Full Text
UDC 517.5
MATHEMATICS
P. E. SOBOLEVSKII
ON FRACTIONAL INTEGRATION BY PARTS
(Presented by Academician I. N. Vekua on March 1, 1968)
Consider the problem
\[ v'(t)+Av(t)=f(t)\quad (0\leq t\leq 2\pi),\qquad v(0)=v_0 \tag{1} \]
in a Hilbert space \(H\) with a positive definite self-adjoint operator \(A\). By its solution we shall mean a function \(v(t)\) satisfying (1) such that the functions \(v'(t)\) and \(Av(t)\) are continuous. If a solution of problem (1) exists, then it is given by the formula
\[ v(t)=\exp\{-tA\}v_0+\int_0^t \exp\{-(t-s)A\}f(s)\,ds =\exp\{-tA\}v_0+Qf(t), \tag{2} \]
where \(\exp\{-tA\}\) is the semigroup generated by the operator \(A\). Necessary conditions for the existence of a solution of problem (1) are the continuity of the function \(f(t)\) and the membership of \(v_0\) in the domain \(D(A)\) of the operator \(A\). Are these conditions sufficient for the existence of a solution of problem (1)? If \(v_0\in D(A)\), then problem (1) has a solution in the case when the problem
\[ v'(t)+Av(t)=f(t)\quad (0\leq t\leq 2\pi),\qquad v(0)=0. \tag{3} \]
has a solution.
In the present paper, the solvability of problem (3) is proved under the assumption that \(f(t)\) belongs to the abstract space \(W_2^{1/2}\) of L. N. Slobodetskii. We note that \(f(t)\in W_2^{1/2}\) need not satisfy any Hölder condition. This result is based on the method of estimating integrals of the form
\[ I_i=\int_\Omega u\,\frac{\partial v}{\partial x_i}\,dx,\qquad i=1,\ldots,n, \tag{4} \]
of products of smooth scalar functions \(u(x)\) and \(v(x)\) (or functions with values in a Hilbert space) defined in an \(n\)-dimensional domain \(\Omega\).
1. Theorem 1. Let the boundary \(S\) of the domain \(\Omega\) be continuously differentiable once. Then the inequality
\[ |I_i|\leq M(\alpha)\|u\|_{W_2^\alpha(\Omega)}\|v\|_{W_2^{1-\alpha}(\Omega)} \tag{5} \]
holds for any \(\alpha\) from \([0,1/2)\), and the inequality
\[ |I_i|\leq M(1/2)\bigl(\|u\|_{W_2^{1/2}(\Omega)}+\|u\|_{L_2(S)}\bigr) \bigl(\|v\|_{W_2^{1/2}(\Omega)}+\|v\|_{L_2(S)}\bigr). \tag{6} \]
We shall prove, for the one-dimensional integral
\[ I=\int_0^{2\pi} u(x)v'(x)\,dx, \]
the one-dimensional analogues of these inequalities,
\[ |I|\leq M(\alpha)\|u\|_{W_2^\alpha}\|v\|_{W_2^{1-\alpha}} \qquad (0\leq \alpha<1/2), \tag{7} \]
\[ |I|\leq M(1/2)\bigl(\|u\|_{W_2^{1/2}}+|u(0)|+|u(2\pi)|\bigr) \bigl(\|v\|_{W_2^{1/2}}+|v(0)|+|v(2\pi)|\bigr). \tag{8} \]
Consider inequality (7). Let the function \(u(x)\) be periodic \((u(0)=u(2\pi))\). Setting \(v(x)=v_1(x)+v_2(x)\), where \(v_2(x)= (t/2\pi)[v(2\pi)-v(0)]\), we obtain
\[ I=I_1+I_2=\int_0^{2\pi} u(x)v_1'(x)\,dx+\frac{1}{2\pi}\int_0^{2\pi}u(x)\,dx\,[v(2\pi)-v(0)]. \]
Let
\[ u(x)=\sum_{|n|=0}^{+\infty} a_n e^{inx},\qquad v_1(x)=\sum_{|n|=0}^{+\infty} b_n e^{inx}. \]
Then
\[ |I_1|\leq 2\pi \sum |n|\,|a_n|\,|b_n| \leq 2\pi\left(\sum |n|^{2\alpha}|a_n|^2\right)^{1/2} \left(\sum |n|^{2-2\alpha}|b_n|^2\right)^{1/2} \]
But
\[ 2\pi\sum(1+|n|^{2\alpha})|a_n|^2=\|u\|_{W_2^\alpha}^{\,2},\qquad 2\pi\sum(1+|n|^{2-2\alpha})|b_n|^2=\|v_1\|_{W_2^{1-\alpha}}^{\,2}, \]
since the functions \(u(x)\) and \(v_1(x)\) are periodic. Therefore
\[
|I_1|\leq \|u\|_{W_2^\alpha}\|v_1\|_{W_2^{1-\alpha}}.
\]
Further, for \(\alpha>0\),
\[ \|v_1\|_{W_2^{1-\alpha}} \leq \|v\|_{W_2^{1-\alpha}}+\|v_2\|_{W_2^{1-\alpha}} \leq \|v\|_{W_2^{1-\alpha}}+C_1(\alpha)|v(2\pi)-v(0)|. \]
By the embedding theorems,
\[
|v(x)|\leq C_2(\alpha)\|v\|_{W_2^{1-\alpha}}
\]
for \(\alpha<1/2\). Finally,
\[ \left|(1/2\pi)\int_0^{2\pi}u(x)\,dx\right|\leq C_3\|u\|_{W_2^\alpha}. \]
Hence inequality (7) follows for a periodic smooth function \(u(x)\). However, for \(\alpha<1/2\) a smooth function \(u(x)\) can be approximated in the metric \(W_2^\alpha\) by smooth periodic functions. Obviously, it is enough to prove this assertion for the linear function \((t/\pi)[u(2\pi)-u(0)]\), which can be approximated in the metric \(W_2^\alpha\), for \(\alpha<1/2\), by its partial Fourier sums (these partial sums are periodic functions, and their Fourier coefficients decrease as \(c/n\)).
To prove inequality (8), the functions \(u(x)\) and \(v(x)\) must be reduced to periodic ones by the method indicated above.
- The closure of the set \(K\) of all smooth functions \(f(t)\), defined on \([0,2\pi]\) with values in \(H\), in the norm
\[ \|f\|_{L_2}=\left(\int_0^{2\pi}\|f\|_H^2(t)\,dt\right)^{1/2} \]
forms the space \(L_2=L_2([0,2\pi],H)\). The closure of the set \(K\) in the norm
\[ \|f\|_{W_2^1}:=(\|f\|_{L_2}^2+\|f'\|_{L_2}^2)^{1/2} \]
forms the space \(W_2^1\). The closure of the set \(K\) in the norm
\[ \|f\|_{W_2^\alpha}= \left(\|f\|_{L_2}^2+ \int_0^{2\pi}\int_0^{2\pi} \|f(t)-f(s)\|_H^2 |t-s|^{-1-2\alpha}\,dt\,ds \right)^{1/2} \qquad (0<\alpha<1) \]
forms the space \(W_2^\alpha\). The spaces \(W_2^l\) are defined analogously for any (integer or fractional) \(l\).
If in the preceding constructions one considers, instead of \(K\), its subset \(\dot K\), consisting of functions \(f(t)\) for which \(f(0)=0\), then we obtain the spaces \(\dot L_2,\dot W_2^1,\dot W_2^\alpha\). Obviously, \(\dot L_2=L_2\). Just as in item 1, it is shown that \(\dot W_2^\alpha=W_2^\alpha\) for \(\alpha<1/2\).
It is known (see, for example, (1)) that the operators \(AQ\) and \(\dfrac{d}{dt}Q\) act boundedly in the spaces \(L_2\) and \(\dot W_2^1\). Hence it follows that
Theorem 2. The operators \(AQ\) and \(\dfrac{d}{dt}Q\) act boundedly in the spaces \(\dot W_2^\alpha\).
For the proof one may use the method applied in (2), p. 158, item 7, for spaces of scalar functions. In doing so, instead of the space \(L_2((-\infty,+\infty),H)\) one must take its subspace \(L_2\), consisting of odd functions, and take into account that the Fourier transform is a unitary operator in \(L_2\), that in it there acts the operator \(J\) of multiplication by the function \((1+|t|^{\frac12})^{1/2}\), and that it is a positive definite self-adjoint operator. Next, the domains of definition of the fractional powers \(J^\alpha\) of the operator \(J\) form the spaces \(\overset{\circ}{W}{}_2^\alpha\). Finally, the traces of functions from these spaces on the segment \([0,2\pi]\) form the spaces \(\overset{\circ}{W}{}_2^\alpha\) introduced above. Therefore between the spaces \(L_2\) and \(\overset{\circ}{W}{}_2^1\) one can stretch a Hilbert scale \(H_\alpha\) so that the norms in \(H_\alpha\) and in \(\overset{\circ}{W}{}_2^\alpha\) are equivalent. Applying the interpolation theorem, we obtain the assertion of Theorem 2.
From Theorem 2 and the interpolation theorem it follows that
Theorem 3. The operator \(A^\alpha Q\) acts boundedly from \(L_2\) to \(W_2^{1-\alpha}\) and from \(\overset{\circ}{W}{}_2^\beta\) to \(\overset{\circ}{W}{}_2^{\beta+1-\alpha}\).
- Denote by \(C^\alpha=C^\alpha([0,2\pi],H)\) \((0\le \alpha\le 1)\) the closure of the set \(K\) in the norm
\[ \|f\|_{C^\alpha}=\max_{0\le t\le 2\pi}\|f(t)\|_H+ \sup_{0\le t,\ s\le 2\pi}|t-s|^{-\alpha}\|f(t)-f(s)\|_H . \]
Theorem 4. For any \(\alpha\) from \([0,1]\), \(T>0\), \(v_0\) from \(H\), the function \(v(t)=\exp\{-tA\}A^{-\alpha}v_0\) belongs to \(C^\alpha([0,T],H)\cap W_2^\beta([0,T],H)\) for \(\beta=(1+\alpha)/2\), and the inequality
\[ \|v(t)\|_{C^\alpha\cap W_2^\beta}\le M_\alpha\|v_0\|_H, \tag{9} \]
holds, where \(M_\alpha\) depends neither on \(T\) nor on \(v_0\). Here \(\|\cdot\|_{C^\alpha\cap W_2^\beta}=\|\cdot\|_{C^\alpha}+\|\cdot\|_{W_2^\beta}\).
Proof. If \(v_0\in D(A^n)\), then the function \(v(t)\) is \(n\) times continuously differentiable. Since \(D(A^n)\) is dense in \(H\), for the proof of the theorem it suffices to establish estimate (9). To prove the estimate \(\|v(t)\|_{C^\alpha}\le M_\alpha\|v_0\|_H\) one must use the spectral decomposition
\[ A=\int_\delta^{+\infty}\lambda\,dE_\lambda,\qquad \delta>0 \]
and the representation, following from it, for the function \(\|v(t)-v(\tau)\|_H^2\)
\[ \|v(t)-v(\tau)\|_H^2= \int_\delta^{+\infty}[e^{-t\lambda}-e^{-\tau\lambda}]^2\lambda^{-2\alpha}\,d(E_\lambda v_0,v_0)_H . \]
Estimation of the integrand leads to the required inequality. Consider the integral
\[ I=\int_0^T\int_0^T\|v(t)-v(\tau)\|_H^2|t-\tau|^{-1-2\beta}\,dt\,d\tau= \]
\[ =\int_\delta^{+\infty} \left[ \int_0^T\int_0^T |e^{-t\lambda}-e^{-\tau\lambda}|^2\lambda^{-2\alpha} |t-\tau|^{-1-2\beta}\,dt\,d\tau \right]d(E_\lambda v_0,v_0). \]
Further,
\[ I=\int_\delta^{+\infty} \left[ \int_0^{T\lambda}\int_0^{T\lambda} |e^{-s}-e^{-z}|^2|s-z|^{-1-2\beta}\,ds\,dz \right]d(E_\lambda v_0,v_0) \le M_\alpha^2\|v_0\|_H^2, \]
since the inner integrals are bounded uniformly in \(\lambda\).
Corollary 1. The operators \(AQ\) and \(\dfrac{d}{dt}Q\) act boundedly from
\[ W_2^{1/2}\cap C^0 \quad \text{to} \quad W_2^{1/2}. \]
Theorem 5. The operator \(A^\beta Q\) acts boundedly from \(W_2^\alpha\) to \(C^{1/2+\alpha-\beta}\) for \(0\le \alpha<1/2\) and \(0\le \beta\le 1/2+\alpha\). The operator \(A^\beta Q\) acts boundedly from \(W_2^{1/2}\cap C^0\) to \(C^{1-\beta}\) for \(0\le \beta\le 1\).
Proof. If \(\alpha<\frac12\) and \(\beta<\frac12+\alpha\), then the result follows from the embedding of the space \(W_2^\alpha\) into the space \(L_{2/(1-2\alpha)}\) and from the estimate
\[
\|A^\alpha[\exp\{-(t+\Delta t)A\}-\exp\{-tA\}]\|_{H\to H}
\le C(\alpha,\beta)\Delta t^\beta t^{-\alpha-\beta}e^{-\delta t}
\]
\[
(0<t<t+\Delta t,\quad 0\le \beta\le 1,\quad \alpha>-\beta),
\tag{10}
\]
which follows from the integral representation standing under the sign of the \(\|\cdot\|_{H\to H}\) operator norm.
If, however, \(\beta=\frac12+\alpha\) and \(\alpha<\frac12\), then to study the operator \(A^\beta Q\) we apply the methods of Sec. 1. Let \(v_0\) be an arbitrary element of \(H\). Put
\[
\varphi(t)=(A^\beta Qf(t),v_0)_H
=\int_0^t (A^\beta\exp\{-(t-s)A\}f(s),v_0)_H\,ds
\]
\[
=\int_0^t (f(s),A\exp\{-(t-s)A\}A^{\beta-1}v_0)_H\,ds.
\]
Here the self-adjointness of the operator \(A\) has been used. Further,
\[ \varphi(t)=\int_0^t \left(f(s),\frac{d}{ds}\,[\exp\{-(t-s)A\}A^{\beta-1}v_0]\right)_H\,ds =\int_0^t (f(s),\psi'(s))_H\,ds. \]
Let \(\psi_1(s)=\psi(s)-\psi_2(s)\), where
\(\psi_2(s)=(s/t)[\psi(t)-\psi(0)]\). Then \(\psi_1(s)\) is already periodic on \([0,t]\). Let also \(f(s)\) be periodic on \([0,t]\). Then for the function
\[ \varphi_1(t)=\int_0^t (f(s),\psi_1'(s))_H\,ds \]
the estimate \(|\varphi_1(t)|\le\|f\|_{W_2^\alpha}\|\psi_1\|_{W_2^{1-\alpha}}\) is valid at the point \(t\). Here the spaces \(W_2\) are considered on the segment \([0,t]\). Further,
\(\|\psi\|_{W_2^{1-\alpha}}\le C_1\|v_0\|_H\). Here Theorem 4 is applied. With the aid of (10) we obtain
\(\|\psi_2\|_{W_2^{1-\alpha}}\le C_2\|v_0\|_H\). Therefore
\(\|\psi_1\|_{W_2^{1-\alpha}}\le C_3\|v_0\|_H\). Hence it follows that
\(|\varphi_1(t)|\le C_4\|f\|_{W_2^\alpha}\|v_0\|_H\). By virtue of (10),
\[ |\varphi(t)-\varphi_1(t)| \le C_5 t^{-1/2-\alpha}\int_0^t \|f(s)\|_H\,ds \le C_6\|f\|_{L_{2/(1-2\alpha)}} \le C_7(\alpha)\|f\|_{W_2^\alpha}. \]
In the last inequality the embedding of \(W_2^\alpha\) in \(L_{2/(1-2\alpha)}\) has been used. Thus, finally, we obtain that
\(|\varphi(t)|\le M(\alpha)\|f\|_{W_2^\alpha}\|v_0\|_H\). By means of passage to the limit we free ourselves (since \(\alpha<\frac12\)) from the assumption of periodicity of \(f(s)\) on \([0,t]\).
Let now \(\alpha=\frac12\) and \(\beta=1\). Then the estimate of the function \(AQf(t)\) is established as above in the case \(\beta=\frac12+\alpha\) with \(\alpha<\frac12\), but first one has to pass to a periodic function. From the estimate of the function \(AQf(t)\) follows the estimate of the function \(\frac{d}{dt}Qf(t)\), since the function \(Qf(t)\) satisfies equation (3). Hence, from the multiplicative inequality for Hölder norms, the second assertion of Theorem 5 follows.
Corollary 2. The operators \(AQ\) and \(\frac{d}{dt}Q\) act boundedly in the space \(W_2^{1/2}\cap C^0\).
Theorem 6. In order that problem (1) have such a solution \(v(t)\) that the functions \(v'(t)\) and \(Av(t)\) belong to the space \(W_2^{1/2}\cap C^0\), it is necessary and sufficient that \(v_0\) belong to \(D(A)\) and \(f(t)\) belong to \(W_2^{1/2}\cap C^0\).
Voronezh State University
Received
23 II 1968
REFERENCES
- P. E. Sobolevskii, DAN, 122, No. 6 (1958).
- S. G. Krein, Yu. I. Petunin, UMN, 21, issue 2 (128), 89 (1966).