MATHEMATICS

D. V. SALEKHOV

ON THE CONVERGENCE OF SINGULAR INTEGRALS AT LEBESGUE–ORLICZ POINTS OF FUNCTIONS OF SYMMETRIC SPACES

(Presented by Academician M. A. Lavrent’ev, September 24, 1966)

Denote by \(E\) a Banach symmetric space of functions measurable on \([0;1]\) \((^{1})\). The formula \(\|\chi_e\|_E=\varphi(me)\), where \(\chi_e(t)\) is the characteristic function of the measurable set \(e\subset[0;1]\), defines the function \(\varphi(t)\), called the fundamental function. We shall assume that the function \(\varphi(t)\) is monotonically increasing, concave, and \(\varphi(t)\to 0\) as \(t\to 0\).

The present paper consists of two parts, naturally connected with each other. In the first part the notion of a Lebesgue–Orlicz point of a function from the symmetric space \(E\) is introduced, and the properties of the set of such points are described. In the second part the question of the representability of functions \(x\in E\) by a singular integral at Lebesgue–Orlicz points is considered.

1. Denote by \(\chi_h^{t_0}\) the characteristic function of the interval \((t_0-h;\,t_0+h)\).

Definition. We shall call a point \(t\in[0;1]\) a Lebesgue–Orlicz point of the function \(x\in E\) if

\[ \lim_{h\to 0}\{\,\|[x(t)-x(t_0)]\chi_h^{t_0}\|_E/\|\chi_h^{t_0}\|_E\,\}=0. \tag{1} \]

If \(E=L_p\) \((p>1)\), then definition (1) coincides with the definition of a Lebesgue point of order \(p\). For \(p=1\), (1) means that the point \(t_0\) is a Lebesgue point of the function \(x(t)\). In the case when the space \(E\) is the Orlicz space \(L_M^*\), definition (1) coincides with the definition of a Lebesgue–Orlicz point of a function \(x\in L_M^*\), introduced in \((^{2})\). This follows from the relation \(u<M^{-1}(u)N^{-1}(u)\le 2u\), valid for all \(u>0\), and from the formula for the norm of a characteristic function in an Orlicz space.

We shall say that the fundamental function \(\varphi(t)\) satisfies condition \((\alpha)\) if \(\varphi[t\alpha(t)]/\varphi(t)\to 0\) as \(t\to 0\) for any function \(\alpha(t)\to 0\).

It is not difficult to prove the following theorem.

Theorem 1. If the fundamental function \(\varphi(t)\) satisfies condition \((\alpha)\), then the union of the set of density points of the set \(e\subset[0;1]\) and the set of density points of the set \(\complement e\) coincides with the set of Lebesgue–Orlicz points of the characteristic function \(\chi_e(t)\).

Theorem 2. In order that every bounded function \(x\in E\) have a set of Lebesgue–Orlicz points of full measure, it is necessary and sufficient that the fundamental function \(\varphi(t)\) satisfy condition \((\alpha)\).

We give the scheme of the proof of necessity. If condition \((\alpha)\) is not satisfied, then for some function \(\alpha(t)\) there exists a sequence \(\{h_n\}\) such that for every \(n\) one has \(\alpha(h_n)<1/(2^{n+1}+1)\) and \(\varphi[h_n\alpha(h_n)]\ge \alpha_0\varphi(h_n)\), where \(\alpha_0>0\). Let a function \(\psi(h)\) be defined on \((0;1)\), monotonically increasing, and \(\psi(h_n)=1/2^{n+2}\). Construct an increasing sequence of integers \(\{k_n\}\) such that, for every \(n\),

\[ (2^{-1}+2^{-n-1})/2^{k_n+1}<\psi^{-1}(2^{-n-2})\le (2^{-1}+2^{-n-1})/2^{k_n}. \]

From the sequence \(\{k_n\}\) we construct in \([0;1]\) sets \(F\) and \(\mathcal{E}\) by the method indicated in \((^{3})\),

p. 73. Here \(m\mathscr E=1/2\), and every point of the set \(\mathscr E\) is not a Lebesgue—Orlicz point of the function \(\chi_F(t)\).

In the case where \(E\) is an Orlicz space, condition \((\alpha)\) is equivalent to the \(\Delta_2\)-condition. Therefore, from Theorem 2 follows the first part of the main theorem in \(\left({}^3\right)\).

Theorem 2 is sharp in the following sense. There exists a symmetric space \(E\), whose fundamental function satisfies condition \((\alpha)\), and which contains a function that is the limit of bounded functions, such that the measure of its set of points that are not Lebesgue—Orlicz points is positive. As an example of such a space one may take the Orlicz space \(L_M^*\), generated by an \(N\)-function \(M(u)\) satisfying the \(\Delta_2\)-condition and not satisfying the \(\Delta'\)-condition \(\left({}^4\right)\). Indeed, on the basis of Theorem 5 \(\left({}^3\right)\), in \(L_M^*\) there exists a function whose set of points not being Lebesgue—Orlicz points has positive measure. It remains to recall that the \(\Delta_2\)-condition is equivalent to condition \((\alpha)\).

Suppose that the function \(\varphi(t)\) does not satisfy condition \((\alpha)\). Let the function \(\alpha(t)\) be such that

\[ \lim_{t\to 0}\{\varphi[t\alpha(t)]/\varphi(t)\}=0. \tag{2} \]

Let \(e\subset[0;1]\), and let \(t_0\) be a density point of the set \(e\). We shall say that the set \(e\) has at the point \(t_0\) a density order of type \((\alpha)\) if there exists an \(h_0>0\) such that for all \(h<h_0\) one has
\[ [m e\cap(t_0-h;t_0+h)]/2h \ge 1-\alpha(2h), \]
where \(\alpha(t)\) satisfies (2).

Theorem 3. In order that the point \(t_0\) be a Lebesgue—Orlicz point of a bounded function \(x\in E\), it is necessary and sufficient, and for an unbounded function it is necessary, that there exist a set \(e\subset[0;1]\) such that:

1. The function \(x(t)\) is continuous with respect to \(e\) at the point \(t_0\).
2. The density order of the set \(e\) at the point \(t_0\) is of type \((\alpha)\).

If the function \(x\in E\) is the limit of bounded functions, then
\[ \lim_{me\to 0}\|x\chi_e\|_E=0. \]
Put \(\gamma(t)=\sup\|x\chi_e\|_E\), where the supremum is taken over all sets \(e\subset[0;1]\) such that \(me\le t\).

Theorem 4. In order that the point \(t_0\) be a Lebesgue—Orlicz point of a function \(x\in E\), which is the limit of bounded functions, it is sufficient that there exist a set \(e\subset[0;1]\) such that:

1. The function \(x(t)\) is continuous with respect to \(e\) at the point \(t_0\).
2. The density order of the set \(e\) at the point \(t_0\) satisfies the condition
   \[ [m e\cap(t_0-h;t_0+h)]/2h \ge 1-\gamma^{-1}[\alpha(2h)\varphi(2h)]/2h, \]
   where \(\alpha(t)\to0\) as \(t\to0\), and the function \(\gamma^{-1}(t)\) is the inverse of the function \(\gamma(t)\).

§ 2. In the paper \(\left({}^5\right)\) B. I. Korenblyum formulated a necessary and sufficient condition for representability at Lebesgue points of a function \(x\in L_p\) \((p>1)\) by a singular integral:

\[ x(t)=\lim_{n\to\infty}\int_0^1 k_n(s,t)x(s)\,ds. \tag{3} \]

In analogous form, C. Tandori \(\left({}^6\right)\) and R. Taberski \(\left({}^7\right)\) found necessary and sufficient conditions for representability by formula (3) of functions from the spaces \(L_p\) and Orlicz at Lebesgue—Orlicz points. Below Theorem 7 is presented, from which the results of R. Taberski follow. The basis of the proof of Theorem 7 is the scheme set forth by B. I. Korenblyum in \(\left({}^5\right)\).

Denote by \(E_0\) the closure in the space \(E\) of the set of bounded functions. It is known that the general form of a functional in \(E_0\) is integral. Denote by \(R_E\) the linear set of functions \(x\in E_0\) such that,

\[ \lim_{h \to 0} \{\|x\chi_h\|_E/\|\chi_h\|_E\}=0, \tag{4} \]

where \(\chi_h(t)\) is the characteristic function of the interval \((0;h)\).

Relation (4) means that the class \(R_E\) contains precisely those functions of the space \(E_0\) which vanish at \(t=0\) and for which the point \(t=0\) is a Lebesgue–Orlicz point.

Let \(x \in R_E\). Put
\[ \|x\|_{R_E}=\sup_{0<h\leqslant 1}\{\|x\chi_h\|_E/\|\chi_h\|_E\}. \]
The space \(R_E\) is complete.

Theorem 5. If the function \(k(t)\) has the property that for every function \(x\in R_E\) there exists
\[ I(x)=\lim_{h\to 0}\int_h^1 x(t)k(t)\,dt, \]
then \(I\) is a linear functional in \(R_E\).

For the proof, note that the function \(k(t)\chi_{(h,1)}(t)\) is a linear functional in \(E_0\), and
\[ |I_h(x)|\leqslant \varphi(1)\|x\|_{R_E}\|k\chi_{(h,1)}\|_{E_0^*}, \]
where
\[ I_h(x)=\int_h^1 x(t)k(t)\,d(t), \]
whence it follows that for every \(h\), \(I_h\) is a linear functional in \(R_E\). It remains to apply the Banach–Steinhaus theorem.

Denote by \(\|I\|_R\) the norm of the functional \(I\) in \(R_E\). Let \(h_0=1\) and let \(q<1\). Denote by \(h_n\) \((n=0,1,2,\ldots)\) the numbers defined by the formulas
\[ h_{n+1}=\varphi^{-1}[q\varphi(h_n)], \tag{5} \]
where \(\varphi^{-1}(t)\) is the function inverse to \(\varphi(t)\). Since \(\varphi(t)\) is concave, it follows from (5) that \(h_n\leqslant q^n\). Denote by \(C(k)\) the expression
\[ C(k)=\sum_{n=0}^{\infty}\varphi(h_n)\|k\chi_{(h_{n+1},\,h_n)}\|_{E_0^*}. \]

Lemma. The following inequality holds:
\[ q(1-q)C(k)\leqslant \|I\|_R\leqslant C(k). \tag{6} \]

We indicate the main points of the proof. Let the function \(k(t)\) be such that \(C(k)<\infty\). Let \(x\in R_E\). Then
\[ |I(x)|\leqslant \sum_{n=0}^{\infty}\left|\int_{h_{n+1}}^{h_n}x(t)k(t)\,dt\right|\leqslant \]
\[ \leqslant \sum_{n=0}^{\infty}\|x\chi_{(h_{n+1},\,h_n)}\|_E\|k\chi_{(h_{n+1},\,h_n)}\|_{E_0^*} \leqslant \sum_{n=0}^{\infty}\varphi(h_n)\|x\|_{R_E}\|k\chi_{(h_{n+1},\,h_n)}\|_{E_0^*}, \]
whence the right-hand side of inequality (6) follows. Let \(\|I\|_R<\infty\). Then
\[ \int_{h_{n+1}}^{h_n}x(t)k(t)\,dt<\infty \]
for every \(x\in E_0\). Therefore \(k(t)\chi_{(h_{n+1},\,h_n)}(t)\in E_0^*\) \((n=0,1,2,\ldots)\). Let \(x_n\in E_0\) be such functions that, for every \(n\), \(\|x_n\chi_{(h_{n+1},\,h_n)}\|_E=1\) and

\[ \left|\int_{h_{n+1}}^{h_n} x_n(t)\,k(t)\,dt\right|\geq \left\|k\chi_{(h_{n+1},\,h_n)}\right\|_{E_0^*}-\varepsilon. \]

Take the function

\[ x_p^*= \begin{cases} \varphi(h_n)\,|x_n|\,\operatorname{sign} k(t), & t\in (h_{n+1},h_n),\quad n=0,1,2,\ldots,p,\\ 0, & t\in (0,h_{p+1}). \end{cases} \]

It turns out that
\[ \|x_p^*\chi_h\|_E\leq \varphi(h)/q(1-q), \]
whence it follows that \(x_p^*\in R_E\) and
\[ \|x_p^*\|_{R_E}\leq 1/q(1-q) \]
for every \(p\). Next we find

\[ \|I\|_R\|x_p^*\|_{R_E}\geq \left|\int_0^1 x_p^*(t)\,k(t)\,dt\right|= \]

\[ =\sum_{n=0}^{p}\varphi(h_n)\int_{h_{n+1}}^{h_n}|x_n(t)k(t)|\,dt \geq \sum_{n=0}^{p}\varphi(h_n)\left\|k\chi_{(h_{n+1},\,h_n)}\right\|_{E_0^*} -\varepsilon\sum_{n=0}^{p}\varphi(h_n), \]

from which the left-hand side of inequality (6) follows.

The following theorem follows from the lemma.

Theorem 6. The general form of a functional in \(R_E\) can be given by the formula

\[ I(x)=\int_0^1 k(t)x(t)\,dt, \]

where \(k(t)\) is such a function that, for every \(h\), \(k(t)\chi_{(h,1)}(t)\in E_0^*\), and the inequality \(C(k)<\infty\) holds.

Denote by
\[ I_n(x)=\int_0^1 x(t)k_n(t)\,dt\quad (n=1,2,\ldots). \]
Theorem 6 allows us to obtain the following theorem.

Theorem 7. For any function \(x\in E_0\) for which the point \(t=0\) is a Lebesgue–Orlicz point, the relation

\[ \lim_{n\to\infty} I_n(x)=x(0) \]

holds if and only if:

1.
\[ \lim_{n\to\infty}\int_0^h k_n(t)\,dt=1 \quad\text{for every }h,\quad 0<h\leq 1. \]

2.
\[ \overline{\lim}_{n\to\infty}\sum_{\nu=0}^{\infty} \varphi(h_\nu)\left\|k_n\chi_{(h_{\nu+1},\,h_\nu)}\right\|_{E_0^*}<\infty, \]

where the numbers \(h_\nu\) are defined by formulas (5); \(q<1\) and \(h_0=1\).

The author expresses sincere gratitude to S. T. Krein for his attention.

Voronezh Civil Engineering Institute

Received
24 VIII 1966

REFERENCES

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