Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1960. Volume 134, No. 6
MATHEMATICS
V. A. KAKICHEV
THE CAUCHY TRANSFORM OF GENERALIZED FUNCTIONS
(Presented by Academician M. A. Lavrent’ev on 28 V 1960)
1°. Let the simple smooth closed contour (L_k) divide the plane of the complex variable (z_k) ((k=1,2,\ldots,n)) into two parts: the interior (D_k^+) and the exterior (D_k^-). Put:
[
\Delta_{p_1\ldots p_m}^{+}
=
D_1^+\times\cdots\times D_{p_1-1}^+\times D_{p_1}^{-}\times D_{p_1+1}^+\times\cdots\times D_{p_m-1}^+\times D_{p_m}^{-}\times D_{p_m+1}^+\times\cdots\times D_n^+,
]
[
\tau_{p_1\ldots p_m}
=
(\tau_1,\ldots,\tau_{p_1-1},\, t_{p_1},\, \tau_{p_1+1},\ldots,\tau_{p_m-1},\, t_{p_m},\, \tau_{p_m+1},\ldots,\tau_n),
]
[
t_{p_1\ldots p_m}
=
(t_1,\ldots,t_{p_1-1},\, t_{p_1+1},\ldots,t_{p_m-1},\, t_{p_m+1},\ldots,t_n),
\quad \text{where } t_j\in L_j,\ \tau_j\in L_j,
]
[
d\tau_{p_1\ldots p_m}
=
d\tau_1\cdots d\tau_{p_1-1}d\tau_{p_1+1}\cdots d\tau_{p_m-1}d\tau_{p_m+1}\cdots d\tau_n
]
[
(0\le m\le n,\quad 1\le p_1<\cdots<p_m\le n),
]
in particular: (t=(t_1,\ldots,t_n)), (\tau=(\tau_1,\ldots,\tau_n)), (d\tau=d\tau_1\cdots d\tau_n), and (z=(z_1,\ldots,z_n));
(L=L_1\times\cdots\times L_n); ({1,2,\ldots,n}\setminus{p_1,p_2,\ldots,p_m}={q_1,q_2,\ldots,q_{n-m}}), (1\le m\le n);
[
\varphi_j=S_j\varphi=\frac{1}{\pi i}\int_{L_j}\frac{\varphi(t\tau_j)\,d\tau_j}{\tau_j-t_j},
\quad
1\le j\le n,
\tag{1}
]
[
\varphi_{p_1\ldots p_m}=S_{p_1\ldots p_m}\varphi=S_{p_1}\cdots S_{p_m}\varphi,\quad
\varphi_{\overline{p_1\ldots p_m}}=\varphi_{q_1\ldots q_{n-m}},\quad
\widetilde{\varphi}=S\varphi=S_{1\ldots n}\varphi
]
and
[
S_0\varphi=\varphi.
]
If the function (\varphi(t)) on (L) satisfies a Hölder condition with exponents (\alpha_1,\ldots,\alpha_n), then we shall write (\varphi(t)\in H(\alpha)); if, moreover, (\varphi(t)) is differentiable (k_j) times with respect to (t_j), and
[
D_{k_1\ldots k_n}\varphi(t)
=
\frac{\partial^{k_1+\cdots+k_n}\varphi(t)}{\partial t_1^{k_1}\cdots \partial t_n^{k_n}}
\in H(\alpha),
]
then we shall write (\varphi(t)\in H_k(\alpha)). If (\varphi(t)\in H(\alpha)), then the limiting values, as (z\to t\in L) and (z\in\Delta_{p_1\ldots p_m}^{-}), of the Cauchy-type integral
[
\Phi(z)=\frac{1}{(2\pi i)^n}\int_L
\frac{\varphi(\tau)\,d\tau}{\prod_{k=1}^n(\tau_k-t_k)},
\tag{2}
]
which we shall denote by (\Phi_{p_1\ldots p_m}^{-}(t)), exist ({}^{(1)}) and are computed by the Sokhotski formulas
[
2^n\Phi_{p_1\ldots p_m}^{-}(t)
=
\left{
\prod_{\mu=1}^{m}(S_0+S_{p_\mu})
\prod_{\nu=1}^{\,n-m}(-S_0+S_{q_\nu})
\right}\varphi,
]
where the singular operators (1) are multiplied according to the rules:
[
S_0S_j=S_j,\quad
S_kS_j=S_jS_k=S_{jk},\quad
S_j^2=S_0,\quad
j=1,2,\ldots,n.
]
2°. Theorem 1. Let (\varphi(t)\in H(\alpha)); then the values of the integral (2) near (L) satisfy the condition
[
\left|\Phi(z)-\Phi_{p_1\ldots p_m}^{-}(t)\right|
\le
M\sum_{k=1}^{n}|t_k-z_k|^{\beta_k},
\quad
M>0,\quad 0<\beta_k<\alpha_k,\quad k=1,2,\ldots,n.
]
Theorem 1 is proved in the same way as the analogous theorem in the case of one variable ({}^{(3)}).
Lemma*. Let (\varphi(t)\in H_k(\alpha)); then
[
\lim_{z\to t} D_{l_1\ldots l_n}\Phi(z)
=
\left[D_{l_1\ldots l_n}\Phi(t)\right]^{-}{p_1\ldots p_m}
=
D(t),}\Phi^{-}_{p_1\ldots p_m
]
where (z\in \Delta^{-}_{p_1\ldots p_m}), (t\in L), (0\le m\le n), (1\le p_1<\cdots<p_m\le n), (0\le l_j\le k_j,\ j=1,2,\ldots,n).
The lemma is proved by mathematical induction, using the fundamental Cauchy theorem, the Sokhotski formulas, and Theorem 1.
Theorem 2. If (\varphi(t)\in H_k(\alpha)), then the equalities
[
D_{l_1\ldots l_n}S_{r_1\ldots r_k}\varphi
=
S_{r_1\ldots r_k}D_{l_1\ldots l_n}\varphi,
]
[
1\le k\le n,\qquad
1\le r_1<\cdots<r_k\le n,\qquad
0\le l_j\le k_j
\tag{3}
]
hold.
Theorem 2 is a consequence of the lemma, since the functions (\varphi_{r_1\ldots r_k}(t)), by the Sokhotski formulas, are linear combinations of the functions (\Phi^{-}_{p_1\ldots p_m}(t)), (0\le m\le n,\ 1\le p_1<\cdots<p_m\le n).
3°. By (H) we denote the class of functions satisfying the Hölder condition with arbitrary exponents (\alpha_j), (0<\alpha_j\le 1,\ j=1,2,\ldots,n). The operators (S_{p_1\ldots p_m}) map the class (H) into itself (\left({}^{1}\right)), and (S_{p_1\ldots p_m}=S_0). Let (\varphi(t)\in H). We define the Cauchy transform by the equality (S\widetilde{\varphi}=\varphi); then (\widetilde{\varphi}(t)\in H), and the inversion formula has the form (S\widetilde{\varphi}=\varphi).
If (\varphi(t)\in H) and (\psi(t)\in H), then
[
(\varphi_{p_1\ldots p_m}(t),\psi(t))
=
(-1)^m(\varphi(t),\psi_{p_1\ldots p_m}(t)),
\tag{4}
]
[
(\varphi,\psi)=\int_L \varphi(\tau)\psi(\tau)\,d\tau.
\tag{5}
]
We denote by (H') the class of generalized functions (\left({}^{2}\right)) generated by the functional (5) over the basic class (H). The operator (S_{p_1\ldots p_m}) on functions of the class (H') is defined by the equality (4). Let (\varphi(t)\in H') and (\psi(t)\in H); then
[
(S_{p_1\ldots p_m}\varphi_{p_1\ldots p_m},\psi)
=
(-1)^m(\varphi_{p_1\ldots p_m},\psi_{p_1\ldots p_m})
=
(\varphi,S^2_{p_1\ldots p_m}\psi)
=
(\varphi,\psi).
]
Hence for (m=n) the validity of the inversion formula in (H') follows.
By (H_k) we denote the class of functions (\varphi(t)) such that (D_{k_1\ldots k_n}\varphi(t)\in H). The corresponding class of generalized functions will be denoted by (H'_k). Using the formula for differentiating generalized functions in (3), we have
[
D_{l_1\ldots l_n}S\varphi
=
\frac{l_1!\cdots l_n!}{(\pi i)^n}
\int_L
\frac{\varphi(\tau)\,d\tau}
{\displaystyle\prod_{k=1}^{n}(\tau_k-t_k)^{l_k+1}}.
\tag{6}
]
We shall say that (L) belongs to the class (P_{k_1\ldots k_n}(E)), where (E\subset L), if there exists a neighborhood (\widetilde E\subset L) of the manifold (E), differentiable (k_j+1) times with respect to (t_j), (j=1,2,\ldots,n). Suppose that on (L\in P_{k_{p_1}\ldots k_{p_m}}(\Gamma_{p_1\ldots p_m})) a surface
(\Gamma_{p_1\ldots p_m}) is given:
[
t_{p_j}=\tilde t_{p_j}(t_{p_1\ldots p_m}),\qquad
j=1,2,\ldots,m,\qquad
1\le m\le n,
]
on which the delta-function (\delta(\Gamma_{p_1\ldots p_m})) is concentrated, and let (\psi(t)\in H_k); then, by virtue of (4)—(6), for (1\le l_j\le k_{p_j}) we have (\left({}^{2}\right))
[
\left(
S\frac{\partial^{l_1+\cdots+l_m}}
{\partial \tau_{p_1}^{l_1}\cdots \partial \tau_{p_m}^{l_m}}
\delta(\Gamma_{p_1\ldots p_m}),\psi(t)
\right)
=
(-1)^{\,n+\sum_{\alpha=1}^{m}l_\alpha}
\left(
\delta(\Gamma_{p_1\ldots p_m}),
\frac{\partial^{l_1+\cdots+l_m}\widetilde{\psi}(t)}
{\partial \tau_{p_1}^{l_1}\cdots \partial \tau_{p_m}^{l_m}}
\right)
=
]
* The proof of the lemma for (n=1) was communicated to the author by S. Ya. Al’per.
[
= (-1)^{\,n+\sum_{\alpha=1}^{m}(l_\alpha+p_\alpha)-\frac{m(m-1)}{2}}
\frac{l_1!\ldots l_m!}{(\pi i)^n}
\int_{\Gamma_{p_1\ldots p_m}} d t_{p_1\ldots p_m}\,
\frac{\displaystyle \prod_{k=1}^{n-m}(t_{q_k}-\tau_{p_k})^{-1}\psi(t)\,dt}
{\displaystyle \prod_{j=1}^{m}[t_{p_j}-f_{p_j}(\tau_{p_1\ldots p_m})]^{l_j+1}} .
]
Hence
[
S\,\frac{\partial^{l_1+\cdots+l_m}}
{\partial \tau_{p_1}^{l_1}\ldots \partial \tau_{p_m}^{l_m}}
\delta(\Gamma_{p_1\ldots p_m})
=
(-1)^{\sum_{\alpha=1}^{m}(l_\alpha+p_\alpha)-\frac{m(m+1)}{2}}
\frac{l_1!\ldots l_m!}{(\pi i)^n}
\times
]
[
\times
\int_{\Gamma_{p_1\ldots p_m}}
\frac{\displaystyle \prod_{k=1}^{n-m}(\tau_{q_k}-t_{q_k})^{-1}\,d\tau_{p_1\ldots p_m}}
{\displaystyle \prod_{j=1}^{m}[t_{p_j}-f_{p_j}(\tau_{p_1\ldots p_m})]^{l_j+1}} .
\tag{7}
]
(4^\circ). Let (a(t)\in H,\ \varphi(t)\in H); then the Poincaré—Bertrand formula (1) holds, which, using the operators (1), can be written as follows:
[
Sa\widetilde{\varphi}
=
(-1)^n\widetilde{S}a\varphi+a\varphi+
\sum_{k=1}^{n}
\left(
\sum_{r_1=1}^{n}\ldots\sum_{r_k=k}^{n}
a_{r_1\ldots r_k}\varphi_{r_1\ldots r_k}
\right)
+
]
[
+
\sum_{m=1}^{n-1}(-1)^m
\left{
\sum_{\substack{p_1=1\ p_1<\cdots<p_m}}^{n}
\ldots
\sum_{p_m=m}^{n}
S_{p_1\ldots p_m}
\left[
\varphi a_{p_1\ldots p_m}
+
\right.\right.
]
[
\left.\left.
+
\sum_{j=1}^{n-m}
\left(
\sum_{\substack{q_1=1\ q_1<\cdots<q_j}}^{n-m}
\ldots
\sum_{q_j=j}^{n-m}
\varphi_{q_1\ldots q_j}a_{p_1\ldots p_m q_1\ldots q_j}
\right)
\right]
\right}.
]
Replacing here (a(t)) by (\widetilde{a}(t)) and taking into account that (S_{p_1\ldots p_m}\widetilde{a}=a_{p_1\ldots p_m}), we obtain
[
S\widetilde{a}\widetilde{\varphi}
=
\frac{1}{(\pi i)^n}
\int
\left[
(-1)^n a(\tau)\varphi(\tau)+a(\tau)\varphi(t)+
\right.
]
[
\left.
+
\sum_{k=1}^{n}
\left(
\sum_{\substack{r_1=1\ r_1<\cdots<r_k}}^{n}
\ldots
\sum_{r_k=k}^{n}
a(t_{t_{r_1\ldots r_k}})\,
\varphi(t_{t_{r_1\ldots r_k}})
\right)
+
\right.
]
[
\left.
+
\sum_{m=1}^{n-1}(-1)^m
\left[
\sum_{\substack{p_1=1\ p_1<\cdots<p_m}}^{n}
\ldots
\sum_{p_m=m}^{n}
a(\tau)\varphi(t_{\tau_{p_1\ldots p_m}})
+
\right.\right.
]
[
\left.\left.
+
\sum_{j=1}^{n-m}
\left(
\sum_{\substack{q_1=1\ q_1<\cdots<q_j}}^{n-m}
\ldots
\sum_{q_j=j}^{n-m}
a(\tau_{t_{q_1\ldots q_j}})
\varphi(t_{\tau_{p_1\ldots p_m q_1\ldots q_j}})
\right)
\right]
\right]
\frac{d\tau}{\displaystyle \prod_{k=1}^{n}(\tau_k-t_k)}
\equiv a(t)*\varphi(t).
\tag{8}
]
The expression (a\varphi) with respect to the Cauchy transform plays the same role as convolutions in the theory of integral transforms ((^4)), and we shall call it the convolution for the Cauchy transform*.
Theorem 3. If (a(t)\in H,\ \varphi(t)\in H); (\widetilde{a}(t),\ \widetilde{\varphi}(t)) are their Cauchy transforms, then
[
S\widetilde{a}\widetilde{\varphi}=a\varphi,\qquad
Sa\varphi=\widetilde{a}\widetilde{\varphi}.
\tag{9}
]
If (a(t)\in H,\ \varphi(t)\in H'), then formulas (9) hold in the class (H').
For (a(t)\in H,\ \varphi(t)\in H), formulas (9) are obvious. In the case (\varphi(t)\in H'), (a(t)\in H,\ \psi(t)\in H), we have
[
(S\widetilde{a}\widetilde{\varphi},\psi)
=
(-1)^n(\widetilde{a}\widetilde{\varphi},\widetilde{\psi})
=
(-1)^n(\varphi,a\widetilde{\psi})
=
(\varphi,Sa\psi).
]
Replacing in the last equality on the right (S\widetilde{a}\widetilde{\psi}) by formula (8) and using (4), we find that
[
(S\widetilde{a}\widetilde{\varphi},\psi)=(a\varphi,\psi).
]
Hence (S\widetilde{a}\widetilde{\varphi}=a\varphi).
5°. An equation of convolution type
[
\lambda\varphi(t)+a(t)*\varphi(t)=f(t),\qquad t\in L,
\tag{10}
]
where (a(t)\in H) and (f(t)\in H) are known functions, and (\lambda) is a parameter, for (\lambda+\tilde a(t)\ne0) on (L), has in the class (H) the unique solution:
[
\varphi(t)=\frac{1}{(\pi i)^n}\int_L
\frac{\tilde f(\tau)}{\lambda+\tilde a(\tau)}
\frac{d\tau}{\displaystyle\prod_{k=1}^{n}(\tau_k-t_k)} .
\tag{11}
]
Applying the operator (S) to both sides of (10) and taking (9) into account, we find (\tilde\varphi(t)). Applying the operator (S) to (\tilde\varphi(t)), we obtain (11).
Suppose now that (\lambda+\tilde a(t)) vanishes on surfaces of the form (\Gamma_{p_1\ldots p_m}\subset L), together with its derivatives with respect to (t_{p_j}) of order (k_{p_j}), and (L\in P_{k_{p_1}\ldots k_{p_m}}(\Gamma_{p_1\ldots p_m})).* In this case the solution of ([\lambda+\tilde a(t)]\tilde\varphi(t)=0) in the class (H') is ({}^{(5)}) a linear combination of the delta-function (\delta(\Gamma_{p_1\ldots p_m})) and its derivatives up to the derivative
[
\partial^{\,k_{p_1}+\cdots+k_{p_m}-m}/\partial t_{p_1}^{\,k_{p_1}-1}\cdots \partial t_{p_m}^{\,k_{p_m}-1}.
]
From the validity of the inversion formula in the class (H') and Theorem 3 it follows that the solution of the equation
[
\lambda\varphi(t)+a(t)*\varphi(t)=0
]
in the class (H') is a linear combination of the functions standing on the right in (7).
For the solvability of the nonhomogeneous equation in this case it is necessary and sufficient that the function (\tilde f(t)) vanish together with its derivatives on the same surfaces (\Gamma_{p_1\ldots p_m}) as the function (\lambda+\tilde a(t)), and moreover the order of the derivatives of the function (\tilde f(t)) that vanish on these surfaces must be no lower than that of the corresponding derivatives of the function (\lambda+\tilde a(t)).
Let now (L_j) ((j=1,2,\ldots,n)) be the circles (|z_j|=\rho_j), (\alpha_j=\pi/k_j^{-1}), (l_j) and (k_j) integers ((k_j>0)), (te^{i\alpha}=(t_1e^{i\alpha_1},\ldots,t_ne^{i\alpha_n})), (a(t)\in H) and (f(t)\in H); then, applying to the equation
[
\varphi(t)+a(te^{i\alpha})*\varphi(te^{i\alpha})=f(t),\qquad t\in L,
\tag{12}
]
the operator (S), we obtain
[
\tilde\varphi(t)+\tilde a(te^{i\alpha})\tilde\varphi(te^{i\alpha})=\tilde f(t).
\tag{13}
]
Substituting in (13) successively instead of (t) the values (te^{i\alpha},\ldots,te^{i(k-1)\alpha}), where (k) is equal to twice the least common multiple of the numbers (k_1,\ldots,k_n), and eliminating from the resulting chain of equalities the functions (\varphi(te^{ip\alpha})) ((p=1,2,\ldots,k-1)), we find (\tilde\varphi(t)). If
[
1-\prod_{p=1}^{k}\tilde a(te^{ip\alpha})\ne0
]
on (L), then, applying the operator (S) to (\tilde\varphi(t)), we find the unique solution in the class (H) of equation (12)
[
\varphi(t)=\frac{1}{(\pi i)^n}\int_L
\frac{
\tilde f(\tau)+\displaystyle\sum_{p=1}^{n}(-1)^p \tilde f(\tau e^{ip\alpha})\prod_{q=1}^{p}\tilde a(\tau e^{iq\alpha})
}{
\left[1-\displaystyle\prod_{p=1}^{k}\tilde a(\tau e^{ip\alpha})\right]\displaystyle\prod_{j=1}^{n}(\tau_j-t_j)
}\,d\tau .
]
Shakhty Pedagogical Institute
Received
18 V 1960
References
- V. A. Kakichev, Uch. zap. Shakhtinsk. ped. inst., 2, 6 (1959).
- I. M. Gel'fand, G. E. Shilov, Generalized Functions, Vol. 1, 1958.
- N. I. Muskhelishvili, Singular Integral Equations, 1946.
- E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 1948.
- L. Schwartz, Théorie des distributions, Paris, 1950—1951.
* If these surfaces intersect one another along manifolds of analogous structure (and only such are studied here) of lower dimension, then these manifolds are regarded as independent.