemann mapping function", Math. Ann., 236: 85–93, doi:10.1007/bf01420257, S2CID 121336615 <a href="https://doi.org/10.1007%2Fbf01420257" target="_blank">https://doi.org/10.1007%2Fbf01420257</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Shapiro 1992, p. 65 - Shapiro, H. S. (1992), The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9, Wiley-Interscience, ISBN 0-471-57127-X <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Bell 1992 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Shapiro 1992, pp. 66–67 - Shapiro, H. S. (1992), The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9, Wiley-Interscience, ISBN 0-471-57127-X <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Duren 1970, p. 168 - Duren, Peter L. (1970), Theory of Hp spaces, Pure and Applied Mathematics, vol. 38, Academic Press <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Bell 1992, pp. 17–18 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Bell 1992, pp. 19–20 - Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>See: Gakhov 1990 Mikhlin & Prössdorf 1986 - Gakhov, F. D. (1990), Boundary value problems. Reprint of the 1966 translation, Dover Publications, ISBN 0-486-66275-6 <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> </ol>