UDC 517.5

MATHEMATICS

P. E. SOBOLEVSKII

ON FRACTIONAL POWERS OF WEAKLY POSITIVE OPERATORS

(Presented by Academician M. A. Lavrent’ev, June 9, 1965)

1. In (¹) a new method was proposed for investigating fractional powers of a certain class of operators acting in the spaces \(L_p\). It was established in which spaces \(L_{p_\alpha}\) the operators \(A^{-\alpha}\) act from \(L_{p_0}\), if it is known in which \(L_{p_1}\) the operator \(A^{-1}\) acts from \(L_{p_0}\). The proof is based on the moment inequality for fractional powers and the interpolation theorem from (²). In the present paper another method is proposed, which makes it possible to study the operators \(BA^{-\alpha}\). The operators \(BA^{-\alpha}\) were studied earlier in (³). However, restrictions were imposed there on the operator \(A\), the validity of which in important cases of elliptic operators in domains has not yet been established.

2. An operator \(A\), acting in a Banach space \(E\), is called weakly positive (abbrev. w.p. \(E\) operator) if \(D(A)\) is dense in \(E\), the operator \(A+tI\) has a bounded inverse for all \(t \geqslant 0\), and \(\|(A+tI)^{-1}\|_E \leqslant M(t+1)^{-1}\). Arbitrary powers of such operators are defined. In particular,

\[ A^{-a}=\frac{\sin \pi(1-a)}{\pi(1-a)}\int_0^\infty t^{1-a}(A+tI)^{-2}\,dt \qquad (0<a<2). \tag{1} \]

The moment inequality is valid

\[ \|A^\alpha v\|_E \leqslant \frac{\sin \pi\alpha}{\pi\alpha(1-\alpha)} M_1^{1-\alpha}M_2^\alpha \|Av\|_E^\alpha \|v\|_E^{1-\alpha} \qquad (0\leqslant \alpha\leqslant 1,\ v\in D(A)), \tag{2} \]

where

\[ M_1=\sup_{t>0}\|A(A+tI)^{-1}\|_E,\qquad M_2=\sup_{t>0}\|t(A+tI)^{-1}\|_E. \]

Fractional powers of w.p. \(E\) operators were first introduced and studied in (⁴) (see also (⁵)).

Theorem 1. Let \(A\) be a w.p. \(E\) operator and let \(F(v)\) be a seminorm continuous on \(D(A)\). Suppose that for some \(\alpha\in(0,1)\) the inequality

\[ F(v)\leqslant c\|Av\|_E^\alpha\|v\|_E^{1-\alpha} \qquad (v\in D(A)) \tag{3} \]

holds. Then for every \(\beta\in(\alpha,1]\) the seminorm \(F(v)\) can be extended to a seminorm \(\overline F(v)\) continuous on \(D(A^\beta)\), and for every \(\gamma\in[0,\alpha)\) the inequality

\[ \overline F(v)\leqslant c\,\frac{\beta-\gamma}{(\beta-\alpha)(\alpha-\gamma)} \frac{\sin \pi(1-\beta)}{\pi(1-\beta)} \left[ \frac{\sin \pi(\beta-\gamma)} {\pi(\beta-\gamma)(1-\beta+\gamma)} \right]^{(\beta-\alpha)/(\beta-\gamma)} \times \]

\[ \times M_1^{\alpha+(\beta-\alpha)/(\beta-\gamma)} M_2^{2-\alpha} \|A^\beta v\|_E^{(\alpha-\gamma)/(\beta-\gamma)} \|A^\gamma v\|_E^{(\beta-\alpha)/(\beta-\gamma)} \qquad (v\in D(A^\beta)). \tag{4} \]

Proof. Using successively (1), (3), and (2), we obtain that for every \(N>0\) on \(D(A)\) the inequality

\[ F(v)\leqslant \frac{\sin \pi(1-\beta)}{\pi(1-\beta)} \int_0^\infty t^{1-\beta} F\bigl([A+tI]^{-1}A^\beta[A+tI]^{-1}v\bigr)\,dt \leqslant \]

\[ \leqslant c\,\frac{\sin \pi(1-\beta)}{\pi(1-\beta)} M_1^\alpha M_2^{1-\alpha} \left[ \int_0^N t^{\alpha-\gamma-1} \frac{\sin \pi(\beta-\gamma)} {\pi(\beta-\gamma)(1-\beta+\gamma)} M_1M_2\|A^\gamma v\|_E\,dt + \right. \]

\[ \left. +\int_N^\infty t^{\alpha-\beta-1}M_2\|A^\beta v\|_E\,dt \right]. \]

Minimizing the square bracket with respect to \(N\), we obtain (4).

Remark 1. If for some \(0\leq \gamma<\alpha<\beta\leq 1\) (4) is satisfied, then (3) follows easily from (2).

It follows from (4) that \(\overline F(A^{-\beta}v)\) is continuous in \(E\) and

\[ \overline F(A^{-\beta}v)\leq c(\alpha-\gamma,\beta-\gamma) \|v\|_E^{(\alpha-\gamma)/(\beta-\gamma)} \|A^{-(\beta-\gamma)}v\|_E^{(\beta-\alpha)/(\beta-\gamma)} . \tag{5} \]

The estimate of \(\|A^{-\alpha}v\|_E\) follows from (2), if an estimate of \(\|A^{-1}v\|_E\) is known (see (1)). In a more general situation one applies

Theorem 2. For arbitrary \(0\leq \gamma<\alpha<\beta\leq 1\) the inequality

\[ \|A^{-\alpha}v\|_E\leq \frac{\sin\pi\alpha}{\pi}\, \frac{\beta-\gamma}{(\beta-\alpha)(\alpha-\gamma)} [\varphi_{\beta-\beta}(v)]^{(\alpha-\gamma)/(\beta-\gamma)} [\varphi_{1-\gamma}(v)]^{(\beta-\alpha)/(\beta-\gamma)}, \tag{6} \]

\[ \varphi_\delta(v)=\sup_{t>0}\|t^\delta(A+tI)^{-1}v\|_E . \]

Theorem 3. Let \(\delta\in(0,1)\), and let \(E_\delta\) be a Banach space containing \(E\). Let \(A\) be a w.p. \(E_\delta\) operator, and suppose the inequality

\[ \|v\|_E\leq c\|Av\|_{E_\delta}^{\delta}\|v\|_{E_\delta}^{1-\delta} \quad (v\in D(A)\subset E) \tag{7} \]

holds. Then the inequality

\[ \varphi_{1-\delta}(v)\leq c[M_1(E_\delta)]^\delta [M_2(E_\delta)]^{1-\delta}\|v\|_{E_\delta} \tag{8} \]

holds.

1. Below we consider w.p. \(L_p(\Omega)\) operators \(A\), where \(\Omega\) is a domain of \(n\)-dimensional space with boundary \(S\). Inequality (5) makes it possible to estimate \(\overline F(A^{-\beta}v_e)\) on characteristic functions of measurable sets \(e\subset\Omega\), if an estimate of \(\|A^{-(\beta-\gamma)}v_e\|_{L_p}\) is known. It is easy to see that, in any case,

\[ \|A^{-(\beta-\gamma)}v_e\|_{L_p}\leq c(\operatorname{mes} e)^{1/p}. \]

Such estimates, for the particular case of seminorms \(\|BA^{-\beta}v\|_{L_q}\), where \(L_q=L_q(G)\), and \(G\) is a domain of \(m\)-dimensional space and \(B\) is an operator from \(L_p(\Omega)\) into \(L_q(G)\), together with the interpolation theorem from \((^2)\), make it possible to prove the boundedness of the operators \(BA^{-\beta}\).

Theorem 4. Let \(A\) be a w.p. \(L_{p_i}(\Omega)\) operator, \(i=1,2\). Suppose that for some \(\varepsilon_0,\delta_{0,i}\in(0,1)\)

\[ \|A^{-\varepsilon_0}v_e\|_{L_{p_i}}\leq c_i(\operatorname{mes} e)^{\delta_{0,i}} . \tag{9} \]

Let \(B\) be a linear closed operator from \(L_{p_i}(\Omega)\) into \(L_{q_i}(G)\), \(q_1\ne q_2\), \(D(B)\supset D(A)\), and

\[ \|Bv\|_{L_{q_i}}\leq D_i\|Av\|_{L_{p_i}}^{\alpha_0}\|v\|_{L_{p_i}}^{1-\alpha_0} \quad (v\in D(A)), \tag{10} \]

where \(\alpha_0\) is some number in \((0,1)\). Suppose \(\beta_0\) satisfies the inequalities

\[ \alpha_0<\beta_0<\min\{1,\alpha_0+\varepsilon_0\},\qquad \frac1{r_i}\equiv \frac1{p_i}+\frac{\beta_0-\alpha_0}{\varepsilon_0} \left(\delta_{0,i}-\frac1{p_i}\right)\geq \frac1{q_i}, \]

and the numbers \(q_t\) and \(r_t\), for arbitrary \(t\in(0,1)\), are defined by the equalities
\(1/q_t=t/q_1+(1-t)/q_2\), \(1/r_t=t/r_1+(1-t)/r_2\). Then the operator \(BA^{-\beta_0}\) admits a closure to a bounded operator from \(L_{r_t}(\Omega)\) into \(L_{q_t}(G)\).

Hence follows

Theorem 5. Let \(A\) satisfy the conditions of Theorem 4, and let \(R\) be a w.p. \(L_{q_i}(G)\) operator. Suppose \(T\) and \(RTA^{-1}\) are bounded operators from \(L_{p_i}(\Omega)\) into \(L_{q_i}(G)\). Then the operator \(R^{\alpha_0}TA^{-\beta_0}\) admits a closure to a bounded operator from \(L_{r_t}(\Omega)\) into \(L_{q_t}(G)\).

The proof is based on inequality (2), by means of which inequalities (10) are established for the operator \(B=R^{\alpha_0}T\).

Remark 2. Let \(Q\) be a projection operator in \(L_{p_i}(\Omega)\), and let \(A\) be a w.p. \(E_i=QL_{p_i}(\Omega)\) operator. Then the operator \(BA^{-\beta_0}Q\) from Theorem 4 and the operator \(R^{\alpha_0}TA^{-\beta_0}Q\) from Theorem 5 admit a closure to a bounded operator from \(L_{r_t}(\Omega)\) into \(L_{q_t}(G)\).

1. Consider the boundary-value problem

\[ Av+tv \equiv a(x;1/iD)v(x)+(t_0+t)v(x)=f(x)\quad (x\in\Omega), \tag{11} \]

\[ B_jv\equiv B_j(x;1/iD)v(x)=0\quad (x\in S,\ j=1,\ldots,k). \]

Let \(a_0(x;\xi)\) and \(B_{j0}(x;\xi)\) be the collections of the highest-order terms of the polynomials \(a(x;\xi)\) and \(B_j(x;\xi)\), of degrees \(2k\) and \(m_j\), respectively. Let \(a_0(x;\xi)+t^{2k}\neq 0\) for every real vector \(\xi\) and \(t>0\). Let, for any \(x\in S\), vectors \(\xi\) and \(\eta\) (tangent and normal to \(S\) at the point \(x\)), the polynomial
\[ p(z)=a_0(x;\xi+z\eta)+t^{2k} \]
have \(k\) roots \(z_i^+\) with \(\operatorname{Im} z_i^+>0\). Let the polynomials \(B_{j0}(x;\xi+z\eta)\) be linearly independent modulo the polynomial \((z-z_1^+)\cdots(z-z_k^+)\). Finally, let the system of operators \(B_j\) be normal \(\left({}^{13}\right)\), and \(m_j\leq 2k-1\). Then \(\left({}^{6,7}\right)\) the following holds.

Theorem 6. For sufficiently large \(t_0>0\) and for every \(t\geq 0\), problem (11) is uniquely solvable in every \(L_p(\Omega)\), and

\[ t\|v\|_{L_p}+\|v\|_{W_p^{2k}}\leq c_p\|(A+tI)v\|_{L_p}. \tag{12} \]

From (12), first, it follows that \(A\) is a s.p. \(L_p(\Omega)\) operator. Secondly, from (12) and the multiplicative inequalities for norms in the spaces of S. L. Sobolev—L. N. Slobodetskii (see \(\left({}^{8,9}\right)\)) there follows the inequality

\[ \|v\|_{W_p^{2k\gamma}}\leq c\|Av\|_{L_{p_\delta}}^\delta \|v\|_{L_{p_\delta}}^{1-\delta} \quad (v\in D(A)\subset L_{p_\delta}(\Omega)) \tag{13} \]

for any
\[ 0\leq\gamma<1,\quad \gamma<\delta<\min\left\{1,\frac{n}{2k}\left(1-\frac1p\right)+\gamma\right\}, \]
\[ \frac1{p_\delta}=\frac1p+\frac{2k(\delta-\gamma)}{n}. \]

For \(\gamma=0\), from (13) we obtain an inequality of type (7), and for \(\gamma\geq 0\), an inequality of type (10). When \(2k\gamma=l+r\), \(l\) is an integer and \(0<r<1\), one must consider the operator
\[ Bv=|x-y|^{-n-r/p}\bigl[D_x^l v(x)-D_y^l v(y)\bigr], \]
acting from \(D(A)\subset L_{p_\delta}(\Omega)\) into \(L_p(\Omega\times\Omega)\). Thus one establishes

Theorem 7. \(A^{-\delta}\) acts from \(L_{p_\delta}(\Omega)\) into \(W_p^{2k\gamma}(\Omega)\) and is bounded.

Only in the case when \(A\) is a self-adjoint operator in \(L_2\) does the theory of scales \(\left({}^{10}\right)\) give a stronger result: the operator \(A^{-\delta}\) acts from \(L_2\) into \(W_2^{2k\delta}\) and is bounded.

Remark 3. The conditions given above are satisfied by the operators \(A\) generated by the three classical boundary-value problems for elliptic operators of second order. The weak positivity of such operators was proved earlier in \(\left({}^{11}\right)\).

Let \(A\) be an operator acting in \(\dot H_p\), generated by the stationary Navier—Stokes equations (see \(\left({}^{12}\right)\)). From Theorems 7 and 5 (see Remark 2) it follows that

Theorem 8. \(A^{-\delta}\) acts from \(H_{p_\delta}\) into \(W_p^{2\gamma}\) and is bounded.

1. The operator \(A^*\), acting in \(L_{p'}(\Omega)\) \((1/p+1/p'=1)\) and adjoint to the operator \(A\) acting in \(L_p(\Omega)\), is generated by the adjoint differential expression \(a^*\) and the adjoint system of boundary conditions \(B_j^*\), \(j=1,\ldots,k\), which have the same properties as the differential operators \(a\) and \(B_j\) (see \(\left({}^{13}\right)\)). Let \(W_p^r(A)\) be the collection of functions \(v(x)\in W_p^r(\Omega)\) satisfying the boundary conditions \(B_jv(x)=0\) for all \(m_j\leq r-1\) (\(r\) integer). Define \(W_{p'}^r(A^*)\) analogously.

Theorem 9. Let \(v\in W_p^{2k}(A)\), \(u\in W_{p'}^r(A^*)\), \(r=0,1,\ldots,2k\). Then

\[ \left|\int_\Omega Av\bar u\,dx\right| \leq c(p,r)\|v\|_{W_p^{2k-r}}\|u\|_{W_{p'}^r}. \tag{14} \]

In the proof, constructions from \(\left({}^{13}\right)\) are used. Inequalities of the form (14) make it possible (see \(\left({}^{5,14,15}\right)\)) to investigate \(D(A^\alpha)\).

Theorem 10. \(W_p^{2k}(A)\supset D(A^\delta)\) in \(L_{p_\delta}\) for integral \(2k\gamma\), and \(W_{p_\delta}^{2k\delta}(A)\subset D(A^\gamma)\) in \(L_p\) for integral \(2k\delta\).

Here \(\gamma, \delta, p, p_\delta\) are the same as in (13). Only in the case when \(A\) is a self-adjoint operator in \(L_2\), by means of the theory of scales \({}^{10}\), is a stronger result established: \(D(A^\delta)=W_2^{2k\delta}(A)\) for integer \(2k\delta\).

Theorem 11. Let \(l\) and \(m\) be integers,
\(0\leq l\leq 2k-1,\ 1\leq m\leq n\),

\[ \frac{l}{2k}<\alpha<1,\quad 0<\delta<\left(\alpha-\frac{l}{2k}\right)\left(1-\frac{l}{2k}\right)^{-1},\quad \nu_\delta=\left(\alpha-\frac{l}{2k}\right)-\left(1-\frac{l}{2k}\right)\delta, \]

\[ p>\max\left\{1,\frac{n-m}{2k\nu\delta}\right\},\quad \frac{1}{p_\delta}=\frac{1}{p}\frac{m}{n}+\frac{2k\nu_\delta}{n}, \]

and let \(G\) be a smooth manifold of dimension \(m\) lying in \(\Omega\). Then

\[ \|A^{-\alpha}v\|_{W_p^l(G)} \leq c\,\|A^{-1}v\|_{W_{p_\delta}^{\,l}(\Omega)}^{\delta}\, \|v\|_{L_{p_\delta}(\Omega)}^{1-\delta}. \tag{15} \]

For the proof, in view of (1), one must estimate
\(\|(A+tI)^{-2}v\|_{W_p^l(G)}\) in two ways. First, from (13) there follows the estimate

\[ \|(A+tI)^{-2}\|_{L_{p\delta}(\Omega)\to W_p^l(G)}. \]

Second,

\[ \|(A+tI)^{-2}v\|_{W_p^l(G)} \leq \|(A+tI)^{-1}\|_{L_{p\delta}(\Omega)\to W_p^l(G)} \|(A+tI)^{-1}v\|_{L_{p\delta}(\Omega)}. \]

The first factor is estimated by (13), and the second (see (14)) as follows:

\[ \|(A+tI)^{-1}v\|_{L_{p\delta}(\Omega)} = \sup_u\left|\int_\Omega AA^{-1}v\,(A^*+tI)^{-1}u\,dx\right| \leq \]

\[ \leq c\|A^{-1}v\|_{W_{p\delta}^{\,l}(\Omega)} \|(A^*+tI)^{-1}u\|_{W_{p'_\delta}^{\,2k-l}(\Omega)} \left( \frac{1}{p_\delta}+\frac{1}{p'_\delta}=1,\quad \|u\|_{L_{p'_\delta}}=1 \right). \]

Next (13) is applied again.

Inequalities of type (15) make it possible to estimate \(D^lA^{-\alpha}\), if an estimate is known for \(D^lA^{-1}\). Attention is drawn to the latter circumstance in \({}^{3}\).

1. To clarify the question of which spaces \(L_{p_\alpha}\) from \(L_{p_0}\) the operators \(A^{-\alpha}\) act in, it is convenient to use the following rule. The operators \(A^{-\alpha}\) are similar in this sense to integral operators with kernel \(|x-y|^{2k\alpha-n}\) (for \(\alpha>n/2k\)); the operators \(D^lA^{-\alpha}\) (\(l<2k\alpha\)) are similar to \(A^{-\alpha+\beta}\), \(\beta=l/2k\). For elliptic operators of second order this is not a formal rule, since it has been established \({}^{16}\) that their fractional powers and derivatives of fractional powers are integral operators with kernels of the indicated type. Apparently, this fact also holds for operators of arbitrary order \(2k\).

Voronezh Agricultural Institute

Received
9 VI 1965

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