MATHEMATICS

Kh. Sh. Mukhtarov

ON SOME MULTIPLICATIVE INEQUALITIES AND THEIR APPLICATION TO LINEAR SINGULAR INTEGRAL OPERATORS

(Presented by Academician N. I. Muskhelishvili on 11 VII 1967)

Let the function \(u(x_1,\ldots,x_n)\) be given in the \(n\)-dimensional cube \(D:\ \{a \leq x_1, x_2,\ldots,x_n \leq b\}\).

Definition. By the class \(H_\delta\) we shall mean the collection of functions \(u(x_1,\ldots,x_n)\), defined in \(D\), if for any \(M\) and \(\widetilde M\) from \(D\) the inequality holds

\[ |u(M)-u(\widetilde M)| \leq R_u \sum_{i=1}^{n}|x_i-\widetilde x_i|^\delta, \]

\[ 0<\delta \leq 1,\quad M=M(x_1,\ldots,x_n),\quad \widetilde M=\widetilde M(\widetilde x_1,\ldots,\widetilde x_n), \]

where \(R_u\) is the Hölder constant.

Theorem 1. For the class \(H_\delta\), for any \(p>0\) there exists \(g>0\) such that

\[ \|u\|_C \leq q \|u\|^{(n+\delta p)/(n+2\delta p)} \|u\|_p^{\delta p/(n+2\delta p)}, \tag{1} \]

where

\[ \|u\|_C=\max_{M\in D}|u(M)|,\quad \|u\|_\delta=\|u\|_C+ \sup_{M,\widetilde M\in D} \left\{ \frac{|u(M)-u(\widetilde M)|}{\sum_{i=1}^{u}|x_i-\widetilde x_i|^\delta} \right\}, \]

\[ \|u\|_p^p=\int_D |u(M)|^p\,dv,\quad dv=dx_1\ldots dx_n, \]

\[ q \leq l=\max\left\{n(b-a)^{\alpha_1}/2^\delta;\ 2^{n/p}/(b-a)^{\alpha_3}\right\}^{1/(1+\alpha_3)}, \]

\[ \alpha_1=\delta^2p/(n+\delta p),\quad \alpha_2=\alpha_1/\delta,\quad \alpha_3=n/p. \]

Proof. Let \(M_0(x_1^0,\ldots,x_n^0)\) be a fixed point from \(D\) and \(u(M)\in H_\delta\) \((u(M)\neq 0)\).

Consider the set

\[ E_u^{(i)}(M_0)= \begin{cases} [x_i^0,\ x_i^0+c], & \text{if } a\leq x_i^0\leq (a+b)/2,\\ [x_i^0-c,\ x_i^0], & \text{if } (a+b)/2<x_i^0\leq b, \end{cases} \]

where

\[ c=l_1\|u\|_p^{\widetilde\beta},\quad l_1=(b-a)^{\alpha_2}/2\|u\|_C^{p/(n+\delta p)},\quad \widetilde\beta=p/(n+\delta p),\quad i=1,2,\ldots,n. \]

It is not hard to notice that \(E_u^{(i)}(M_0)\subset [a,b]\).

Denote by \(E_u(M_0)\) the collection of points \(\{x_1,\ldots,x_n\}\), where each of the components respectively ranges over the sets \(E_u^{(1)}(M_0),\ldots,E_u^{(n)}(M_0)\). Then it is clear that \(E_u(M_0)\subset D\). Consequently,

\[ \|u\|_{L_p}\geq \int_{E_u(M_0)} |u(M)|^p\,dv = |u(M')|^p l_1^n \|u\|_{L_p}^{n\widetilde\beta}, \]

\[ M'\in E_u(M_0),\quad M'=\{\xi_1,\xi_2,\ldots,\xi_n\},\quad \xi_i\in E^{(i)}(M_0),\quad i=1,\ldots,n. \]

Thus,

\[ |u(M)| \leq l_1^{-n/p}\|u\|_{L_p}^{\alpha_2}. \]

Since \(u(M)\in H_\delta\) and \(|\xi_i-x_i^0|\leq c\), we have

\[ |u(M_0)|\leq R_u\sum_{i=1}^{n}|\xi_i-x_i^0|+|u(\widetilde M)| \leq (R_u n l_1^\delta+l_1^{-n/p})\|u\|_{L_p}^{\alpha_2}, \]

where

\[ R_u=\sup_{M',\,M''\in D} \left\{ \frac{|u(M')-u(M'')|} {\displaystyle\sum_{i=1}^{n}|x_i'-x_i''|^\delta} \right\}. \]

If instead of \(l_1\) we substitute its expression, we obtain (1).

Corollary. Let \(\rho(x)\geq 0\) \((a\leq x\leq b)\), and let, for \(p>1\), the integral converge:

\[ \int_a^b \rho^{1/(1-p)}(x)\,dx<\infty . \]

Then for any \(u(x)\in H_\delta\) \((0<\delta\leq 1)\) the following holds:

\[ \|u\|_C\leq q'\|u\|_\delta^{(1+\delta)/(1+2\delta)} \|u\|_{L_p(\rho)}^{\delta/(1+2\delta)}; \tag{1'} \]

\(q'\) is a constant independent of \(u(x)\).

Let \(\Gamma\) be a closed or open smooth Lyapunov curve in the plane of the complex variable. For functions \(u(\tau)\) given on the contour \(\Gamma\) and belonging to the Hölder class \(H_\delta\) \((0<\delta\leq 1)\), the following holds.

Theorem 2. For any \(u(\tau)\in H_\delta\) and for any \(p>0\), the inequality

\[ \|u\|_C\leq q_1\|u\|_\delta^{(1+\delta p)/(1+2\delta p)} \|u\|_{L_p}^{\delta p/(1+2\delta p)} \tag{2} \]

holds, where

\[ \|u\|_C=\max_{\tau\in\Gamma}|u(\tau)|;\qquad \|u\|_{L_p}^{p}=\int_\Gamma |u(\tau)|^p\,ds, \]

\[ \|u\|_\delta=\|u\|_C+ \sup_{\tau_1,\tau_2\in\Gamma} \left\{ \frac{|u(\tau_1)-u(\tau_2)|}{|\tau_1-\tau_2|^\delta} \right\}, \qquad q_1=\mathrm{const}. \]

Inequalities of the type (1) and (2) have also been obtained in the case when \(u(\tau)\) belongs to the class \(H(\varphi)\) (for the definition see (1)).

For the Guseinov classes \(H_{\alpha,\beta,\delta}\) (for the definition see (1)) the following result has been obtained:

Theorem 3. For any \(u(x)\in H_{\alpha,\beta,\delta}\) \((a<x<b)\) and for any \(p>0\), there exists a constant number \(q_2>0\) such that

\[ \|u\|_{C(\rho)}\leq q_2 \|u\|_{\alpha,\beta,\delta,\delta_0}^{(1+\delta_0p)/(1+2\delta_0p)} \|u\|_{L_p(\rho_1)}^{\delta_0p/(1+2\delta_0p)}, \tag{3} \]

where

\[ 0<\delta_0\leq\delta,\qquad \rho_1(x)=(x-a)^{(\alpha+\delta)p}(b-x)^{(\beta+\delta)p}, \]

\[ \|u\|_{C(\rho)}=\sup_{a<x<b}\{|u(x)|\rho(x)\},\qquad \rho(x)=(x-a)^{\alpha+\delta}(b-x)^{\beta+\delta}, \]

\[ \|u\|_{\alpha,\beta,\delta,\delta_0} = \|u\|_{C(\rho)} + \sup_{a<x,y<b} \left\{ \frac{|W(x)-W(y)|}{|x-y|^{\delta_0}} \right\}, \qquad W(x)=u(x)\rho(x). \]

Now consider linear singular operators with Hilbert and Cauchy kernels

\[ S_1u=-\frac{1}{2\pi}\int_{-\pi}^{\pi}K(x,s)u(s)\operatorname{ctg}\frac{s-x}{2}\,d\tau, \]

\[ S_2f=\frac{1}{2\pi i}\int_\Gamma \frac{h(t,\tau)f(\tau)}{\tau-t}\,d\tau, \]

where \(K(x,s)\) is a \(2\pi\)-periodic function in \(x\) and \(s\), satisfying a Hölder condition with exponent \(\delta_1>0\) in \(x\) and \(\delta>0\) \((\delta<\delta_1)\) in \(s\), while \(h(t,\tau)\) is a function defined on the closed Lyapunov contour \(\Gamma\), satisfying the same condition as \(K(x,s)\).

If we take into account that \((^2,^3)\) the operators \(S_1\)* and \(S_2\) are bounded in the spaces \(H_\delta\) and \(L_p\) \((0<\delta<1,\ p>1)\), then, by virtue of inequalities (1) and (2), we have

Theorem 4. The operators \(S_1\) and \(S_2\) act from \(H_\delta\) into \(H_\delta\) and satisfy the inequalities

\[ \|S_1u\|_C \leq q_1^*\|u\|_\delta^{(1+\delta p)/(1+2\delta p)}\|u\|_{L_p}^{\delta p/(1+2\delta p)}, \tag{4} \]

\[ \|S_2f\|_C \leq q_2^*\|f\|_\delta^{(1+\delta p)/(1+2\delta p)}\|f\|_{L_p}^{\delta p/(1+\delta p)}; \tag{5} \]

\[ 0<\delta<1;\qquad 1<p;\qquad q_i^*=\mathrm{const};\qquad i=1,2. \]

For the singular operator

\[ S_{\widetilde{\alpha}}u=(x-a)^{\widetilde{\alpha}}(b-x)^{\widetilde{\alpha}} \int_a^b \frac{u(s)\,ds}{(s-a)^{\widetilde{\alpha}}(b-s)^{\widetilde{\alpha}}(s-x)} \]

it has been established \((^4)\) that it acts boundedly from \(H_\delta^0\)** into \(H_\delta^0\) \((0<\delta<\widetilde{\alpha})\). Therefore, by virtue of Khvedelidze’s theorem \((^3)\) and inequality (1), we prove

Theorem 5. The operator \(S_{\widetilde{\alpha}}\) acts from \(H_\delta^0\) into \(H_\delta^0\) and satisfies the inequality

\[ \|S_\alpha u\|_C \leq q_3\|u\|_\delta^{(1+\delta p)/(1+2\delta p)}\|u\|_{L_p}^{\delta p/(1+2\delta p)}, \tag{6} \]

where \(0<\delta<\widetilde{\alpha},\ p>1/(1-\widetilde{\alpha}),\ q_3=\mathrm{const}\).

Finally, let us consider the operator

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

which acts boundedly in the Banach space \(H_{\alpha,\beta,\delta}\) with norm

\[ \|u\|_{\alpha,\beta,\delta} = \sup_{a<x<b}\{|u(x)|\gamma(x)\} + \sup_{a<x,y<b} \left\{ \frac{|W(x)-W(y)|}{|x-y|^\delta} \right\}, \]

\[ W(x)=u(x)\gamma(x),\qquad \gamma(x)=(x-a)^{\alpha+\delta}(b-x)^{\beta+\delta}. \]

Theorem 6. The operator \(S\) acts from \(H_{\alpha,\beta,\delta}\) into \(H_{\alpha,\beta,\delta}\) and satisfies the inequality

\[ \|Su\|_{C(\gamma)} \leq q_4\|u\|_{\alpha,\beta,\delta}^{(1+\delta p)/(1+2\delta p)} \|u\|_{L_p(\gamma)}^{\delta p/(1+2\delta p)}, \tag{7} \]

where

\[ 0<\alpha+\delta,\qquad \beta+\delta<1,\qquad p>\max\{1/(1-\alpha-\delta),\ 1/(1-\beta-\delta)\}, \]

\[ q_4=\mathrm{const}. \]

Now let us consider a sequence \(\{u_n\}\in H_\delta\), satisfying the condition

\[ \lim_{n,m\to\infty} \|u_n-u_m\|_\delta^{\alpha_1} \|u_n-u_m\|_{L_p}^{1-\alpha_1} =0,\qquad p>1, \]

\[ \alpha_1=(1+\delta p)/(1+2\delta p),\qquad 0<\delta<1. \tag{8} \]

By virtue of (1) \((n=1)\) or (2) or (3)*** (depending on where it is defined—

* We mean \(2\pi\)-periodic functions from \(H_\delta\).

** \(H_\delta^0\) contains all functions from \(H_\delta\) that vanish at the endpoints of the interval \([a,b]\).

*** In this case \(\{u_n(x)\}\in H_{\alpha,\beta,\delta}\), and under (8) one must understand

\[ \lim_{n,m\to\infty} \|u_n-u_m\|_{\alpha,\beta,\delta}^{\alpha_1} \|u_n-u_m\|_{L_p(\gamma)}^{1-\alpha_1}=0. \]

chosen $\{u_n\}$, this sequence is fundamental in the metric of the space of continuous functions (in the case (3) with weight $\rho(x)=(x-a)^{\alpha+\delta}(b-x)^{\beta+\delta}$). By virtue of the completeness of the space $C$ there exists $u_0\in C$ for which $\lim_{n\to\infty}\|u_n-u_0\|=0$. The sets of all such $u_0$, corresponding in the sense of fundamentality to the right-hand sides of inequalities (2), (3), (4), and (6), will be denoted respectively by $C_{\delta,1}^p$, $C_{\delta,2}^p$, $C_{\delta,3}^p$, and $C_{\delta,4}^p$. For these sets, using inequalities (1), (2), (3), (4), (5), (6), (7), one proves

Theorem 7. The sets $C_{\delta,1}^p$, $C_{\delta,2}^p$, $C_{\delta,3}^p$, and $C_{\delta,4}^p$ are invariant respectively with respect to the operators $S_1$, $S_2$, $S_{\tilde{\alpha}}$ $(\tilde{\alpha}<1)$, and $S$.

We note that the Hölder or Huseynov classes corresponding to these sets are proper subsets of these sets.

Dagestan State University
named after V. I. Lenin

Received
20 VI 1967

CITED LITERATURE

$^{1}$ Kh. Sh. Mukhtarov, Dokl. AN AzerbSSR, 21, No. 4 (1965).
$^{2}$ N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
$^{3}$ B. V. Khvedelidze, Tr. Tbilissk. matem. inst. AN GruzSSR, 23, 3 (1957).
$^{4}$ Kh. Sh. Mukhtarov, Abstracts of the All-Union Interuniversity Conference on the Application of Methods of Functional Analysis to the Solution of Nonlinear Problems, Baku, 1965.