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).