Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1969. Vol. 186, No. 6
MATHEMATICS
I. A. KIPRIYANOV, M. I. KLYUCHANTSEV
ON THE BOUNDEDNESS OF ONE CLASS OF SINGULAR INTEGRAL OPERATORS
(Presented by Academician S. L. Sobolev, October 7, 1968)
Singular integral operators, considered by S. G. Mikhlin \((^{1,2})\) and A. P. Calderón and A. Zygmund \((^3)\), play an increasingly important role in the theory of differential equations with partial derivatives. With their help, general existence and uniqueness theorems have been proved. In the present note we indicate a class of singular integrals whose kernels have a singularity lying on a hypersurface and which are closely connected with the generalized shift operator \((^4)\).
- Let \(E_{n+2}^{+}\) be the Euclidean half-space of points \((s,z_1,z_2)\) of dimension \(n+2\) \((s=(s_1,\ldots,s_n), z_2 \ge 0)\).
Consider a singular integral of the form
\[ v(x,y)=C_k \int_{E_{n+2}^{+}} \frac{f(\tilde{\theta})\varphi\left(s,\sqrt{z_1^2+z_2^2}\right) z_2^{k-1}\,ds\,dz_1\,dz_2} {\left[\sum (x_i-s_i)^2+(y-z_1)^2+z_2^2\right]^{(n+k+1)/2}}, \tag{1} \]
where \(\tilde{\theta}=(\tilde{\theta}_1,\ldots,\tilde{\theta}_{n+1})\),
\(\tilde{\theta}_i=(x_i-s_i)/r\), \(\tilde{\theta}_{n+1}=\sqrt{(y-z_1)^2+z_2^2}/r\),
\(r^2=\sum (x_i-s_i)^2+(y-z_1)^2+z_2^2\),
\(C_k=\Gamma(k+1/2)/\Gamma(1/2)\Gamma(k)\). The number \(k\) is positive and the coordinate \(y\) is nonnegative.
By the definition of the singular integral we set
\[ v(x,y)=\lim_{\varepsilon\to 0} C_k \int_{r>\varepsilon} \frac{f(\tilde{\theta})\varphi\left(s,\sqrt{z_1^2+z_2^2}\right)} {r^{\,n+k+1}} z_2^{k-1}\,ds\,dz_1\,dz_2 . \tag{2} \]
We shall assume that the density \(\varphi\) and the characteristic \(f\) satisfy, as usual, the following conditions: 1) in any bounded part of the half-space \(E_{n+2}^{+}\) the function \(\varphi\) satisfies the Hölder condition with exponent \(\alpha>0\); 2) at infinity \(\varphi(x,y)=O(r^{-\beta})\), where \(r^2=|x|^2+y^2\), \(\beta>0\); 3) the characteristic \(f(\theta)\) is bounded and continuous. Under these conditions, for the singular integral (1) to exist it is necessary and sufficient that the condition
\[ \int_{\Omega_{n+2}^{+}} f(\tilde{\theta}) z_2^{k-1}\,d\omega_{n+2}=0, \tag{3} \]
be satisfied, where \(\Omega_{n+2}^{+}\) is the unit hemisphere in \(E_{n+2}^{+}\) with center at the point \((x,y,0)\).
The singular integral (1) can also be written in another form. Put \(z\cos\gamma=z_1\), \(z\sin\gamma=z_2\); then
\[ v(x,y)=C_k \int_{E_{n+1}^{+}} \int_{0}^{\pi} \frac{f(\bar{\theta})}{\tilde{r}^{\,n+k+1}} \varphi(s,z) z^k \sin^{k-1}\gamma \, ds\,dz\,d\gamma, \tag{4} \]
where \(\bar{\theta}_i=(x_i-s_i)/\tilde{r}\),
\(\bar{\theta}_{n+1}=\sqrt{y^2+z^2-2yz\cos\gamma}/\tilde{r}\),
\(\tilde{r}=\bigl(|x-s|^2+y^2+z^2-2yz\cos\gamma\bigr)^{1/2}\), and \(E_{n+1}^{+}\) is the \((n+1)\)-dimensional half-space
\((s, z \geqslant 0)\). Taking into account the explicit expression of the generalized shift operator (4),
\[ T_{x,y}^{s,z} f(x,y)= \frac{\Gamma(k+1/2)}{\Gamma(1/2)\Gamma(k)} \int_0^\pi f\left(x-s,\sqrt{y^2+z^2-2yz\cos\gamma}\right)\sin^{k-1}\gamma\,d\gamma, \tag{5} \]
we now write the integral (4) in the form
\[ v(x,y)= \int_{E_{n+1}^{+}} T_{x,y}^{s,z}\frac{f(\theta)}{r^{n+k+1}}\varphi(s,z)z^k\,ds\,dz, \tag{6} \]
where \(\theta_i=x_i/r,\ \theta_{n+1}=y/r,\ r^2=|x|^2+y^2\).
Since \(\varphi\) is summable over any finite domain, and the operator \(T_{x,y}^{s,z}\) is formally self-adjoint (4), the integral (6) can also be written in the form
\[ v(x,y)= \int_{E_{n+1}^{+}} \frac{f(\theta)}{r^{n+k+1}}T_{x,y}^{s,z}\varphi(x,y)z^k\,ds\,dz, \tag{7} \]
where \(\theta_i=s_i/r,\ \theta_{n+1}=z/r,\ r^2=|s|^2+z^2\).
Let us note that if in formula (2) one performs the ordinary shift in the variables \(s\) and \(z_1\), then we obtain
\[ v(x,y)=C_k\lim_{\varepsilon\to0}\int_{r>\varepsilon} \frac{f(\theta)}{r^{n+k+1}} \varphi\left(x-s,\sqrt{(y-z_1)^2+z_2^2}\right)z_2^{k-1}\,ds\,dz_1\,dz_2, \]
where \(\theta_i=s_i/r,\ \theta_{n+1}=\sqrt{z_1^2+z_2^2}/r,\quad r^2=|s|^2+z_1^2+z_2^2\). The substitution \(z_1=z\cos\gamma,\ z_2=z\sin\gamma\) transforms this integral, taking (5) into account, to the form
\[ v(x,y)=\lim_{\varepsilon\to0}\int_{r>\varepsilon} \frac{f(\theta)}{r^{n+k+1}}T_{x,y}^{s,z}\varphi(x,y)z^k\,ds\,dz, \tag{8} \]
where \(\theta_i,\ \theta_{n+1}\), and \(r\) have the same meaning as in (7).
- For the singular integral considered, theorems of the type of the theorems of I. I. Privalov hold (see, for example, \((^2,^5)\)).
For \(0<\alpha<1\) introduce the seminorms
\[ [f]_\alpha=\sup \frac{|f(x,y)-f(s,z)|}{[|x-s|^2+(y-z)^2]^{\alpha/2}}, \qquad [u]_\alpha=\sup\frac{|u(P)-u(Q)|}{|P-Q|^\alpha}, \]
where \(P=P(x,y,t),\ Q=Q(c,z,t)\). Denote by \(\mathscr L_{p,k}\) the set of functions summable in the half-space \(E_{n+1}^{+}\) to degree \(p\geqslant1\) with weight \(y^k\) \((y\geqslant0,\ k>0)\).
Theorem 1. Let
\[ K(x,y,t)=\frac{\Omega(x/r,y/r)}{(r^2+t^2)^{(n+k+1)/2}}, \qquad r^2=|x|^2+y^2, \tag{9} \]
where \(\Omega(x,y)\) satisfies on \(|x|^2+y^2=1\) the Hölder condition with exponent \(\alpha'\) and constant \(\chi\) such that \(|\Omega(x,y)|\leqslant\chi\), and, moreover,
\[ \int_{r=1}\Omega(x,y)y^k\,dx\,dy=0. \tag{10} \]
Let \(f(x,y)\in\mathscr L_{p,k}\) and \([f]_\alpha<\infty\).
Then for the integral operator
\[ u(x,y,t)= \int_{E_{n+1}^{+}} T_{x,y}^{s,z}K(x,y,t)f(s,z)z^k\,ds\,dz \tag{11} \]
the estimate
\[ [u]_\alpha\leqslant C\chi[f]_\alpha \]
is valid, where the constant \(C\) depends only on \(\alpha,\alpha'\), and \(n+k+1\).
The following theorem is also true, following directly from Theorem 1:
Theorem 2. Let the function \(f(x,y)\) belong to \(\mathscr L_{p,k}\) and satisfy the Hölder condition
\[ |f(x,y)-f(s,z)| \le M\bigl(|x-s|^2+(y-z)^2\bigr)^{\alpha/2}. \]
Let
\[ u(x,y)=\lim_{\varepsilon\to0}\int_{r>\varepsilon} \frac{\Omega(s/r,z/r)}{r^{\,n+k+1}}\, T_{x,y}^{s,z} f(x,y)\,z^k\,ds\,dz, \]
where \(\Omega(x,y)\) satisfies, on \(|x|^2+y^2=1\), a Hölder condition with exponent \(\alpha'\) and with constant \(\chi\) such that \(|\Omega|\le \chi\), and condition (10). Then
\[ |u(x,y)-u(s,z)| \le C\chi M\bigl(|x-s|^2+(y-z)^2\bigr)^{\alpha/2}, \]
where \(C\) depends only on \(\alpha,\alpha'\), and \(n+k+1\).
- In conclusion we note that theorems of the type of the well-known theorems of A. P. Calderon and A. Zygmund (see, for example, \((3,5)\)) on boundedness also hold for our singular integral operator in the space \(\mathscr L_{p,k}\).
Theorem 3. Let an operator (11) be given, where the kernel \(K(x,y,t)\) is defined by formula (9) and the angular function \(\Omega(x,y)\) satisfies the Hölder condition and condition (10). Then, if \(f(x,y)\in \mathscr L_{p,k}\), \(p>1\), then for any \(t>0\), \(u(x,y,t)\), as a function of \((x,y)\), belongs to \(\mathscr L_{p,k}\), and the inequality
\[ \|u(x,y,t)\|_{\mathscr L_{p,k}} \le C\|f\|_{\mathscr L_{p,k}}, \]
holds, where the constant \(C\) does not depend on \(f\) or \(t\).
Theorem 4. Let the kernel \(K(x,y)\) be given by the formula
\[ K(x,y)=\frac{\Omega(x/r,y/r)}{r^{\,n+k+1}},\qquad r^2=|x|^2+y^2, \]
where the function \(\Omega(x,y)\) satisfies condition (10) and the condition
\[ \left\{\int_{r=1}|\Omega(x,y)|^q y^k\,dx\,dy\right\}^{1/q}=R_q<\infty \qquad (q>1). \]
Then for any function \(f(x,y)\in\mathscr L_{p,k}\) there exists, in the sense of convergence in \(\mathscr L_{p,k}\), the limit
\[ u(x,y)=\lim_{\varepsilon\to0}\int_{r>\varepsilon} K(s,z)\,T_{x,y}^{s,z}f(x,y)\,z^k\,ds\,dz, \tag{12} \]
belonging to \(\mathscr L_{p,k}\) and satisfying the inequality
\[ \|u\|_{\mathscr L_{p,k}}\le CR_q\|f\|_{\mathscr L_{p,k}}, \]
where \(C\) does not depend on \(f\).
We note that in the course of the proof of Theorems 3 and 4 it becomes clear that, in the sense of convergence in \(\mathscr L_{p,k}\), there exists the limit \(u(x,y,t)\) as \(t\to0\), equal to \(u(x,y)\), belonging to \(\mathscr L_{p,k}\) and being a certain regularization of the divergent integral
\[ \int_{E_{n+1}^{+}} T_{x,y}^{s,z}K(x,y)f(s,z)z^k\,ds\,dz. \tag{13} \]
Formula (12), however, gives another regularization of the divergent integral (13). It is shown that this regularization leads to the very same function \(u(x,y)\).
Voronezh State University
Received
2 X 1968
CITED LITERATURE
- S. G. Mikhlin, UMN, 3, 25 (1948).
- S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Moscow, 1962.
- A. P. Calderon, H. Zygmund, Am. J. Math., 78, No. 2, 310 (1956).
- B. M. Levitan, UMN, 6, No. 2 (42) (1951).
- S. Agmon, A. Douglis, L. Nirenberg, Estimates of Solutions of Elliptic Equations near the Boundary, IL, 1962.