UDC 517.513

MATHEMATICS

S. K. ABDULLAEV, A. A. BABAEV

SOME ESTIMATES FOR A SINGULAR INTEGRAL WITH SUMMABLE DENSITY

(Presented by Academician I. N. Vekua, January 7, 1969)

Let the function \(u(x)\) be summable on \((a,b)\) and belong to \(L_p\) \((p>1)\) on every segment \([a+\xi,b-\eta]\) \((\xi,\eta>0)\). Introduce the functions

\[ \Omega_p(u,\xi,\eta)=\left\{\int_{a+\xi}^{b-\eta}|u(x)|^p\,dx\right\}^{1/p}, \]

\[ \omega_p(u,\tau,\xi,\eta):=\sup_{h\in A}\left\{\int_{a+\xi}^{b-\eta-h}|u(x+h)-u(x)|^p\,dx\right\}^{1/p}, \]

where \(\xi,\eta,\tau>0\); \(\xi+\eta\le b-a=l\); \(1<p\le\infty\); \(A=\{h;\ 0<h\le \min\{\tau,l-\xi-\eta\}\}\).

For \(p=+\infty\)

\[ \Omega_p(u,\xi,\eta)=\max_{x\in[a+\xi,b-\eta]}|u(x)|=\Omega(u,\xi,\eta), \]

\[ \omega_p(u,\tau,\xi,\eta)= \max_{\substack{x,y\in[a+\xi,b-\eta]\\ |x-y|\le\tau}} |u(x)-u(y)|=\omega(u,\tau,\xi,\eta). \]

Denote

\[ \widetilde{u}(x)=\int_a^b \frac{u(s)}{s-x}\,ds =\lim_{\varepsilon\to+0}\left(\int_a^{x-\varepsilon}+\int_{x+\varepsilon}^{b}\right)\frac{u(s)}{s-x}\,ds. \]

In the present work we consider the question of the relation between the ordered pairs
\((\Omega_p(\widetilde{u},\xi,\eta),\omega_p(\widetilde{u},\tau,\xi,\eta))\) and \((\Omega_p(u,\xi,\eta),\omega_p(u,\tau,\xi,\eta))\).

This problem in the case \(p=+\infty\) (in the class of functions continuous on \((a,b)\)) was posed in \((^1)\) and solved in the works \((^{1,2})\).

Theorem 1. Let \(1<p\le\infty\). If the integrals

\[ \int_0 \frac{\Omega_p(u,t,t)}{t^{1/p}}\,dt, \qquad \int_0 \frac{\omega_p(u,t,\xi/2,\eta/2)}{t}\,dt, \]

converge, then for \(0<\xi,\eta\le b/2\), \(\delta>0\), the estimates

\[ \Omega_p(\widetilde{u},\xi,\eta)\le cq\left\{ \int_0^{l/2}\frac{\Omega_p(u,t,l/4)}{t^{1/p}(t+\xi)^{1/q}}\,dt + \int_0^{l/2}\frac{\Omega_p(u,l/4,t)}{t^{1/p}(t+\eta)^{1/q}}\,dt +\right. \]

\[ \left. +\int_0^{\xi/2}\frac{\omega_p(u,t,\xi/2,l/4)}{t}\,dt + \int_0^{\eta/2}\frac{\omega_p(u,t,l/4,\eta/2)}{t}\,dt \right\}, \tag{1} \]

\[ \omega_p(\widetilde{u},\delta,\xi,\eta)\le cq\left\{ \frac{\delta}{\xi+\delta}\int_0^{l/2}\frac{\Omega_p(u,t,l/4)}{t^{1/p}(t+\xi)^{1/q}}\,dt + \frac{\delta}{\eta+\delta}\int_0^{l/2}\frac{\Omega_p(u,l/4,t)}{t^{1/p}(t+\eta)^{1/q}}\,dt +\right. \]

\[ \left. +\delta\int_0^{\xi/2}\frac{\omega_p(u,t,\xi/2,l/4)}{t(t+\delta)}\,dt + \delta\int_0^{\eta/2}\frac{\omega_p(u,t,l/4,\eta/2)}{t(t+\delta)}\,dt \right\}, \tag{2} \]

where \(1/p+1/q=1\); \(c\) is a constant depending only on \(l\).

Note that for \(p=+\infty\) these estimates turn into the estimates of the work \((^2)\), which is a refinement and development of the work \((^1)\).

Let us consider some constructions based on the preceding estimates. Denote by \(G\) \((^2)\) the set of ordered pairs of functions \((\varphi(\xi,\eta),\psi(\delta,\xi,\eta))\), defined respectively on \(\{0<\xi,\eta\mid \xi+\eta\leq l\}\), \(\{0<\delta,\xi,\eta\mid \delta+\xi+\eta\leq l\}\), and satisfying the conditions:

1) \(\varphi(\xi,\eta)\), \(\psi(\delta,\xi,\eta)/\delta\) are positive and almost decreasing* in each of the arguments uniformly with respect to the others;

2) \(\displaystyle \lim_{\delta\to+0}\psi(\delta,\xi,\eta)=0\).

By definition, a function \(u(x)\), given on \((a,b)\), belongs to the set \(H_{\varphi\psi}^{p}\) if there exist constants \(c_1(u),c_2(u)>0\) such that

\[ \Omega_p(u,\xi,\eta)\leq c_1(u)\varphi(\xi,\eta),\qquad \omega_p(u,\delta,\xi,\eta)\leq c_2(u)\psi(\delta,\xi,\eta), \]

where \((\varphi,\psi)\in G\).

By introducing the norm

\[ \|u\|_{\varphi\psi}^{p}= \max\left\{\sup_{\xi,\eta}\frac{\Omega_p(u,\xi,\eta)}{\varphi(\xi,\eta)},\ \sup_{\xi,\eta,\delta}\frac{\omega_p(u,\delta,\xi,\eta)}{\psi(\delta,\xi,\eta)}\right\}, \]

\(H_{\varphi\psi}^{p}\) is turned into an infinite-dimensional Banach space.

Theorem 2. Let \((\varphi_1,\psi_1),(\varphi_2,\psi_2)\in G\).

Then:

a) if \(\varphi_1\sim\varphi_2\), \(\psi_1\sim\psi_2\), then \(H_{\varphi_1\psi_1}^{p}\) and \(H_{\varphi_2\psi_2}^{p}\) coincide**;

b) if the limiting relations

\[ \lim_{\delta\to+0}\frac{\psi_1(\delta,\xi,\eta)}{\psi_2(\delta,\xi,\eta)}=0,\qquad \lim_{\xi\to+0}\frac{\psi_1(\delta,\xi,\eta)}{\psi_2(\delta,\xi,\eta)}=0, \]

\[ \lim_{\eta\to+0}\frac{\psi_1(\delta,\xi,\eta)}{\psi_2(\delta,\xi,\eta)}=0,\qquad \lim_{\xi\to+0}\frac{\varphi_1(\xi,\eta)}{\varphi_2(\xi,\eta)}=0,\qquad \lim_{\eta\to+0}\frac{\varphi_1(\xi,\eta)}{\varphi_2(\xi,\eta)}=0, \]

are satisfied uniformly, then \(H_{\varphi_1\psi_1}^{p}\) is a proper part of \(H_{\varphi_2\psi_2}^{p}\), and the embedding is completely continuous.

Denote by \(\Phi\) the set of ordered pairs of functions \((\varphi,\psi)\in G\) satisfying the conditions:

1) \(\psi(\delta,\xi,\eta)\) almost increases with respect to \(\delta\);

2) \(\psi(\delta_1+\delta_2,\xi,\eta)=O(\psi(\delta_1,\xi,\eta)+\psi(\delta_2,\xi,\eta))\)***;

3) \(\psi(\delta,\xi,\eta)=O(\varphi(\xi,\eta))\).

Following \((^2)\), introduce the set \(H_p\) of ordered pairs of functions \((\varphi(\xi),\psi(\delta,\xi))\) satisfying the conditions:

1) \(\varphi(\xi)>0,\ \psi(\delta,\xi)>0\);

2)

\[ \int_0^{1/2}\frac{\varphi(t)}{t^{1/p}(t+\xi)^{1/q}}\,dt=O(\varphi(\xi)); \]

3)

\[ \delta\int_0^{\xi}\frac{\psi(t,\xi/2)}{t(t+\delta)}\,dt=O(\psi(\delta,\xi)); \]

4)

\[ \frac{\delta}{\xi+\delta}\varphi(\xi)=O(\psi(\delta,\xi)). \]

By definition \((\varphi,\psi)\in \Phi H_p\) if \((\varphi,\psi)\in\Phi\) and \((\varphi(\xi,l/4),\psi(\delta,\xi,l/4))\), \((\varphi(l/4,\eta),\psi(\delta,l/4,\eta))\in H_p\).

Theorem 3. Let \((\varphi,\psi)\in\Phi H_p\). Then the operator

\[ Au=\int_a^b \frac{u(s)}{s-x}\,ds \]

acts in \(H_{\varphi\psi}^{p}\) and is bounded.

* A nonnegative function \(f(x)\), defined on a set \(\chi\subset(-\infty,+\infty)\), is called almost increasing (almost decreasing) if there exists a constant \(c>0\) such that the inequality \(x_1\leq x_2\), \(x_1,x_2\in\chi\), implies the inequality \(f(x_1)\leq cf(x_2)\) \((f(x_1)\geq cf(x_2))\).

** Nonnegative functions \(f(x)\) and \(g(x)\), defined on \(\chi\), are called equivalent \((f\sim g)\) if there exist constants \(B_1,B_2>0\) such that for every \(x\in\chi\) the inequalities \(B_1 f(x)\leq g(x)\leq B_2 f(x)\) hold.

*** Here and in what follows, uniform satisfaction of the \(O\)-relation is assumed.

This theorem, in the case \(p=+\infty\), was proved in \({}^{2}\).
It is easy to verify that the pair of functions

\[ \varphi(\xi,\eta)=\frac{1}{\xi^\alpha}+\frac{1}{\eta^\beta},\qquad \psi(\delta,\xi,\eta)=\frac{\delta^\alpha}{\xi^\alpha(\xi+\delta)^\alpha} +\delta^\gamma+ \frac{\delta^\beta}{\eta^\beta(\eta+\delta)^\beta} \]

\[ (0<\alpha,\beta<1/q;\; 0<\gamma<1) \]
belongs to \(\Phi H_p\).

The authors express their gratitude to V. V. Salaev for valuable comments.

Azerbaijan State University
named after S. M. Kirov
Baku

Received
25 XII 1968

REFERENCES

\({}^{1}\) A. A. Babaev, DAN, 170, No. 5 (1966).
\({}^{2}\) V. V. Salaev, Scientific Notes of Azerbaijan State University, ser. phys.-math. sciences, No. 6 (1966).