UDC 517.948.32

MATHEMATICS

L. G. MIKHAILOV

ON SOME MULTIDIMENSIONAL INTEGRAL OPERATORS WITH HOMOGENEOUS KERNELS

(Presented by Academician A. N. Tikhonov, 26 XI 1966)

In the present paper we study the properties of linearity and complete continuity of certain classes of integral operators with kernels homogeneous of order \(-n\), where \(n\) is the dimension of the integral. In the one-dimensional case their linearity in \(L^p\) was established by Hardy \((^3)\), and in a number of spaces of types \(M\) and \(C\)—in \((^4)\); a number of theorems on the solvability of integral equations was also obtained \((^2)\). Multidimensional operators with kernels of the form \(T(x,y)=|x|^{-\alpha}|x-y|^{\alpha-n}\), \(0<\alpha<n\), were studied in \((^1)\). Such operators arise from differential equations with singular coefficients.

Let \(x\) and \(y\) be points of the \(n\)-dimensional Euclidean space \(E_n\). A measurable function \(\theta(x,y)\) is called homogeneous of order \(-n\) if, for all \(t>0\),

\[ \theta(tx,ty)=t^{-n}\theta(x,y). \tag{1} \]

We pose the problem of investigating the operators

\[ \theta\varphi=\int_D \theta(x,y)\varphi(y)\,dy,\qquad x\in D, \tag{2} \]

where \(D\) is an arbitrary bounded or unbounded domain in \(E_n\) containing the origin.

Adopting the usual notation for the Banach spaces \(L^p\), \(M\), \(C\), we shall also consider the subclass \(M^0\) \((M^\infty)\) of functions from \(M\) that are continuous at the point \(x=0\) \((x=\infty)\). If \(\Phi(x)\) belongs to one of the indicated classes, then by \(L_\beta^p\), \(M_\beta\), \(M_\beta^0\) \((M_\beta^\infty)\), and \(C_\beta\) we shall denote the corresponding isometric Banach spaces of functions (for more details see \((^1)\))

\[ \varphi(x)=|x|^{-\beta}\Phi(x),\qquad \|\varphi\|_\beta=\|\Phi\|. \tag{3} \]

Theorem 1. Let \(\theta(x,y)\) be homogeneous of order \(-n\), satisfy the summability condition*

\[ Q(\beta)=\int_{E_n}|\theta(j,u)|\,|u|^{-\beta}\,du<+\infty,\qquad j=(1,0,\ldots,0), \tag{4} \]

and be invariant under arbitrary rotations \(\gamma\) of the space \(E_n\) in the sense that

\[ \theta[\gamma(x),\gamma(y)]=\theta(x,y). \tag{5} \]

Then the operator \(\theta\varphi\) is linear in \(M_\beta(D)\), and if \(x=0\) \((x=\infty)\) is an interior point of \(D\), then also in \(M_\beta^0(D)\) \((M_\beta^\infty(D))\). If neither of the points \(x=0\), \(x=\infty\) is a boundary point for \(D\), then \(\theta\varphi\) is linear in \(C_\beta(D)\). If, finally, in addition to (4), the condition

\[ Q_1(\beta)=\int_{E_n}|\theta(v,j)|\,|u|^{\beta-n}\,du<+\infty, \tag{6} \]

is satisfied, then the operator is linear in \(L_{\beta-n/p}^{p}(D)\) for every \(p\ge 1\).

* Multidimensional singular integrals \((^5)\) do not satisfy this condition.

Proof.

1) Linearity in \(M_\beta\). Introducing the function \(\Phi(x)\) in accordance with (3) and putting \(\theta\varphi=|x|^{-\beta}\Omega(x)\), we have

\[ \Omega(x)=\int_D \left\{\frac{|x|}{|y|}\right\}^{\beta}\theta(x,y)\Phi(y)\,dy . \tag{7} \]

Let \(u=\gamma_x^{-1}(y)\) be such a rotation of the coordinate system \(oy_1y_2\ldots y_n\) after which the first axis \(oy_1\) lies on the straight line \(ox\), and let \(y=\gamma_x(u)\) be the inverse rotation; let \(u=\delta_x(v)\) be the similarity transformation \(u_i=|x|v_i,\ i=1,2,\ldots,n\), and \(y=l_x(v)\), \(l_x=\gamma_x\delta_x\), their composition. Changing variables \(y=l_x(v)\) and using (5) and (1), we have

\[ \Omega(x)=\int_{D_x}|v|^{-\beta}\theta(j,v)\Phi[l_x(v)]\,dv, \tag{8} \]

where \(D_x=l_x^{-1}(D)\) and \(\operatorname{mes}D_x=|x|^{-n}\operatorname{mes}D\). Hence it follows easily that \(\|\Omega\|_M\le Q(\beta)\|\Phi\|_M\), or \(\|\theta\|_{M_\beta}\le Q(\beta)\), and our first assertion is proved.

2) Linearity in \(M_\beta^0\). If \(x=0\) is an interior point of the domain \(D\), then as \(x\to0\), \(D_x\to E_n\).

Applying to (8) the procedure of subtracting \(\Phi(0)\), it is not difficult to show that \(\lim\Omega(x)=\Omega(0)\) exists and

\[ \Omega(0)=q(\beta)\Phi(0),\qquad q(\beta)=\int_{E_n}|v|^{-\beta}\theta(j,v)\,dv . \tag{9} \]

An analogous property belongs to the point \(x=\infty\), if it is interior.

3) Linearity in \(C_\beta\). Let \(x\to x_0\), \(x_0\in\overline D\), and \(x_0\ne0,\infty\). Then
\[ \Omega(x)-\Omega(x_0)=\Omega_1(x)+\Omega_2(x), \]
where

\[ \Omega_1(x)=\left(\int_{D_x}-\int_{D_{x_0}}\right)|v|^{-\beta}\theta(j,v)\Phi[l_x(v)]\,dv, \]

\[ \Omega_2(x)=\int_{D_{x_0}}|v|^{-\beta}\theta(j,v)\{\Phi[l_x(v)]-\Phi[l_x(v_0)]\}\,dv . \]

Since \(\lim_{x\to x_0}\Phi[l_x(v)]=\Phi[l_x(v_0)]\), we have \(\lim_{x\to x_0}\Omega_2(x)=0\). Passing to \(\Omega_1(x)\), we note that

\[ \int_{D_x}-\int_{D_{x_0}} = \int_{(D_x-D_{x_0})+(D_{x_0}-D_x)} . \]

Assuming additionally that the boundary of the domain \(D\) has zero measure, one can prove that as \(x\to x_0\), \(\operatorname{mes}(D_x-D_{x_0})\to0\) and \(\operatorname{mes}(D_{x_0}-D_x)\to0\). Therefore \(\lim_{x\to x_0}\Omega_1(x)=0\), and the continuity of \(\Omega(x)\) is proved.

4) Linearity in \(L_{\beta-n/p}^{p}\). Putting \(\varphi(x)=|x|^{n/p-\beta}\Phi(x)\), \(\theta\varphi=|x|^{n/p-\beta}\Omega(x)\), for \(\Omega(x)\) we obtain the form (7) with exponent \(\beta-n/p\) instead of \(\beta\). Applying Hölder’s inequality, raising to the power \(p\) and integrating, we obtain

\[ \|\Omega\|_{L^p}\le Q^{1/p'}(\beta)Q_1^{1/p}(\beta)\|\Phi\|_{L^p}, \quad\text{or}\quad \|\theta\|_{L_{\beta-n/p}^{p}}\le Q^{1/p'}(\beta)Q^{1/p}(\beta). \]

The theorem is proved.

Remark 1. If \(x=0\) is a boundary point of the domain \(D\), then the operator does not act in \(M_\beta^0(D)\) and \(C_\beta(D)\). An analogous role of a special point is played by \(x=\infty\), if the domain \(D\) is unbounded.

A large class of kernels satisfying condition (5) is constituted by

\[ \theta(x,y)=h\{|x|,|y|,|x-y|,|x+y|,(x,y/|y|)\}, \]

where \(h\) denotes an arbitrary function of 5 variables satisfying the summability condition (4). If, moreover, \(h\) does not change under interchange of the first two arguments, then (6) can be reduced to (4) by the transformation with inverse radius-vectors and, thus, linearity in \(L_{\beta-n/p}^{p}\) will be ensured by only one summability condition. Kernels of this type include, for example, the previously studied \(T(x,y)\) \({}^{1}\) (except for cases where special manifolds are present) and the kernel \(\theta_0(x,y)=\rho^{-n}\), where \(\rho^2=|x|^2+|y|^2\). Condition (4) for \(\theta_0\) will be fulfilled for all \(\beta\), \(0<\beta<n\).

Relying on this, we prove

Theorem 2. Let \(P(x,y)\) be homogeneous of degree \(n\), positive definite, i.e. \(P(x,y)>0\) for all \(x,y\) for which \(\rho^2=|x|^2+|y|^2>0\), and continuous for the same \(x,y\). Then the operator

\[ P\varphi=\int_D \frac{\varphi(y)\,dy}{P(x,y)} \]

is linear in \(M_\beta(D)\) and \(L_{\beta-n/p}^{p}(D)\) for all \(\beta\), \(0<\beta<n\), and \(p\geqslant 1\).

Theorem 3. Let the singular point \(x=0\) be interior. If, in addition to the assumptions of Theorem 1, it is assumed that \(q(\beta)\ne0\), where \(q(\beta)\) is given by formula (9), then the operator \(\theta\varphi\) will not be completely continuous in \(M_\beta\), \(M_\beta^0\), and \(C_\beta\). It will not be completely continuous in \(L_{\beta-n/p}^{p}\), \(p\geqslant 1\), if it is assumed that \(q_1(\beta)\ne0\), where

\[ q_1(\beta)=\int_{|u|\leqslant 1} |u|^{-\beta}\theta(j,u)\,du . \]

The proof of Theorem 3 in \(M_\beta\), \(M_\beta^0\), and \(C_\beta\) is based on property (9) and is analogous to the proofs of the corresponding theorems in \({}^{1}\), while for \(L_{\beta-n/p}^{p}\) it is analogous to \({}^{6}\). It is possible that the theorem is true without any assumptions of the type \(q(\beta)\ne0\), as was the case in the one-dimensional case \({}^{4}\) (see also \({}^{7}\)).

Umarov Physico-Technical Institute
Academy of Sciences of the Tajik SSR

Received
16 XI 1966

REFERENCES

\({}^{1}\) L. G. Mikhailov, A new class of singular integral equations and its applications to differential equations with singular coefficients, Dushanbe, 1963.
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\({}^{3}\) G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Moscow, 1959.
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