UDC 517.51

MATHEMATICS

L. V. ZHIZHIASHVILI

FOURIER SERIES AND MAXIMAL THEOREMS

(Presented by Academician I. N. Vekua on 8 VII 1966)

Consider a function of two variables \(f(x,y)\). Suppose that it is periodic with respect to each of the variables and that \(f(x,y)\in L(R)\), where \(R=[-\pi,\pi;-\pi,\pi]\). Denote by \(\bar f_i(x,y)\) \((i=1,2,3)\) the conjugate functions of two variables

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

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

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

By \(\sigma[f]\) we shall denote the double Fourier series of the function \(f(x,y)\), and by the symbols \(\bar\sigma[f;x]\), \(\bar\sigma[f;y]\), and \(\bar\sigma[f;x,y]\) the double conjugate trigonometric series to the series \(\sigma[f]\), respectively in the variable \(x\), in the variable \(y\), and in the totality of the variables \(x\) and \(y\). Further, let \(\sigma_{mn}^{(i,\alpha,\beta)}(x,y)\) \((i=0,1,2,3)\), \(\alpha,\beta>0\), be the Cesàro \((C;\alpha,\beta)\)-means respectively of the series \(\sigma[f]\), \(\bar\sigma[f;x]\), \(\bar\sigma[f;y]\), \(\bar\sigma[f;x,y]\), and put

\[ \varphi_i(x,y)=\sup_{m,n\ge0}\left|\sigma_{mn}^{(i,\alpha,\beta)}(x,y)\right|\qquad (i=0,1,2,3), \]

\[ \varphi_4(x,y)=\sup_{0<\varepsilon,\eta\le\pi} \left|\int_{\varepsilon}^{\pi}\int_{\eta}^{\pi} \frac{f(x+s,y+t)-f(x-s,y+t)-f(x+s,y-t)+f(x-s,y-t)} {\operatorname{tg}s/2\,\operatorname{tg}t/2}\,dt\,ds\right|, \]

\[ \varphi_5(x,y)=\sup_{0<\varepsilon\le\pi} \left|\int_{\varepsilon}^{\pi} \frac{f(x+s,y)-f(x-s,y)}{\operatorname{tg}s/2}\,ds\right|, \]

\[ \varphi_6(x,y)=\sup_{0<\eta\le\pi} \left|\int_{\eta}^{\pi} \frac{f(x,y+t)-f(x,y-t)}{\operatorname{tg}t/2}\,dt\right|. \]

Further, if \(s_{ij}(x,y)\) \((i,j=0,1,\ldots)\) denote the partial sums of the series \(\sigma[f]\), then the expressions

\[ \sigma_n^\alpha(x,y)=\sum_{k=0}^{n} A_{n-k}^{\alpha-1}s_{kk}(x,y),\qquad \alpha>0, \]

will be called the \((C,\alpha)\)-means of the series \(\sigma[f]\).

In the present note we give assertions concerning questions of the existence of the function \(\bar f_3(x,y)\) and the summability of the functions \(\varphi_i(x,y)\) \((i=0,\ldots,6)\); we also study the behavior of the \((C,\alpha)\)-means of the series \(\sigma[f]\).

As A. Zygmund showed \((^1)\), if \(f(x,y)\in L\log^+ L\), then the function \(\bar f_3(x,y)\) exists almost everywhere. In \((^2)\) he also posed the question: if \(f(x,y)\in\)

\(\in L(R)\), then does there exist almost everywhere the function \(f_3(x,y)\), where

\[ \bar f_3(x,y)=\lim_{(\varepsilon,\eta)_\lambda\to 0} \int_{\varepsilon}^{\pi}\int_{\eta}^{\pi} \frac{f(x+s,y+t)-f(x-s,y+t)-f(x+s,y-t)+f(x-s,y-t)} {\operatorname{tg}s/2+\operatorname{tg}t/2}\,dt\,ds, \tag{1} \]

where the symbol \((\varepsilon,\eta)_\lambda\to 0\) means that \(\varepsilon\to 0\), \(\eta\to 0\), and \(1/\lambda \leq \varepsilon/\eta \leq \lambda\), \(\lambda\geq 1\). E. Stein (see \((^3)\), Theorem 5) showed that, in contrast to the case of a function of one variable, the limit (1) may fail to exist on a set of positive planar measure even in the case when \(\lambda=1\).

Below we shall show that the scheme of the proof (see \((^3)\), Theorem 5) given by E. Stein contains an error. To verify this, we shall need the following

Lemma. Let \((x,y)\in [0,1/3;0,1/3]\equiv I\) and \(\varepsilon>0\). Then, if

\[ E^\varepsilon(\alpha)\equiv E(\alpha)= \left\{(x,y)\in I:\frac{1}{xy|\ln x|^\varepsilon|\ln y|^\varepsilon}\geq \alpha\right\}, \]

then for every \(\alpha\geq \alpha_0(\varepsilon)>0\) the measure

\[ \operatorname{mes}E(\alpha)\ll \begin{cases} \dfrac{A}{\alpha}(\ln \alpha)^{1-2\varepsilon}, & \text{if } 1-2\varepsilon>0,\\[6pt] \dfrac{A'}{\alpha}, & \text{if } 1-2\varepsilon<0, \end{cases} \]

where \(A\) and \(A'\) are certain positive constants.

Proof. Put

\[ E_1(\alpha)= \left\{(x,y)\in I:\frac{1}{xy|\ln x|^\varepsilon|\ln y|^\varepsilon}\geq \alpha,\ y\leq x\right\}, \]

\[ E_2(\alpha)= \left\{(x,y)\in I:\frac{1}{xy|\ln x|^\varepsilon|\ln y|^\varepsilon}\geq \alpha,\ y>x\right\}, \]

\[ E_3(\alpha)= \left\{(x,y)\in I:\frac{1}{xy|\ln x|^{2\varepsilon}}\geq \alpha\right\}, \]

\[ E_4(\alpha)= \left\{(x,y)\in I:\frac{1}{xy|\ln y|^{2\varepsilon}}\geq \alpha\right\}. \]

It is not difficult to see that \(E(\alpha)\subset E_1(\alpha)+E_2(\alpha)\subset E_3(\alpha)+E_4(\alpha)\). Consequently, the measure

\[ \operatorname{mes}E(\alpha)\leq \operatorname{mes}E_3(\alpha)+\operatorname{mes}E_4(\alpha). \tag{2} \]

But the set

\[ E_3(\alpha)= \left\{(x,y)\in I:\ y\leq \frac{1}{\alpha x|\ln x|^{2\varepsilon}}\right\}. \tag{3} \]

Let the number \(x_0\equiv x_0(\alpha,\varepsilon)\) be chosen so that \(1/\alpha x_0|\ln x_0|^{2\varepsilon}=1/3\). It is clear that

\[ 1/\alpha^2<x_0<1/\alpha,\qquad \alpha\geq \alpha_0(\varepsilon). \tag{4} \]

Thus, taking into account (3) and (4), we shall have

\[ \operatorname{mes}E_3(\alpha)\ll \frac{1}{\alpha}\int_{1/\alpha^2}^{1/3}\frac{dx}{x|\ln x|^{2\varepsilon}} +\frac{1}{\alpha} \ll \begin{cases} \dfrac{A}{2\alpha}(\ln\alpha)^{1-2\varepsilon}, & 1-2\varepsilon>0,\\[6pt] A'/2\alpha, & 1-2\varepsilon<0;\ \alpha\geq \alpha_0(\varepsilon). \end{cases} \tag{5} \]

Similarly,

\[ \operatorname{mes}E_4(\alpha)\ll \begin{cases} \dfrac{A}{2\alpha}(\ln\alpha)^{1-2\varepsilon}, & 1-2\varepsilon>0,\\[6pt] A'/2\alpha, & 1-2\varepsilon<0,\ \alpha\geq \alpha_0(\varepsilon). \end{cases} \tag{6} \]

From (2), (5), and (6) the validity of the lemma follows.

Analyzing the proof of the lemma, it is not difficult to see that if \(1-2\varepsilon=0\), i.e. \(\varepsilon=1/2\), then

\[ \operatorname{mes}E(\alpha)\ll \frac{A''}{\alpha}\ln\ln\alpha,\qquad \alpha\geq \alpha_0(\varepsilon). \]

Let us note here also that the following estimate is valid:

\[ \operatorname{mes} E(\alpha) \geqslant \frac{B}{\alpha}(\ln \alpha)^{1-2\varepsilon},\quad 1-2\varepsilon>0,\ \alpha\geqslant \alpha_0(\varepsilon), \tag{7} \]

where \(B\) is a positive constant. But the proof of relation (7) is more complicated than the proof of the lemma, and therefore we do not give it here.

The assertion that

\[ \operatorname{mes} E(\alpha)\geqslant B'/\alpha,\quad 1-2\varepsilon<0,\quad \alpha\geqslant \alpha_0(\varepsilon),\quad B'>0, \]

is, generally speaking, false. Thus, according to the lemma and (7), we shall have

\[ \operatorname{mes} E(\alpha)\sim (\ln \alpha)^{1-2\varepsilon}/\alpha,\quad 1-2\varepsilon>0\quad (\alpha\to\infty). \]

Now, analyzing the proof of E. Stein (see \({}^{3}\), Theorem 5, pp. 159–160), one can conclude that the function \(f_\delta(x,y)\) belongs to the class \(L(\log L)^{1-\varepsilon}\) (\(\varepsilon>0\), arbitrarily small) only when \(\delta>1-\varepsilon>1/2\); that is, \(1-2\delta<0\), and, according to the lemma,

\[ \operatorname{mes} E^\delta(\alpha)\leqslant A'/\alpha,\quad \alpha\geqslant \alpha_0(\delta), \]

which contradicts assertion (13.6) of E. Stein (see \({}^{3}\), p. 160). Consequently, E. Stein (see \({}^{3}\), p. 154) obtains no contradiction, and thereby he has not proved that if \(f(x,y)\in L(\log L)^\alpha\) for some \(\alpha\in[0,1)\), then, generally speaking, the double conjugate series \(\bar\sigma[f;x,y]\) is not summable almost everywhere by the Poisson method. Consequently, Stein’s conclusion on the nonexistence of \(\bar f_3(x,y)\), made in the conclusion and referring to Theorem 5, is insufficiently convincing.

We now give results connected with questions of summability of the functions \(\varphi_i(x,y)\) \((i=0,\ldots,6)\).

Theorem 1. If \(f(x,y)\in L\log L\), then

\[ \varphi_i(x,y)\in L^q(R)\quad \text{for all }q\in(0,1)\quad (i=0,\ldots,4). \tag{8} \]

If, however, \(f(x,y)\in L(\log L)^2\), then

\[ \varphi_i(x,y)\in L(R)\quad (i=0,\ldots,4). \tag{9} \]

Let us note that for functions \(f(x,y)\in L(\log L)^\alpha\) for all \(\alpha\in[0,1)\), assertion (8) is, generally speaking, false; and for the validity of assertion (9) the condition \(f(x,y)\in L(\log L)^2\) is also essential, since the following theorem is true.

Theorem 2. There exists a nonnegative \(2\pi\)-periodic function \(f_0(x,y)\in L(\log L)^{2-\varepsilon}\) for all \(\varepsilon\in(0,2]\), however \(\varphi_i(x,y)\notin L(R)\) \((i=0,\ldots,4)\).

For the summability of the functions \(\varphi_i(x,y)\) \((i=5,6)\), the condition \(f(x,y)\in L\log L\) is sufficient, which follows from a known theorem of A. Zygmund \({}^{4}\).

Theorem 3. Let \(f(x,y)\in L(R)\). If \(\bar f_i(x,y)\) \((i=1,2)\) are summable, then the series \(\bar\sigma[f;x]\) and \(\bar\sigma[f;y]\) are Fourier series respectively of the functions \(\bar f_1(x,y)\) and \(\bar f_2(x,y)\); but if \(f(x,y)\in L\log L\) and \(\bar f_3(x,y)\in L(R)\), then the series \(\bar\sigma[f;x,y]\) is also the Fourier series of the function \(\bar f_3(x,y)\).

J. Marcinkiewicz \({}^{5}\) proved that if \(f(x,y)\in L\log L\), then almost everywhere \(\sigma_n^1(x,y)\to f(x,y)\) as \(n\to\infty\). In fact, the following is true.

Theorem 4. If \(f(x,y)\in L(R)\), then for every \(\alpha>0\), almost everywhere

\[ \lim_{n\to\infty}\sigma_n^\alpha(x,y)=f(x,y). \]

The author expresses deep gratitude to Prof. P. L. Ulyanov for valuable advice.

Received
8 VII 1966

REFERENCES

\({}^{1}\) A. Zygmund, Fund. Math., 36, 207 (1949).
\({}^{2}\) A. Zygmund, Rend. Mat. e applic., 17, fasc. 2—4, 468 (1957).
\({}^{3}\) E. M. Stein, Ann. Math., 74, No. 1, 140 (1961).
\({}^{4}\) A. Zygmund, Fund. Math., 13, 284 (1929).
\({}^{5}\) J. Marcinkiewicz, Ann. Scuola norm. super Pisa, 8, 149 (1939).