UDC 517.51

MATHEMATICS

L. V. ZHIZHIASHVILI

ON CONJUGATE FUNCTIONS

(Presented by Academician I. N. Vekua on June 22, 1965)

1. Let the \(2\pi\)-periodic function \(f(x)\) be summable on \([-\pi,\pi]\). Denote by \(\sigma[f]\) the Fourier series of the function \(f(x)\), and by \(s_n(x,f)\) the partial sums of the series \(\sigma[f]\). Further, as usual, by the symbol \(\bar f(x)\) we shall denote the function conjugate to \(f(x)\), i.e.

\[ \bar f(x)=-\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x+t)\operatorname{ctg}\frac{t}{2}\,dt, \]

which exists almost everywhere for every summable (see \((^{1})\), p. 528) function \(f(x)\). It is known (see \((^{1})\), p. 557) that if \(f(x)\) is a bounded \(2\pi\)-periodic function, then \(\bar f(x)\) need not be bounded. However, as was shown by P. Turán \((^{2})\) and M. Kinukawa \((^{3})\), if \(f(x)\) is an even \(2\pi\)-periodic function bounded on \((-\infty,+\infty)\), then the functions

\[ \varphi(x)=\frac{1}{\operatorname{tg} x/2}\int_{0}^{x}\bar f(t)\,dt,\qquad \psi(x)=\int_{x}^{\pi}\frac{\bar f(t)}{2\operatorname{tg} t/2}\,dt \quad (|x|\leqslant \pi) \]

are bounded on \((-\infty,+\infty)\)*. The behavior of the partial sums of the series \(\sigma[\varphi]\) and \(\sigma[\psi]\) was studied by M. and S. Izumi \((^{4})\). In particular, they showed that if \(f(x)\) is an even bounded function and \(\|s_n(x,f)\|_C=O(1)\), then \(\|s_n(x,\varphi)\|_C=O(1)\), \(\|s_n(x,\psi)\|_C=O(1)\). The question arises: what can be said about the continuity of the functions \(\varphi(x)\) and \(\psi(x)\), or about the uniform convergence of the series \(\sigma[\varphi]\) and \(\sigma[\psi]\)?

Theorem 1. If \(f(x)\) is an even \(2\pi\)-periodic continuous function, then the function \(\varphi(x)\) is also continuous. Moreover, if

\[ f(0)=\int_{0}^{\pi} f(t)\,dt=0, \]

then the function \(\psi(x)\) is also continuous, and its modulus of continuity is

\[ \omega(\delta,\psi)=O\left\{\omega(\delta,f)+\delta\int_{\delta}^{1}\frac{\omega(t,f)}{t^2}\,dt\right\}, \]

where \(\omega(\delta,u)\) is the modulus of continuity of the function \(u(x)\in C(-\pi,\pi)\).

Theorem 2. Let \(f(x)\) be an even \(2\pi\)-periodic continuous function and \(|s_n(0,f)|\leqslant M\) \((n=1,2,\ldots)\). Then the series \(\sigma[f]\) converges uniformly. Moreover, if

\[ f(0)=\int_{0}^{\pi} f(t)\,dt=0, \]

then the series \(\sigma[\psi]\) also converges uniformly.

Analogous assertions for the functions \(\varphi(x)\) and \(\psi(x)\) in the case when \(f(x)\) is odd are, generally speaking, false.

1. Let us now consider a function of two variables \(f(x,y)\). Suppose that it is periodic with respect to each of the variables and \(f(x,y)\in\)

* Here it is assumed that the functions \(\varphi(x)\) and \(\psi(x)\) are extended periodically with period \(2\pi\) from the interval \([-\pi,\pi]\) to the entire line.

\(\in L(R)\), where \(\bar R=[-\pi,\pi,-\pi,\pi]\). Consider the conjugate functions of two variables

\[ \bar f_1(x,y)=-\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x+s,y)\operatorname{ctg}\frac{s}{2}\,ds, \]

\[ \bar f_2(x,y)=-\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x,y+t)\operatorname{ctg}\frac{t}{2}\,dt, \]

\[ \bar f_3(x,y)=\frac{1}{4\pi^2}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi} f(x+s,y+t)\operatorname{ctg}\frac{s}{2}\operatorname{ctg}\frac{t}{2}\,ds\,dt. \]

If \(f(x,y)\) is an even function with respect to the aggregate of the two variables, i.e. \(f(-x,-y)=f(x,y)\), then, generally speaking, the assertions of P. Turán \(\left({}^{2}\right)\) and M. Kinukawa \(\left({}^{3}\right)\) are false \(\left({}^{5}\right)\) for the functions \(\bar f_i(x,y)\) \((i=1,2,3)\).

Let us now suppose that \(f(x,y)\) is an even function with respect to each variable, i.e. \(f(-x,y)=f(x,-y)=f(x,y)\). Further, let

\[ \varphi(x,y)=\frac{1}{\operatorname{tg}x/2\,\operatorname{tg}y/2} \int_{0}^{x}\int_{0}^{y}\bar f_3(s,t)\,ds\,dt, \qquad \psi(x,y)=\int_{x}^{\pi}\int_{y}^{\pi} \frac{\bar f_3(s,t)}{4\operatorname{tg}s/2\,\operatorname{tg}t/2}\,ds\,dt, \]

\((x,y)\in R\), and
\[ \varphi(x+2\pi,y)=\varphi(x,y+2\pi)=\varphi(x,y),\qquad \psi(x+2\pi,y)=\psi(x,y+2\pi)=\psi(x,y) \]
for all \(x,y\).

Lemma 1. If \(f(x,y)\) is a bounded function, even with respect to each variable, then the relations

\[ \varphi(x,y)=\frac{1}{\pi^2\operatorname{tg}x/2\,\operatorname{tg}y/2} \int_{0}^{\pi}\int_{0}^{\pi} f(s,t)\log\left| \frac{\sin(x+s)/2\,\sin(x-s)/2}{\sin^2 s/2} \right| \]
\[ \times \log\left| \frac{\sin(y+t)/2\,\sin(y-t)/2}{\sin^2 t/2} \right|\,ds\,dt, \]

\[ \psi(x,y)=\frac{1}{4\pi^2}\int_{0}^{\pi}\int_{0}^{\pi} f(s,t)\operatorname{ctg}\frac{s}{2}\operatorname{ctg}\frac{t}{2} \log\left|\frac{\sin(x+s)/2}{\sin(x-s)/2}\right| \log\left|\frac{\sin(y+t)/2}{\sin(y-t)/2}\right|\,ds\,dt \]
\[ -\frac{\pi-x}{4\pi^2}\int_{0}^{\pi}\int_{0}^{\pi} f(s,t)\operatorname{ctg}\frac{t}{2} \log\left|\frac{\sin(y+t)/2}{\sin(y-t)/2}\right|\,ds\,dt \]
\[ -\frac{\pi-y}{4\pi^2}\int_{0}^{\pi}\int_{0}^{\pi} f(s,t)\operatorname{ctg}\frac{s}{2} \log\left|\frac{\sin(x+s)/2}{\sin(x-s)/2}\right|\,ds\,dt \]
\[ +\frac{(\pi-x)(\pi-y)}{16\pi^2} \int_{0}^{\pi}\int_{0}^{\pi} f(s,t)\,ds\,dt. \]

hold.

On the basis of this lemma Theorems 3 and 4 are proved.

Theorem 3. If \(f(x,y)\) is a bounded even function (with respect to each variable), then the functions \(\varphi(x,y)\) and \(\psi(x,y)\) are also bounded.

Theorem 4. If \(f(x,y)\) is a continuous even function (with respect to each variable), then the function \(\varphi(x,y)\) is continuous. Moreover, if

\[ f(0,0)=\int_{0}^{\pi}\int_{0}^{\pi} f(s,t)\,ds\,dt=0, \]

then the function \(\psi(x,y)\) is continuous and its modulus of continuity

\[ \omega(\delta,\delta';u)= O\left\{ \omega_1(\delta,f)+\omega_2(\delta',f) +\delta\int_{\delta}^{1}\frac{\omega_1(s,f)}{s^2}\,ds +\delta'\int_{\delta'}^{1}\frac{\omega_2(t,f)}{t^2}\,dt \right\}, \]

where \(\omega(\delta,\delta';u)\), \(\omega_1(\delta,u)\) and \(\omega_2(\delta',u)\) are, respectively, the full and partial moduli of continuity of the function \(u(x,y)\in C(R)\).

Analogous assertions are also valid for the functions

\[ \bar f_i(x,y)\quad (i=1,2). \]

3. Let now \(f(x,y)\in L(R)\) and

\[ F(x,y)=\int_0^x\int_0^y f(s,t)\,ds\,dt. \]

Put

\[ \Delta_x(F;x,y,s)=F(x+s,y)+F(x-s,y)-2F(x,y), \]

\[ \Delta_y(F;x,y,t)=F(x,y+t)+F(x,y-t)-2F(x,y), \]

\[ \Delta^2(F;x,y,s,t)=\Delta_x(\Delta_y(F;x,y,t))=\Delta_y(\Delta_x(F;x,y,s)); \]

\[ \widetilde F_1(x,y)=\lim_{\varepsilon\to 0+}\int_\varepsilon^\pi \frac{\Delta_x(F;x,y,s)}{2\sin^2 s/2}\,ds,\qquad \widetilde F_2(x,y)=\lim_{\eta\to 0+}\int_\eta^\pi \frac{\Delta_y(F;x,y,t)}{2\sin^2 t/2}\,dt, \]

\[ \widetilde F_3(x,y)=\lim_{\varepsilon,\eta\to 0+} \int_\varepsilon^\pi\int_\eta^\pi \frac{\Delta^2(F;x,y,s,t)}{4\sin^2 s/2\,\sin^2 t/2}\,ds\,dt. \]

The question is: under what conditions do the functions \(F_i(x,y)\) \((i=1,2,3)\) exist, and what is the relation between them?

The functions \(\widetilde F_i(x,y)\) \((i=1,2)\) exist almost everywhere for every summable \(2\pi\)-periodic function \(f(x,y)\), which may be obtained from the corresponding result of A. Plessner \((^6)\). For the function \(\widetilde F_3(x,y)\) the analogous assertion is, generally speaking, false, since the following holds.

Theorem 5. There exists a function \(f(x,y)\in L(\log^+L)^\alpha\) for all \(\alpha\in[0,1)\), for which almost everywhere on \(R\)

\[ \lim_{\varepsilon\to 0+} \int_\varepsilon^\pi\int_\varepsilon^\pi \frac{\Delta^2(F;x,y,s,t)}{4\sin^2 s/2\,\sin t/2}\,ds\,dt \]

does not exist.

If, however, \(f(x,y)\in L(\log^+L)^\alpha\), \(\alpha\ge 1\), then such an assertion is no longer valid, for the following is true.

Theorem 6. If \(f(x,y)\in L\log^+L\), then \(\widetilde F_3(x,y)\) exists almost everywhere and, for almost all \(x\) and \(y\),

\[ \widetilde F_3(x,y)=\widetilde F_{i,j}(x,y)=\bar f_{i,j}(x,y)\qquad (i,j=1,2;\ i\ne j)^*, \]

where

\[ \widetilde F_{i,j}(x,y)=\lim_{\varepsilon\to 0+}\int_\varepsilon^\pi \frac{\Delta_x(\widetilde F_i;x,y,s)}{2\sin^2 s/2}\,ds \quad\text{for } i=2,\ j=1. \]

The results given here can be generalized also to the case of functions of \(n\) variables. We shall confine ourselves to considering the analogue of Theorem 6.

Let \(f(x_1,\ldots,x_n)\in L(R')\), where \(R'=[-\pi,\pi;\ldots;-\pi,\pi]\). Put

\[ F(x_1,\ldots,x_n)=\int_0^{x_1}\cdots\int_0^{x_n} f(s_1,\ldots,s_n)\,ds_1\cdots ds_n, \]

\[ \Delta_k F\equiv \Delta_k(F;x_1,\ldots,x_n,s_k) =F(x_1,\ldots,x_k+s_k,\ldots,x_n)+ \]

\[ +F(x_1,\ldots,x_k-s_k,\ldots,x_n)-2F(x_1,\ldots,x_n), \]

\[ \Delta^n(F;x_1,\ldots,x_n;s_1,\ldots,s_n) =\Delta_n\bigl(\Delta_{n-1}(\cdots(\Delta_1 F)\bigr)=\cdots =\Delta_1\bigl(\Delta_2(\cdots(\Delta_n F)\cdots)\bigr). \]

\[ {}^*\ \text{The symbol }\bar f_{i,j}(x,y)\text{ means that the conjugation operation is applied to the function }f(x,y)\text{ first in the }i\text{-th variable, and then in the }j\text{-th.} \]

Theorem 7. Let \(f(x_1,\ldots,x_n)\in L(\log^{+}L)^{n-1}\). Then almost everywhere on \(R'\) there exists

\[ \lim_{(\varepsilon_1,\ldots,\varepsilon_n)\to 0+} \int_{\varepsilon_1}^{\pi}\cdots\int_{\varepsilon_n}^{\pi} \frac{\Delta^n(F;x_1,\ldots,x_n;s_1,\ldots,s_n)} {2^n\sin^2 s_1/2\ldots \sin^2 s_n/2}\, ds_1\ldots ds_n, \]

and almost everywhere

\[ \widetilde F_{2^n-1}(x_1,\ldots,x_n) = \widetilde F_{i_1,\ldots,i_n}(x_1,\ldots,x_n) \]

\[ (i_1,\ldots,i_n=1,2,\ldots,n,\ \text{in the order } i_j\ne i_k), \]

where

\[ \widetilde F_{i_k}(x_1,\ldots,x_n) = \lim_{\varepsilon_{i_k}\to 0+} \int_{\varepsilon_{i_k}}^{\pi} \frac{\Delta_{i_k}(F;x_1,\ldots,x_n,s_{i_k})} {2\sin^2 s_{i_k}/2}\,ds_{i_k}, \]

\[ \widetilde F_{i_k,i_j}(x_1,\ldots,x_n) = \lim_{\varepsilon_{i_j}\to 0+} \int_{\varepsilon_{i_j}}^{\pi} \frac{\Delta_{i_j}(\widetilde F_{i_k};x_1,\ldots,x_n,s_{i_j})} {2\sin^2 s_{i_j}/2}\,ds_{i_j}, \]

and the assertion loses its force if, in the hypothesis of the theorem, the exponent \(n-1\) is replaced by a smaller one.

I take this opportunity to express my deep gratitude to P. L. Ulyanov for valuable advice in carrying out this work.

Tbilisi
State University

Received
10 VI 1965

REFERENCES

\({}^{1}\) N. K. Bari, Trigonometric Series, 1961.
\({}^{2}\) P. Turan, Ann. Soc. Polon. Math., 25, 155 (1952).
\({}^{3}\) M. Kinukawa, Dissertation Northwestern Univ., 1960.
\({}^{4}\) M. Izumi, S. Izumi, Acta Math. Sci. Hung., 13, No. 1–2, 133 (1962).
\({}^{5}\) L. V. Zhizhiashvili, DAN, 155, No. 3, 521 (1964).
\({}^{6}\) A. Plessner, J. Math., 159, 219 (1927).