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MATHEMATICS
V. R. PORTNOV
A THEOREM ON THE DENSITY OF FINITE FUNCTIONS IN THE SPACE \(L_{p,a}^{(m)}(E_n)\)
(Presented by Academician S. L. Sobolev on 10 IX 1963)
Let \(E_n\) be the \(n\)-dimensional Euclidean space of points \(x=(x_1,\ldots,x_n)\). Consider the set of functions \(f(x)\) for which the relation
\[ \|f\|_{L_{p,a}^{(m)}(E_n)}^p = \int_{E_n} a(x)\left[\sum_{|\alpha|=m}|D^\alpha f|^2\right]^{p/2}\,dx<\infty, \tag{1} \]
holds, where \(a(x)\) is a measurable, almost everywhere finite and almost everywhere positive function, called the weight; \(D^\alpha f=\partial^m f/\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}\) is the generalized partial derivative of order \(m>0\).
It is clear that (1) is equivalent to the following condition:
\[ \int_\sigma d\sigma\left[ \int_0^\infty b(x)\left[\sum_{|\alpha|=m}|D^\alpha f|^2\right]^{p/2}\,dr \right]<\infty, \tag{2} \]
where \(\sigma\) is the unit sphere of the space \(E_n\),
\[ b(x)=r^{n-1}a(x)=\left[\sum_{i=1}^n x_i^2\right]^{(n-1)/2}a(x). \tag{3} \]
Introduce the notation:
\[ B(x)=b^{1/(1-p)}(x)=a^{1/(1-p)}(x)\,r^{(n-1)/(1-p)}. \tag{4} \]
For any \(m>0\) we shall subject the weight \(a(x)\) to the following two conditions:
-
\[ \operatorname*{vrai\,inf}_{|x|\le R} a(x)=\varepsilon(R)>0 \quad \text{for all } \quad 0<R<\infty, \tag{5} \]
\[ \operatorname*{vrai\,sup}_{|x|\le R} a(x)=\varepsilon_1(R)>0 \quad \text{for all } \quad 0<R<\infty. \] -
There exists an \(R_0>0\) such that, for \(r\ge R_0\), \(a(x)\) depends only on \(r\), i.e.
\[ a(x)=a(r). \tag{6} \]
In addition to conditions (5) and (6), we shall assume that: for \(m=1\), \(a(r)\) is infinitely differentiable if \(r\ge R_0\); for \(m>1\), \(a(r)\) satisfies the conditions
\[ \int_{R_0}^r B(t)\,dt<\infty \quad (R_0<r<\infty); \tag{7} \]
\[ \lim_{r\to\infty} \frac{B(r)}{\displaystyle \int_{R_0}^r B(t)\,dt} = \delta_1>0, \quad \text{if } \int_{R_0}^{\infty} B(t)\,dt=\infty; \tag{8} \]
\[ \lim_{r\to\infty} \frac{B(r)}{\displaystyle \int_r^\infty B(t)\,dt} = \delta_2>0, \quad \text{if } \int_{R_0}^{\infty} B(t)\,dt<\infty. \tag{9} \]
The set \(L_{p,a}^{(m)}(E_n)\) is a normed space of classes of functions. The zero class consists of polynomials of degree \(m-1\). Classes
are composed in the usual way. We shall call a class finitely infinitely differentiable if it contains a function possessing the indicated property.
Our task, taking into account the restrictions imposed on the weight above, is to prove the following theorem.
Theorem. In the space \(L_{p,a}^{(m)}(E_n)\), finitely infinitely differentiable classes form a dense set.
We first prove one lemma.
Lemma. Let \(X\) and \(Y\) be spaces with \(\sigma\)-finite measure, and let \(b(x)\), \(m(x,y)\), \(K(x,y)\) be measurable functions such that the following relations hold almost everywhere:
\[ 0<b(x)<\infty; \tag{10} \]
\[ 0<m(x,y)<\infty; \tag{11} \]
\[ 0\leq K(x,y)<\infty; \tag{12} \]
\[ 0<\int_Y K(x,y)m(x,y)\,dy<\infty; \tag{13} \]
\[ 0<\int_X b^{1/(1-p)}K^{p'/q'}m^{-p'/q} \left[\int_Y Km\,dy\right]^{p'/q}dx<\infty. \tag{14} \]
Then, for any nonnegative function \(f(x)\) measurable on \(X\), the inequality
\[ \left\{\int_Y\left[\int_X b^{1/(1-p)}(x)K^{p'/q'}(x,y)m^{-p'/q}(x,y) \left[\int_Y K(x,y)m(x,y)\,dy\right]^{p'/q}dx\right]^{-q/p'}\right. \]
\[ \left.\times \left[\int_X K(x,y)f(x)\,dx\right]^qdy\right\}^{q/p} \leq \left[\int_X bf^pdx\right]^{q/p}, \tag{15} \]
where
\[ 1<p\leq q<\infty. \tag{16} \]
If we now prove the density of finite classes in the space \(L_{p,a}^{(m)}(E_n)\), the assertion of the theorem will follow from this, since every function \(f(x)\in L_{p,a}(E_n)\) finite on \((E_n)\) can be obtained as the limit, in the metric of \(L_{p,a}(E_n)\), of a sequence of infinitely differentiable finite functions, analogously to how this is done in \((^1)\), p. 20.
We now pass directly to the proof of the theorem.
First consider the case \(m=1\). Obviously, only two possibilities can arise:
\[ \text{1.}\qquad \int_{R_0}^{\infty} b^{1/(1-p)}(r)\,dr = \int_{R_0}^{\infty} B(r)\,dr <\infty. \tag{17} \]
\[ \text{2.}\qquad \int_{R_0}^{\infty} B(r)\,dr=\infty. \tag{18} \]
If (17) holds, then it can be shown that \(f(x)\) is representable in the form
\[ f(r,\gamma)=C-\int_r^{\infty}\frac{\partial f(r,\gamma)}{\partial r}\,dr, \tag{19} \]
where \(r=|x|\), \(\gamma=x/|x|\), and \(C\) does not depend on \(\gamma\). Setting \(C=0\), we obtain the function
\[ f(r,\gamma)=-\int_r^{\infty}\frac{\partial f(r,\gamma)}{\partial r}\,dr, \tag{20} \]
equivalent to the original one.
For the proof of the theorem it is sufficient to approximate the function (20) by finite functions in the metric of the space \(L_{p,a}^{(1)}(E_n)\).
We introduce the cutoff function
\[ \psi_\eta(r)=\psi(\eta\cdot\mu(r)), \tag{21} \]
where \(\psi(\alpha)\) is an arbitrarily smooth function satisfying the condition
\[ \psi(\alpha)= \begin{cases} 1, & \text{if } \alpha<1/2,\\ 0, & \text{if } \alpha>1; \end{cases} \tag{22} \]
\[ \mu(r)=\left[\int_r^\infty B(t)\,dt\right]^{-1} \qquad (r>R_0). \tag{23} \]
We shall show that for the function (20) the equality
\[ \lim_{\eta\to0}\|f-f\cdot\psi_\eta(r)\|_{L_{p,a}^{(1)}(E_n)}=0 \tag{24} \]
holds. We have:
\[ \|f-f\cdot\psi_\eta(r)\|_{L_{p,a}^{(1)}(E_n)} \leq K\sum_{|\alpha|=1} \|D^\alpha f-\psi_\eta(r)\cdot D^\alpha f-fD^\alpha\psi_\eta(r)\|_{L_{p,a}(E_n)} \leq \]
\[ \leq K\left[ \sum_{|\alpha|=1}\|(1-\psi_\eta(r))D^\alpha f\|_{L_{p,a}(E_n)} + \sum_{|\alpha|=1}\|fD^\alpha\psi_\eta(r)\|_{L_{p,a}(E_n)} \right]. \tag{25} \]
Denote by \(\mu^{-1}(r)\) the function inverse to \(\mu(r)\) \((r>R_0)\). We shall have:
\[ \lim_{\eta\to0}\|(1-\psi_\eta(r))D^\alpha f\|_{L_{p,a}(E_n)} = \lim_{\eta\to0} \int_{|x|>\mu^{-1}(1/2\eta)} a(x)|D^\alpha f|^p\,dx =0. \tag{26} \]
Introduce the function
\[ \gamma(r)=\left[\frac{\mu'(r)}{\mu(r)}\right]^p . \tag{27} \]
Then
\[ \|fD^\alpha\psi_\eta(r)\|_{L_{p,a}(E_n)} = \int_{|x|>R_0} a(r)|f(x)|^p|D^\alpha\psi_\eta(r)|^p\,dx \leq \]
\[ \leq K_0\int_\sigma d\sigma \left[ \int_{\mu^{-1}(1/\eta)}^\infty b(r)\gamma(r) \left(\int_r^\infty \left|\frac{\partial f}{\partial r}\right|\,dr\right)^p dr \right]. \tag{28} \]
The right-hand side of inequality (28) tends to \(0\) as \(\eta\to0\). Indeed, putting in (15)
\(X=Y=(R_0,\infty)\), \(p=q\),
\[ K(x,y)= \begin{cases} 1, & \text{if } x\geq y,\\ 0, & \text{if } x<y; \end{cases} \]
\[ m(x,y)= \frac{d}{dy} \left[ \left(\int_y^\infty B(t)\,dt\right)^{-1/p} - \left(\int_{R_0}^\infty B(t)\,dt\right)^{-1/p} \right]^{p-1}, \tag{29} \]
we obtain
\[ \int_{R_0}^\infty b(r)\gamma(r) \left[\int_r^\infty\left|\frac{\partial f}{\partial r}\right|\,dr\right]^p dr \leq N\int_{R_0}^\infty b(r)\left|\frac{\partial f}{\partial r}\right|^p\,dr. \tag{30} \]
Integrating both sides of inequality (30) over the unit sphere and taking into account that \(f\in L_{p,a}^{(1)}(E_n)\), we obtain
\[ \int_\sigma d\sigma \left[ \int_{R_0}^\infty b(r)\gamma(r) \left|\int_r^\infty \frac{\partial f}{\partial r}\,dr\right|^p dr \right] <\infty. \tag{31} \]
From inequality (31) it follows that the right-hand side of inequality (28) tends to \(0\).
Thus, the assertion of the theorem for the case \(m=1\) and
\[ \int_{R_0}^{\infty} B(t)\,dt < \infty \]
has been proved. The remaining cases are proved on the basis of inequality (15) in an analogous way.
In the case \(m=1\) and when relation (18) is satisfied, we set
\[ \psi_\eta(r)=\psi\left[\eta\cdot\int_{R_0}^{r} B(t)\,dt\right]. \tag{32} \]
In the case \(m>1\), we set
\[ \psi_\eta(r)=\psi(\eta\cdot e^r). \tag{33} \]
The density theorem for finite functions in the space \(L_{p,a}^{(m)}(E_n)\) for \(a(x)\equiv 1\) was proved by S. L. Sobolev.
The author expresses gratitude to S. L. Sobolev for posing the problem and for his attention to the work.
Institute of Mathematics with Computing Center
Siberian Branch of the Academy of Sciences of the USSR
Received
16 VIII 1963
REFERENCES CITED
- S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Publ. Siberian Branch, Academy of Sciences of the USSR, 1962.
- S. L. Sobolev, Siberian Mathematical Journal, 4, 3 (1963).