Full Text
UDC 513.88:513.83
MATHEMATICS
V. A. DIKAREV
IMBEDDING THEOREMS FOR A CLASS OF FUNCTION SPACES
(Presented by Academician S. L. Sobolev on 19 X 1965)
If a function \(F\) belongs to the space \(W_p^l\), then, according to S. L. Sobolev’s imbedding theorem \({}^{(1)}\), its derivatives of order \(l'\) \((l'<l)\) are summable to the power \(p'\), where \(l' - n/p' = l - n/p\) (\(n\) is the dimension of the space). This theorem cannot be improved while remaining within the framework of the spaces \(L_p\); however, it turns out that one can obtain more information about derivatives of order \(l'\) if the space \(L_{p'}\) is replaced by another space with a norm invariant under rearrangements of functions. Below is the minimal space with this property.
- Let \(E^n\) be \(n\)-dimensional real space. For a measurable (complex) function \(f\) on \(E_n\), introduce \(f^*(t)\) \((0<t<\infty)\), the decreasing, right-continuous equimeasurable rearrangement of \(|f|\):
\[ \operatorname{mes}\{t:f^*(t)>N\}=\operatorname{mes}\{x:|f|>N\} \quad \text{for every } N . \]
We also introduce the function \(f^{**}(t)\):
\[ f^{**}(t)=\frac{1}{t}\int_0^t f^*(s)\,ds, \]
the functional \(\|\cdot\|_{\lambda,p}\)
\[ \|f\|_{\lambda,p}= \left(\int_0^\infty t^{p\lambda-1} f^{*p}(t)\,dt\right)^{1/p}, \qquad 0<\lambda<1;\quad 1\le p<\infty, \]
\[ \|f\|_{\lambda,\infty}=\sup_{0<t<\infty} t^\lambda f^*(t), \qquad 0\le \lambda<1, \tag{1} \]
and the functional \(\langle f\rangle_{\lambda,p}\), which is defined as \(\|f\|_{\lambda,p}\), but with \(f^{**}(t)\) substituted for \(f^*(t)\) on the right-hand side of (1).
The inequality holds
\[ C_1(\lambda,p)\|f\|_{\lambda,p}\le \langle f\rangle_{\lambda,p}\le C_2(\lambda,p)\|f\|_{\lambda,p}, \qquad 0<C_1<C_2<\infty . \]
Following \({}^{(2-4)}\), by \(L(\lambda,p,E_n)\) we shall denote the set of functions \(r\) for which \(\|f\|_{\lambda,p}<\infty\); \(L(\lambda,p,E_n)\) is a Banach space with norm \(\langle\cdot\rangle_{\lambda,p}\).
Next, introduce the space \(L(1,1,E_n)\), coinciding with \(L_1(E_n)\), by putting \(\langle f\rangle_{1,1}=\|f\|_{1,1}=\|f\|_{L_1}\). It is obvious that for \(\lambda=1/p\) the space \(L(\lambda,p,E_n)\) coincides with \(L_p(E_n)\).
We shall also consider the space*
\[ L(\lambda_1,\lambda_2,p,E_n) = L(\lambda_1,p,E_n)\cap L(\lambda_2,p,E_n), \qquad \lambda_1\le \lambda_2 . \]
\(L(\lambda_1,\lambda_2,p,E_n)\) is a Banach space with norm
\[ \|f\|_{\lambda_1,\lambda_2,p} = \max\{\|f\|_{\lambda_1,p};\|f\|_{\lambda_2,p}\}. \]
* The necessity of considering the spaces \(L(\lambda_1,\lambda_2,p,E_n)\) is due to the fact that below imbedding theorems are studied on the whole space. For a bounded domain the spaces \(L(\lambda,p,E_n)\) suffice.
Theorem 1. Let one of the two conditions be satisfied:
a) \(\lambda_1' > \lambda_1,\ \lambda_2' < \lambda_2;\)
b) \(\lambda_1' \ge \lambda_1,\ \lambda_2' \le \lambda_2,\ p' \ge p.\)
Then \(L(\lambda_1,\lambda_2,p,E_n) \subseteq L(\lambda_1,\lambda_2,p',E_n)\).
There are no other embedding relations for the spaces \(L(\lambda_1,\lambda_2,p,E_n)\).
- Lemma 1 is a strengthening of an inequality of V. P. Il’in \((^{6,7})\) (in the form given to it by P. I. Lizorkin).
Lemma 1. Let
\[ Jf=\int_{E_n}\frac{f(t)e^{-a\rho}}{\rho^\alpha}\,dt, \qquad f\in L(\lambda_1,\lambda_2,p,E_n),\qquad a>0. \]
\(\rho\) is the distance between the points \(x\) and \(t\).
If the conditions \(0<\alpha<n,\quad 1-\alpha/n<\lambda_1\le\lambda_2<\min\{1,\frac{n-\alpha}{n-m}\}\) are satisfied, then
\(Jf\in L(\mu_1,\lambda_2,p,E_m)\), where \(E_m\) is a subspace of \(E_n\), and
\[ \mu_1=(n/m)(\lambda_1-1+\alpha/n). \]
Put \(\lambda_1=\lambda_2=1/p\). Then \(f\in L_p,\quad Jf\in L(\mu_1,\lambda_2,p,E_m)=L(\mu_1,p,E_m)\cap L(\lambda_2,p,E_m)\). From the conditions of the lemma it follows that \(\mu_1<\lambda_1,\ p<1/\mu_1\). Applying Theorem 1 to \(L(\mu_1,p,E_m)\) with \(\lambda_1'=\mu_1,\lambda_2'=1/\mu_1\), we have
\[ L(\mu_1,p,E_m)\cap L(\lambda_2,p,E_m)\subset L_{1/\mu_1}\cap L_p\subseteq L_{p'}, \]
where \(p\le p'\le 1/\mu_1\). Thus, in this particular case one obtains a strengthening of the Il’in–Lizorkin inequality.
- By \(L^r(\lambda_1,\lambda_2,p,E_m)\) we shall denote the set of functions \(F\) representable in the form
\[ F(x)=\mathcal F^{-1}\bigl[(\mathcal F f)(1+|\lambda|^2)^{-r/2}\bigr], \]
where
\[ \mathcal F f=\int_{E_n} f(t)e^{i(\lambda,t)}\,dt \]
is the Fourier transform operator, \(f\in L(\lambda_1,\lambda_2,p,E_n)\), \(r\ge0\). By \(\|F\|_{\lambda_1,\lambda_2,p}^{(r)}\) we shall denote the norm in \(L^r(\lambda_1,\lambda_2,p,E_n)\) of the function \(F\). Put
\[ \|F\|_{\lambda_1,\lambda_2,p}^{(r)}=\|f\|_{\lambda_1,\lambda_2,p}. \]
As is known \((^{5,6})\), the function \(F\) is representable in the form of a “Bessel potential”
\[ F(x)=\int_{E_n}G_r(x-y)f(y)\,dy, \]
where \(f(y)\in L(\lambda_1,\lambda_2,p,E_n)\),
\[ G_r(x)=\frac{|x|^{(r-n)/2}}{2^{(n+r-2)/2}\pi^{n/2}\Gamma(r/2)}\cdot K_{(n-r)/2}(|x|), \]
and \(K_\alpha(t)\) is the Macdonald function.
We note that for \(\lambda_1=\lambda_2=1/p\) the spaces \(L^r(\lambda_1,\lambda_2,p,E_n)\) coincide with the spaces \(L_p^r\) \((^{5,6})\), while for \(\lambda_1=\lambda_2=1/p\) and integer \(r=l\) they coincide with the spaces \(W_p^l\) of S. L. Sobolev.
We shall consider \(F\in L^r(\lambda_1,\lambda_2,p,E_n)\) on the subspace \(E_m\), \(m\le n\). The question is whether the function that arises in this way can be characterized in terms of the spaces \(L^{r'}(\mu_1,\lambda_2,p,E_m)\). The answer is given by the following
Theorem 2. Let \(F(x)\in L^r(\lambda_1,\lambda_2,p,E_n)\), \(0<\lambda_i<1,\ r'\ge0,\ r'-m\lambda_i<r-n\lambda_i<r'\) \((i=1,2)\). Then the embedding
\[ L^r(\lambda_1,\lambda_2,p,E_n)\subset L^{r'}(\mu_1,\lambda_2,p,E_m) \]
holds, where
\[ r'-m\mu_1=r-n\lambda_1. \]
This theorem is proved with the aid of Lemma 1 and the theorem on multipliers of S. G. Mikhlin \((^8)\).
Put \(\lambda_1=\lambda_2=1/p\). Then \(F(x)\in L_p^r(E_n)\) and
\[ L_p^r(E_n)\subset L^{r'}(\mu_1,\lambda_2,p,E_m) = L^{r'}(\mu_1,p,E_m)\cap L^{r'}(\lambda_2,p,E_m), \]
where
\[ \mu_1=\frac{n}{mp}-\frac{r-r'}{m}. \]
Taking into account that \(p<1/\mu_1\), and applying Theorem 1 to \(L^{r'}(\mu_1,p,E_m)\), we have
\[ L^{r'}(\mu_1,p,E_m)\cap L^{r'}(\lambda_2,p,E_m) \subset L^{r'}_{1/\mu_1}\cap L_p^{r'} \subseteq L_{p'}^{r'}, \]
where \(p \leqslant p' \leqslant 1/\mu_1\). Thus, in this case we obtain a strengthening of P. I. Lizorkin’s theorem\(^6\), which for integral \(r\) and \(r'\) coincides with S. L. Sobolev’s imbedding theorem.
Let again \(E_m\) be a subspace in \(E_n\) \((m \leqslant n)\), and let \(F(x)\) range over the space \(L^r(\lambda_1,\lambda_2,p,E_n)\). If \(r'\) satisfies the conditions of Theorem 2, then the representation
\[
F(x)=\int_{E_m'} G_{r'}(x-y)\,\varphi(y)\,dy,\qquad x\in E_m .
\tag{2}
\]
is possible.
According to Theorem 2, \(\varphi\in L(\mu_1,\lambda_2,p,E_m)\).
Theorem 3. The set of functions \(\{f(x)\}\), defined and measurable on \(E_m\) and such that \(|f(x)|\) is majorized by a rearrangement of the modulus of some function \(\varphi(x)\) occurring in the representation (2), coincides with \(L(\mu_1,\lambda_2,p,E_m)\), where \(\mu_1\) is defined as in Theorem 2.
Theorem 3 shows that Theorem 2 cannot, in a certain sense, be strengthened.
In conclusion I express my gratitude to V. I. Matsaev for posing the problem and for discussing the results.
Kharkov Institute of Mining Machine Building,
Automation and Computer Technology
Received
5 X 1965
CITED LITERATURE
- S. L. Sobolev, Some applications of functional analysis in mathematical physics, L., 1950.
- G. G. Lorentz, Ann. Math., 51, 37 (1950).
- J. Halperin, Canad. J. Math., 5, 273 (1953).
- A. P. Calderon, Sborn. per., matematika, 3, 56 (1965).
- A. P. Calderon, Proc. Symp. Pure Math., Providence, R. I., 4, 1961, p. 33.
- P. I. Lizorkin, Matem. sborn., 60 (102), 3, 325 (1963).
- V. P. Il’in, UMN, 4 (70), 131 (1956).
- S. G. Mikhlin, Vestn. LGU, 7, 143 (1957).
- L. R. Volevich, B. P. Paneiakh, UMN, 20, 1 (121), 3 (1965).