MATHEMATICS

L. V. ZHIZHIASHVILI

CONJUGATE FUNCTIONS OF TWO VARIABLES AND DOUBLE FOURIER SERIES

(Presented by Academician A. N. Kolmogorov, 20 X 1962)

1. It is known (see \((^4)\))* that for every integrable function \(f(x)\) there exists almost everywhere the integral

\[ \bar f(x)=-\frac{1}{2\pi}\int_{0}^{\pi}[f(x+t)-f(x-t)]\operatorname{ctg}\frac{t}{2}\,dt, \]

called the function conjugate to \(f(x)\). Riesz \((^7)\) established that if \(f(x)\in L^p(-\pi,\pi)\), \(p>1\), then \(\bar f(x)\in L^p(-\pi,\pi)\) and \(\|\bar f\|_{L^p}\leq A(p)\|f\|_{L^p}\). If \(p=1\), then Riesz’s theorem does not hold, since \(\bar f(x)\) may fail to be summable (\((^3)\), p. 227). But the following theorem of A. N. Kolmogorov is valid:

If \(f(x)\) is integrable, then \(|\bar f(x)|^p\) is integrable, and

\[ \left\{\int_{-\pi}^{\pi}|\bar f(x)|^p dx\right\}^{1/p} \leq B(p)\int_{-\pi}^{\pi}|f(x)|\,dx, \qquad 0<p<1, \]

where \(B(p)\) is a constant depending only on \(p\) \((^2)\).

A. Zygmund observed \((^9)\) that if \(|f(x)|\log^+|f(x)|\in L(-\pi,\pi)\), then \(\bar f(x)\) is summable and

\[ \int_{-\pi}^{\pi}|\bar f(x)|\,dx \leq A\int_{-\pi}^{\pi}|f(x)|\log^+|f(x)|\,dx+B, \]

where \(A\) and \(B\) are constants.

In the case when \(f(x)\geq 0\), this theorem admits a converse, i.e., if \(f(x)\in L(-\pi,\pi)\), \(f(x)\geq 0\), and \(\bar f(x)\) is summable, then \(|f(x)|\log^+|f(x)|\) is summable \((^7)\).

Let now \(f(x,y)\), \(2\pi\)-periodic with respect to each of the variables, be summable on \(R_0=[-\pi,\pi;-\pi,\pi]\). Consider the functions

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

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

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

Following Cesari \((^8)\), we shall call the functions \(\bar f_1(x,y)\), \(\bar f_2(x,y)\), \(\bar f(x,y)\) conjugate respectively with respect to the variable \(x\), with respect to the variable \(y\), and

* For \(f(x)\in L^2(-\pi,\pi)\) this result was earlier obtained by N. N. Luzin (\((^3)\), p. 217).

with respect to the set of variables \(x\) and \(y\). Sokol-Sokolovskii \((^9)\) established that the Riesz theorem is also valid for conjugate functions of two variables. If, however, \(|f(x,y)|\log^+|f(x,y)|\in L(R_0)\), then, as Zygmund \((^{10})\) proved, the function \(\bar f(x,y)\) exists almost everywhere and

\[ \left\{\iint_{R_0} |\bar f(x,y)|^p\,dx\,dy\right\}^{1/p}\leq \]

\[ \leq A'(p)\iint_{R_0}|f(x,y)|\log^+|f(x,y)|\,dx\,dy+B'(p),\qquad 0<p<1, \]

where \(A'(p)\) and \(B'(p)\) are constants depending only on \(p\).

However, it is unknown whether the condition \(|f(x,y)|\log^+|f(x,y)|\in L(R_0)\) implies that \(\bar f(x,y)\in L(R_0)\).

2. In this section we shall give a complete answer to the question just posed and present a number of other assertions belonging to the same circle of ideas.

Theorem 1. There exists a function \(f(x,y)\), \(2\pi\)-periodic in each of the variables \(x\) and \(y\), such that
\[ |f(x,y)|(\log^+|f(x,y)|)^{2-\varepsilon}\in L(R_0) \]
for every \(0<\varepsilon<2\), but \(\bar f(x,y)\) is not summable.

Theorem 2. There exists a nonnegative \(2\pi\)-periodic function \(f(x,y)\in L(R_0)\) for which \(\bar f_1(x,y)\) is not summable on any interval with respect to \(y\) for any fixed \(x\in E\), where \(|E|>0\).

Theorem 3. If \(|f(x,y)|\log^+|f(x,y)|\in L(R_0)\), then almost everywhere
\[ \bar f(x,y)=\bar f_{1,2}(x,y)=\bar f_{2,1}(x,y). \]

This theorem was published by us in \((^{11})\).

It is easy to prove that if
\[ |f(x,y)|(\log^+|f(x,y)|)^\alpha\in L(R_0),\quad \alpha>0, \]
then
\[ |\bar f_i(x,y)|\log^{\alpha-1}\bigl(2+|\bar f_i(x,y)|\bigr)\in L(R_0)\qquad (i=1,2). \]

Consequently, from Theorems 1 and 3 it follows immediately:

Corollary. If
\[ |f(x,y)|(\log^+|f(x,y)|)^2\in L(R_0), \tag{1} \]
then
\[ \bar f(x,y)\in L(R_0), \tag{2} \]
and assertion (2) loses its force if, in condition (1), the square is replaced by a smaller power.

An assertion of analogous type also holds for functions of \(n\) variables. For example, the following is true.

Theorem 4. If
\[ |f(x_1,x_2,\ldots,x_n)|\bigl(\log^+|f(x_1,x_2,\ldots,x_n)|\bigr)^n\in L(R_0'), \tag{3} \]
then
\[ \bar f(x_1,x_2,\ldots,x_n)\in L(R_0'), \tag{4} \]
and assertion (4) loses its force if, in condition (3), the power \(n\) is replaced by a smaller one,
\[ R_0'=[-\pi,\pi,-\pi,\pi;\ldots;-\pi,\pi]. \]

3. In this section we shall give a number of assertions concerning \((C,\alpha,\beta)\)-summability of double Fourier series and their conjugate series.

Let the series

\[ \sum_{m,n=0}^{\infty}\lambda_{mn}\bigl(a_{mn}\cos mx\cos ny+b_{mn}\sin mx\cos ny+ \]
\[ +c_{mn}\cos mx\sin ny+d_{mn}\sin mx\sin ny\bigr) \tag{5} \]

be the double Fourier–Lebesgue series of the function \(f(x,y)\), where \(\lambda_{00}=1/4,\) \(\lambda_{0n}=\lambda_{m0}=\lambda_{mn}=1,\ m,n>0.\) Denote by \(\sigma_{mn}^{\alpha,\beta}(x,y)\) the Cesàro \((C,\alpha,\beta)\)-means of the series (5). Further, let (cf. (1))

\[ \Delta_{h\eta}f(x,y)=f(x+h,y+\eta)-f(x+h,y)-f(x,y+\eta)+f(x,y), \]
\[ \Delta_h f(x,y)=f(x+h,y)-f(x,y),\qquad \Delta_\eta f(x,y)=f(x,y+\eta)-f(x,y). \]

Theorem 5. If the function \(f(x,y)\), continuous and \(2\pi\)-periodic with respect to each of the variables \(x\) and \(y\), satisfies the conditions:

\[ \left\{\iint_{R_0} |\Delta_{h\eta}f(x,y)|^{p_1}\,dx\,dy\right\}^{1/p_1} =O(h^\alpha\eta^\beta); \]

\[ \left\{\int_{-\pi}^{\pi} |\Delta_h f(x,y)|^{p_2}\,dx\right\}^{1/p_2} =O(h^{\alpha'}) \quad\text{uniformly with respect to }y; \]

\[ \left\{\int_{-\pi}^{\pi} |\Delta_\eta f(x,y)|^{p_3}\,dy\right\}^{1/p_3} =O(h^{\beta'}) \quad\text{uniformly with respect to }x, \]

then

\[ \left\|\sigma_{mn}^{-\lambda,-\delta}(x,y)-f(x,y)\right\|_{C} =O\left(m^{-\alpha+1/p_1}+m^{-\alpha'+1/p_2}+n^{-\beta+1/p_1}+n^{-\beta'+1/p_3}\right), \]

where

\[ 0<\alpha,\beta,\alpha',\beta'<1,\qquad \lambda,\delta<\frac1{p_k}\quad(k=1,2,3), \]

\[ p_1>\max\left\{\frac1\alpha,\frac1\beta\right\},\quad p_2>\frac1{\alpha'},\quad p_3>\frac1{\beta'}. \]

If, however, \(\lambda=\frac1{p_1}\), then

\[ \left\|\sigma_{mn}^{-\lambda,-\delta}(x,y)-f(x,y)\right\|_{C} =O\left(\frac{\ln m^{1-1/p_1}}{m^{\alpha+1/p_1}}\right) \]

\[ {}+m^{-\alpha'+1/p_2}+n^{-\beta+1/p_1}+n^{-\beta'+1/p_3}. \]

Analogous estimates are obtained if \(\lambda=\frac1{p_k}\) or \(\delta=\frac1{p_k}\) \((k=1,2,3)\). The theorem remains valid also for double conjugate trigonometric series in the sense of Cesàro \(({}^{8})\).

Theorem 6. If \(f(x,y)\) is a continuous and \(2\pi\)-periodic function satisfying the condition

\[ \left\{\iint_{R_0}|f(x+h,y+\eta)-f(x,y)|^p\,dx\,dy\right\}^{1/p} =O(h^\alpha+\eta^\beta), \]

\[ 0<\alpha,\beta<1,\qquad p\ge \max\left\{\frac1\alpha,\frac1\beta\right\}, \]

then the series \(\sigma[f]\) is uniformly \((C,-\lambda,-\delta)\)-summable to \(f(x,y)\), i.e.

\[ \sigma_{mn}^{-\lambda,-\delta}(x,y)-f(x,y)\to 0,\qquad \lambda,\delta<\frac{\min(\alpha,\beta)}2. \]

If \(p>\max\left\{\frac1\alpha,\frac1\beta\right\}\), then analogous theorems are valid also for double conjugate trigonometric series.

Theorem 7. If \(f(x,y)\) is a continuous \(2\pi\)-periodic function, then

\[ \left\|\sigma_{mn}^{\alpha,\beta}(x,y)-f(x,y)\right\|_{C}=O(\varphi_{mn}), \]

where

\[ \varphi_{mn}=\omega_1\left(\frac{\ln m}{m}\right)+\omega_2\left(\frac{\ln n}{n}\right), \qquad \alpha,\ \beta>0, \]

\[ \omega_1(\delta)=\max_y\left\{\sup_{|x_2-x_1|\leq\delta}|f(x_2,y)-f(x_1,y)|\right\}, \]

\[ \omega_2(\delta)=\max_x\left\{\sup_{|y_2-y_1|\leq\delta}|f(x,y_2)-f(x,y_1)|\right\}. \]

On the basis of results of V. G. Chelidze \((^5)\), it is proved that under the hypotheses of Theorems 6 and 5 the following relations hold:

\[ \left\|\sigma_{mn}^{\alpha,\beta}(x,y)-f(x,y)\right\|_C \leq A''\left\|\sigma_{mn}^{-\lambda,-\delta}(x,y)-f(x,y)\right\|_C, \]

\[ \alpha>-\lambda,\quad \beta>-\delta. \]

Received
20 X 1962

REFERENCES

\(^{1}\) I. E. Zhak, Matem. sborn., 31 (73), 3, 469 (1952).
\(^{2}\) A. N. Kolmogoroff, Fund. Math., 7, 23 (1925).
\(^{3}\) N. N. Luzin, Integral and Trigonometric Series, 1951.
\(^{4}\) J. J. Privalov, Bull. Soc. Math. de France, 44, 100 (1916).
\(^{5}\) V. G. Chelidze, Some Questions in the Theory of Double Series, Publishing House of Wuhan University, China, 1958.
\(^{6}\) K. Sokol-Sokolowski, Fund. Math., 34, 166 (1947).
\(^{7}\) M. Riesz, Math. Zs., 27, 218 (1927).
\(^{8}\) L. Cesari, Ann. Scuola Norm. Super. di Pisa, 2, 7, 279 (1938).
\(^{9}\) A. Zygmund, Fund. Math., 13, 284 (1929).
\(^{10}\) A. Zygmund, Fund. Math., 36, 207 (1947).
\(^{11}\) L. V. Zhizhiashvili, Reports of the Academy of Sciences of the Georgian SSR, 2, 24 (1960).