Abstract
Full Text
UDC 517.948.32
MATHEMATICS
S. G. SAMKO
A GENERALIZED ABEL EQUATION, THE FOURIER TRANSFORM, AND EQUATIONS OF CONVOLUTION TYPE
(Presented by Academician A. A. Dorodnitsyn on 24 XII 1968)
We consider in \(\mathscr L_p(-\infty,\infty)\) the integral equation of the first kind with a kernel of potential type (the generalized Abel equation)*:
\[ M_\alpha \varphi \equiv \frac{1}{\Gamma(\alpha)} \int_{-\infty}^{\infty} \frac{c_1(x)+c_2(x)\operatorname{sign}(x-t)} {|x-t|^{1-\alpha}}\,\varphi(t)\,dt = f(x), \tag{1} \]
where \(c_1^2(x)+c_2^2(x)\ne0,\ -\infty\le x\le\infty;\ 0<\alpha<1\). Denote by \(I_+^\alpha\varphi,\ I_-^\alpha\varphi\) the fractional integrals, so that
\[ M_\alpha\varphi \equiv u(x)I_+^\alpha\varphi+v(x)I_-^\alpha\varphi=f(x), \tag{2} \]
where \(u(x)=c_1(x)+c_2(x),\ v(x)=c_1(x)-c_2(x)\).
Let us note that the usual inverse of the operator \(I_+^\alpha\),
\[ \varphi(x)=I_+^{-\alpha}\Phi = \frac{1}{\Gamma(1-\alpha)} \frac{d}{dx} \int_{-\infty}^{x} \frac{\Phi(t)\,dt}{(x-t)^\alpha}, \tag{3} \]
which is valid for \(\varphi(x)\in \mathscr L_1(-\infty,\infty)\), no longer holds for \(\varphi(x)\in \mathscr L_p(-\infty,\infty)\), \(p>1\). Therefore, instead of the operator (3), one should consider its extension
\[ I_+^{-\alpha}\Phi= \frac{\alpha}{\Gamma(1-\alpha)} \int_{-\infty}^{x} \frac{\Phi(x)-\Phi(t)}{(x-t)^{1+\alpha}}\,dt, \tag{4} \]
applicable in the range of the operator \(I_+^\alpha\), in particular for such \(\Phi(x)\) that
\[ \omega_p(\Phi;\delta)=O(\delta^\gamma),\qquad \gamma>\alpha, \tag{5} \]
where \(\omega_p(\Phi;\delta)\) is the integral modulus of continuity of the function \(\Phi(x)\). On the other hand, for \(p=1\) the operator (3) is inverse, but another difficulty arises. The known methods of solving equation (1) in one way or another use the singular operator \(S\), for which the space \(\mathscr L_1(-\infty,\infty)\) is not invariant. However, in the case when the coefficients \(c_1(x), c_2(x)\) are constant, the investigation in \(\mathscr L_1(-\infty,\infty)\) can \(\left({}^{8}\right)\) be carried through to the end and, in particular, a description of the range of the operator \(M_\alpha\) can be given in terms of the absolute continuity of a certain auxiliary function.
In the present note, first, the range of the operator \(M_\alpha\) is described in terms of the Fourier transform and an effective solution of the equation \(M_\alpha\varphi=f\) is given; second, the equivalence of equation (1) to a certain equation of convolution type is established.
* The generalized Abel equation was first considered by K. D. Sakalyuk \(\left({}^{6}\right)\) for the case of a finite interval \(a<t,\ x<b\).
§ 1. Let us introduce notation
\[ S\varphi=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\varphi(t)\,dt}{t-x},\qquad P=\frac{p}{1-\alpha p}, \]
\[ \varkappa=\frac{1}{\pi}\int_{-\infty}^{\infty} d\arg\left\{c_1(x)-ic_2(x)\tg\frac{\alpha\pi}{2}\right\}. \]
By \(H^\lambda(-\infty,\infty)\) we denote the class of functions \(h(x)\) such that the functions
\[ h\left(i\,\frac{t+1}{t-1}\right) \]
satisfy on the circle \(|t|=1\) a Hölder condition of order \(\lambda\). Then
\[ |h(x_1)-h(x_2)|\le A|x_1-x_2|^\lambda/(1+|x_1|^\lambda)(1+|x_2|^\lambda). \]
Let also \(\overline{\varphi}(x)=F(\varphi)(x)\) be the Fourier transform of the function \(\varphi(x)\). Denote by \(F_p(-\infty,\infty)\) the class of functions which are Fourier transforms of functions from \(\mathscr L_{p'}(-\infty,\infty)\), \(p'=p(p-1)^{-1}\), \(p>1\).
By the Young—Hausdorff theorem \(F_p(-\infty,\infty)=\mathscr L_p(-\infty,\infty)\) for \(1<p\le2\) and \(F_p(-\infty,\infty)\subset \mathscr L_p(-\infty,\infty)\) for \(p>2\). And, finally, by \(R(-\infty,\infty)\) we denote the Wiener ring of Fourier transforms of functions from \(\mathscr L_1(-\infty,\infty)\).
Theorem 1. Let \(c_1(x),c_2(x)\in H^\lambda(-\infty,\infty)\), where \(\lambda>\alpha\), and let \(\varkappa\ge0\). In order that the function \(f(x)\) be representable in the form \(f(x)=M_\alpha\varphi\), where \(\varphi(x)\in \mathscr L_p(-\infty,\infty)\), \(1<p<1/\alpha\), it is necessary and sufficient that
\[ \text{for }p=2\qquad |x|^\alpha \hat f(x)\in \mathscr L_2(-\infty,\infty), \tag{6'} \]
\[ \text{for }1<p<2\qquad |x|^\alpha \hat f(x)\in F_{p'}(-\infty,\infty), \tag{6''} \]
\[ \text{for }p>2\text{ it is sufficient that}\qquad |x|^\alpha \hat f(x)\in \mathscr L_{p'}(-\infty,\infty). \tag{6'''} \]
When conditions \((6')\)—\((6''')\) are fulfilled, all solutions of equation (1) are determined from Abel’s equation:
\[ I_+^\alpha\varphi=\Phi(x), \tag{7} \]
where \(\Phi(x)\) are solutions of the equation with Cauchy kernel from the class \(\mathscr L_p(-\infty,\infty)\):
\[ a_1(x)\Phi(x)+a_2(x)S\Phi=f(x), \tag{8} \]
\[ a_1(x)=2c_1(x)\cos^2(\alpha\pi/2)+2c_2(x)\sin^2(\alpha\pi/2), \]
\[ a_2(x)=[c_1(x)-c_2(x)]\sin\alpha\pi. \]
Let us add further that the solutions of equation (7) are found by formula (4). If, however, it is known that all solutions \(\Phi(x)\) of equation (8) satisfy the condition
\[ \left(\ctg(\alpha\pi/2)-i\operatorname{sign}x\right)|x|^\alpha \hat\Phi(x)\in R(-\infty,\infty), \tag{9} \]
then the solutions \(\varphi(x)\) are also found by formula (3).
The proof of the theorem is based on the following lemmas:
Lemma 1*. The fractional integration operators \(I_\pm^\alpha\) are equivalent to the operator of division by \(x^\alpha\). Namely, let \(\varphi(x)\in \mathscr L_p(-\infty,\infty)\), \(1\le p\le2\), and \(p\alpha<1\). Then
\[ F(I_\pm^\alpha\varphi)(x)=\frac{e^{\pm i\frac{\alpha\pi}{2}\operatorname{sign}x}}{|x|^\alpha}\,F(\varphi)(x). \tag{10} \]
Equality (10) is also true for \(2<p<\alpha^{-1}\) (for \(\alpha<1/2\)), if one additionally assumes that \(F(\varphi)\in \mathscr L_{p'}(-\infty,\infty)\).
* Lemma 1 follows from an analogous result of G. O. Okikiolu \((^1)\) for the Riesz potential; see also \((^4)\).
The spaces \(I_+^\alpha(\mathscr L_p)\), \(I_-^\alpha(\mathscr L_p)\) of functions \(f(x)\), representable in the form \(f(x)=I_+^\alpha\varphi\) or \(f(x)=I_-^\alpha\varphi\), respectively, where \(\varphi(x)\in\mathscr L_p(-\infty,\infty)\), coincide, and we shall denote them simply by \(I^\alpha(\mathscr L_p)\), \(1<p<1/\alpha\).
Corollary to Lemma 1. In order that \(f(x)\in I_+^\alpha(\mathscr L_p)\), for \(1<p\leqslant 2\) it is necessary and sufficient that conditions \((6')\)—\((6'')\) be satisfied; for \(p>2\) it is sufficient that condition \((6''')\) be satisfied for \(f(x)\); and for \(p=1\) it is necessary and sufficient that (9) be satisfied.
Lemma 2. The class of functions \(I^\alpha(\mathscr L_p)\), for \(1<p<\alpha^{-1}\), is invariant with respect to the operator \(S\) and with respect to multiplication by functions from \(H^\lambda(-\infty,\infty)\) when \(\lambda>\alpha\).
Corollary to Lemma 2. Let \(f(x)\) satisfy condition \((6')\) or \((6'')\). Then the functions \(Sf\), \(h(x)f(x)\) also satisfy the same condition. If, however, \(f(x)\) satisfies condition \((6''')\), then one can only assert that \(h(x)f(x)\) and \(Sf\) belong to \(I^\alpha(\mathscr L_p)\).
Lemma 3. Let \(c_1(x), c_2(x)\in H^\lambda(-\infty,\infty)\). If \(f(x)\in I^\alpha(\mathscr L_p)\), then all solutions \(\Phi(x)\) from \(\mathscr L_p(-\infty,\infty)\) of equation (8) also belong to \(I^\alpha(\mathscr L_p)\).
Equation (8), and in general Theorem 1, we obtain by using the following identities for \(\varphi(x)\in\mathscr L_p(-\infty,\infty)\):
\[ I_-^\alpha\varphi=\cos(\alpha\pi)\,I_+^\alpha\varphi+\sin(\alpha\pi)\,SI_+^\alpha\varphi,\qquad 1\leqslant p<1/\alpha, \]
\[ SI_\pm^\alpha\varphi=I_\pm^\alpha S\varphi,\qquad 1<p<1/\alpha \]
(see \((^{2,7,8})\)).
Comparing Theorem 1 and Lemmas 2, 3, we conclude that the ranges of the operators \(I_\pm^\alpha\) and \(M_\alpha\) coincide:
\[ I^\alpha(\mathscr L_p)=M_\alpha(\mathscr L_p),\qquad 1<p<1/\alpha, \]
provided that \(c_1(x), c_2(x)\in H^\lambda(-\infty,\infty)\), \(\lambda>\alpha\), and \(\varkappa\geqslant 0\).
§ 2. Formulas (8) make it possible to establish a connection between equation (1) and equations of convolution type. We shall assume that
\[ 1<p<2/(1+\alpha), \tag{11} \]
It is known (\((^3)\), p. 147) that \(\widehat{\varphi}(x)|x|^{1/p'-1/r}\in\mathscr L_r(-\infty,\infty)\) for \(\varphi(x)\in\mathscr L_p(-\infty,\infty)\) and \(p\leqslant r\leqslant p'\). In particular,
\[ \widehat{\varphi}(x)|x|^{-\alpha}\in\mathscr L_{P'}(-\infty,\infty); \tag{12} \]
the requirement \(p\leqslant P'\leqslant p'\) is fulfilled by virtue of condition (11). Consequently, the functions \(I_\pm^\alpha\varphi\) and
\[ e^{\pm i\frac{\alpha\pi}{2}\operatorname{sign}x}\,|x|^{-\alpha}\widehat{\varphi}(x) \]
form (with the corresponding choice of signs) pairs of Fourier transforms from the spaces \(\mathscr L_p(-\infty,\infty)\) and \(\mathscr L_{P'}(-\infty,\infty)\), respectively, and we arrive at the following theorem on convolutions. Let \(K(x)=\widehat{k}(x)\).
Theorem 2. If \(\varphi(x)\in\mathscr L_p(-\infty,\infty)\), where \(p\) satisfies condition (11), then
\[ F\bigl(K(x)I_\pm^\alpha\varphi\bigr) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} k(t-x)\, \frac{\widehat{\varphi}(t)}{|t|^\alpha} e^{\pm i\frac{\alpha\pi}{2}\operatorname{sign}t}\,dt \tag{13} \]
provided one of the conditions is fulfilled:
\[ \text{1) }\quad k(x)\in\mathscr L_1(-\infty,\infty); \tag{14} \]
\[ \text{2) }\quad k(x)\in\mathscr L_P(-\infty,\infty),\quad K(x)\in\mathscr L_{P'}(-\infty,\infty); \tag{15} \]
\[ \text{3) }\quad k(x)\in\mathscr L_r(-\infty,\infty),\quad K(x)\in\mathscr L_{r'}(-\infty,\infty),\quad r<P. \tag{16} \]
In case (14), both sides of equality (13) belong to \(\mathscr L_{P'}(-\infty,\infty)\); in case (16), to \(\mathscr L_Q(-\infty,\infty)\), \(Q=Pr(P-r)^{-1}\); and in case (15) one can only assert that they exist almost everywhere.
Let now the coefficients \(c_1(x), c_2(x)\) of equation (1) have the form
\[ c_j(x)=\lambda_j+K_j(x),\qquad K_j(x)=\hat{k}_j(x),\qquad j=1,2, \]
where \(k_j(x)\) satisfies one of the conditions (14)—(16). On the basis of Theorem 2, equation (1)—(2) is transformed into the form
\[ (d_1+d_2\operatorname{sign}x)\Phi_\alpha(x)+ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} [m_1(x-t)+m_2(x-t)\operatorname{sign}t]\Phi_\alpha(t)\,dt= \]
\[ =F(x), \tag{17} \]
where
\[
d_1=2\lambda_1\cos(\alpha\pi/2),\quad
d_2=2\lambda_2 i\sin(\alpha\pi/2),\quad
F(x)=\hat f(x),
\]
\[
m_1(x)=2\cos(\alpha\pi/2)k_1(-x),\quad
m_2(x)=2i\sin(\alpha\pi/2)k_2(-x),\quad
\Phi_\alpha(x)=\hat\varphi(x)/|x|^\alpha .
\]
A convolution-type equation of the form (17) in the space \(\mathscr L_2(-\infty,\infty)\) was investigated by Yu. I. Cherskii \((^5)\).
Equations (1) and (17) are equivalent in the sense that to every solution \(\varphi(x)\) of equation (1) there corresponds, by the formula \(\Phi_\alpha(x)=|x|^{-\alpha}\hat\varphi(x)\), a solution of equation (17), and, conversely, to every solution \(\Phi_\alpha(x)\) of equation (17), representable in the form \(\Phi_\alpha(x)=|x|^{-\alpha}F(\varphi)(x)\), where \(\varphi(x)\in \mathscr L_p(-\infty,\infty)\), there corresponds a solution \(\varphi(x)\) of equation (1). The solutions \(\Phi_\alpha(x)\) of equation (17), therefore, must be sought in the space \(\mathscr L_p(-\infty,\infty)\) or in the space with weight \(\mathscr L_p'(|x|^{\alpha p'})\), and among these solutions one must choose those representable in the form \(|x|^{-\alpha}\hat\varphi(x)\), \(\varphi(x)\in\mathscr L_p(-\infty,\infty)\).
In particular, for the case of constant coefficients \(c_1,c_2\) we obtain the following inversion formula:
\[ \varphi(x)=\frac12 F^{-1}\left\{ \frac{|x|^\alpha\hat f(x)} {c_1\cos(\alpha\pi/2)+ic_2\operatorname{sign}x\sin(\alpha\pi/2)} \right\}. \]
For comparison we note that the summable solutions \(\varphi(x)\in\mathscr L_1(-\infty,\infty)\) are given by the formula
\[ \varphi(x)= \frac{1} {(c_1^2\cos^2(\alpha\pi/2)+c_2^2\sin^2(\alpha\pi/2))\Gamma(1-\alpha)} \frac{d}{dx} \int_{-\infty}^{\infty} \frac{c_2+c_1\operatorname{sign}(x-t)} {|x-t|^\alpha} f(t)\,dt . \]
In conclusion we note that the connection between equations (1) and (17) may be of interest also for convolution-type equations themselves, since solving the latter in the space \(\mathscr L_{p'}(|x|^{\alpha p'})\), \(p'>2/(1-\alpha)\), can be reduced* to solving the generalized Abel equation (1).
Rostov State University
Received
20 XII 1968
CITED LITERATURE
- G. O. Okikiolu, Proc. Cambr. Phil. Soc., 62, No. 1, 73 (1966).
- L. v. Wolfersdorf, Math. Zs., 90, No. 1, 24 (1965).
- E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Moscow, 1948.
- M. A. Perel’man, in the collection Cybernetic Models, Tbilisi, 1965, p. 34.
- Yu. I. Cherskii, Matem. sborn., 41 (83), 3, 277 (1957).
- K. D. Sakalyuk, DAN, 131, No. 4, 748 (1960).
- S. G. Samko, Differential Equations, No. 2, 298 (1968).
* It should, however, be stipulated that along this path one can find, generally speaking, not all solutions of a convolution-type equation, but only those of them which are representable in the form \(|x|^{-\alpha}F(\varphi)\).