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,\ }\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).