Abstract
Full Text
S. V. USPENSKII
ON BOUNDARY PROPERTIES OF FUNCTIONS FROM “WEIGHTED” CLASSES (W_{\vec{\alpha},p}^{1})
(Presented by Academician S. L. Sobolev, 25 I 1961)
In the present paper we determine necessary and sufficient conditions satisfied by a function on the boundary of a domain if each of its first derivatives is summable in the domain with a certain weight degenerating on the boundary, in general different for each derivative.
Definition. Let (R^{n-1}) be the ((n-1))-dimensional space of points ((x_1,\ldots,x_{n-1})), and let (R_0^n) be the half-space ((x_n \ge 0,\ x(x_1,\ldots,x_{n-1}) \in R^{n-1})). We shall say that a function (f) belongs to the class (W_{\vec{\alpha},p}^{1})
[
(1<p<\infty,\quad \vec{\alpha}=(\alpha_1,\ldots,\alpha_n),\quad -1<\alpha_i<p-1,\quad i=1,\ldots,n),
]
if the function (f) has all first generalized Sobolev derivatives ((^{1})) and
[
|f|{W},p}^{1}}^{p
=
\int_{R_0^n} |f|^p\,dR_0^n
+
\int_{R_0^n} \sum_{i=1}^{n} x_n^{\alpha_i}\left|\frac{\partial f}{\partial x_i}\right|^p\,dR_0^n
<\infty .
\tag{1}
]
Following L. N. Slobodetskii ((^{2})), we shall also introduce the classes (W_{x,p}^{r}) ((1<p<\infty,\ x=(x_1,\ldots,x_{n-1}),\ r=(r_1,\ldots,r_{n-1}),\ 0<r_i<1)). We shall consider that (f\in W_{x,p}^{r}) if (f\in L_p(R^{n-1})) and
[
|f|{W}^{r}}^{p
=
\int_{R^{n-1}} |f|^p\,dR^{n-1}
+
\sum_{i=1}^{n-1}\int_0^\infty
\int_{R^{n-1}}
\frac{|\Delta_{i,h}f|^p}{h^{1+pr_i}}\,dR^{n-1}\,dh
<\infty ,
\tag{2}
]
where
[
\Delta_{i,h}f
=
f(x_1,\ldots,x_i,\ldots,x_{n-1})
-
f(x_1,\ldots,x_i+h,\ldots,x_{n-1}).
]
We shall also say that the function (f) assumes the value (\varphi) on (R^{n-1}) ((F|_{R^{n-1}}=\varphi)) if, after a possible modification of (F) on a set of (n)-dimensional measure zero,
[
\lim_{x_n\to 0}\int_{R^{n-1}} |F-\varphi|^p\,dR^{n-1}=0.
\tag{3}
]
The following two theorems generalize the corresponding results of A. A. Vasharin ((^{3})) and P. I. Lizorkin ((^{4})).
Theorem 1 (direct). Let (f\in W_{\vec{\alpha},p}^{1}). Then
[
f\big|{R^{n-1}}=\varphi\in W,},p}^{\vec{\beta}
\qquad
\beta_i=\frac{p-1-\alpha_n}{p-\alpha_n+\alpha_i}
\quad (i=1,\ldots,n-1),
]
[
|\varphi|{W},p}^{\vec{\beta}}
\le
c|f|{W,},p}^{1}
\tag{4}
]
where the constant (c) does not depend on (f).
Theorem 2 (converse). Let (\varphi \in \overset{\to}{W}{}^{\beta}{x,p}); let (\alpha_n) be arbitrary in the interval ((-1,p-1)), and let
[
\alpha_i=\alpha_n-p+\frac{p-\alpha_n-1}{\beta_i}\quad (i=1,\ldots,n-1).
]
Then there exists an infinitely differentiable function (f), defined on (R_0^n), such that
[
f\big|=\varphi,\qquad}
|f|{W^1\leq c|\varphi|}{\overset{\to}{W}{}^{\beta},}
\tag{5}
]
where the constant (c) does not depend on (\varphi).
We give brief proofs of the theorems.
Proof of Theorem 1. The existence of boundary values (\varphi) and their membership in (L_p(R^{n-1})) follow from the corresponding results of L. D. Kudryavtsev ((^5)).
We prove (4). Let
[
\gamma_i=\frac{p}{p-\alpha_n+\alpha_i},\qquad
\beta_i=\frac{p-1-\alpha_n}{p-\alpha_n+\alpha_i}.
]
We have
[
|\varphi|^p_{\overset{\to}{W}{}^{\beta}{x,p}}
=
\sum}^{n-1}\int_0^\infty\int_{R^{n-1}
\frac{|\Delta_{i,h}\varphi|^p}{h^{1+p\beta_i}}\,dR^{n-1}\,dh
\leq
]
[
\leq
c\sum_{i=1}^{n-1}\int_0^\infty\int_{R^{n-1}}
\frac{
|f(x_1,\ldots,x_i,\ldots,x_{n-1},h^{\gamma_i})
-
f(x_1,\ldots,x_i,\ldots,x_{n-1},0)|^p
}{h^{1+p\beta_i}}\,dR^{n-1}\,dh
+
]
[
+
c\sum_{i=1}^{n-1}\int_0^\infty\int_{R^{n-1}}
\frac{
|f(x_1,\ldots,x_i+h,\ldots,x_{n-1},h^{\gamma_i})
-
f(x_1,\ldots,x_i,\ldots,x_{n-1},h^{\gamma_i})|^p
}{h^{1+p\beta_i}}\,dR^{n-1}\,dh
+
]
[
+
c\sum_{i=1}^{n-1}\int_0^\infty\int_{R^{n-1}}
\frac{
|f(x_1,\ldots,x_i+h,\ldots,x_{n-1},h^{\gamma_i})
-
f(x_1,\ldots,x_i+h,\ldots,x_{n-1},0)|^p
}{h^{1+p\beta_i}}\,dR^{n-1}\,dh
\leq
]
[
\leq
c\sum_{i=1}^{n-1}\int_0^\infty\int_{R^{n-1}}
\left[
\left|\int_0^{h^{\gamma_i}}\frac{\partial f(x,u)}{\partial x_n}\,du\right|^p
+
h^p\left|\frac{\partial f(x,h^{\gamma_i})}{\partial x_i}\right|^p
\right]
h^{-1-p\beta_i}\,dR^{n-1}\,dh
=
]
[
\text{(substitution }v=h^{\gamma_i}\text{)}
]
[
=
c_1\sum_{i=1}^{n-1}\int_0^\infty\int_{R^{n-1}}
\left[
\left(\int_0^v\frac{\partial f(x,u)}{\partial x_n}\,du\right)^p
+
v^{p/\gamma_i}\left|\frac{\partial f(x,v)}{\partial x_i}\right|^p
\right]
v^{-(1+p\beta_i/\gamma_i)}\,dR^{n-1}\,dh
\leq
]
[
\text{(Hardy’s inequality ((^6)))}
]
[
\leq
c_2\sum_{i=1}^{n}\int_0^\infty\int_{R^{n-1}}
\left(
x_n^{p-1-p\beta_i/\gamma_i}\left|\frac{\partial f}{\partial x_n}\right|^p
+
x_n^{(p-p\beta_i)/\gamma_i-1}\left|\frac{\partial f}{\partial x_i}\right|^p
\right)\,dR^{n-1}\,dx_n
=
]
[
=
c_2\sum_{i=1}^{n}\int_0^\infty\int_{R^{n-1}}
\left(
x_n^{\alpha_n}\left|\frac{\partial f}{\partial x_n}\right|^p
+
x_n^{\alpha_i}\left|\frac{\partial f}{\partial x_i}\right|^p
\right)\,dR^{n-1}\,dx_n.
]
Proof of Theorem 2. Let (\alpha_n) be arbitrary fixed from the interval ((-1,p-1)). Put
[
\gamma_i=\frac{p-\alpha_n-1}{p\beta_i}>0.
]
Construct the function
[
f=k\int_{R^{n-1}}\varphi(t)\prod_{i=1}^{n-1}
\frac{x_n^{\gamma_i}}{(t_i-x_i)^2+x_n^{2\gamma_i}}\,dR^{n-1},
]
where
[
k=\left[\int_{R^{n-1}}\prod_{i=1}^{n-1}\frac{1}{1+u_i^2}\,dR^{n-1}\right]^{-1}.
]
We shall show that the function (f) satisfies the conditions of the theorem. It is easy to see that (f|_{R^{n-1}}=\varphi). We also have, denoting by (\Pi') the product (\Pi) with one factor omitted,
[
\frac{\partial f}{\partial x_n}
=
k\int_{R^{n-1}}\varphi(t)\sum_{j=1}^{n-1}\prod_{i=1}^{n-1}{}'
\frac{x_n^{\gamma_i}}{(t_i-x_i)^2+x_n^{2\gamma_i}}\,
\frac{\partial}{\partial x_n}
\left[
\frac{x_n^{\gamma_j}}{(t_j-x_j)^2+x_n^{2\gamma_j}}
\right]dR^{n-1}
=
]
[
k\int_{R^{n-1}}\varphi(t)\sum_{j=1}^{n-1}\prod_{i=1}^{n-1}
\frac{x_n^{\gamma_i}}{(t_i-x_i)^2+x_n^{2\gamma_i}}\,
\gamma_j
\frac{\partial^2}{\partial u^2}
\left[
\ln\left[
\frac{1}{(t_j-x_j)^2+u^2}
\right]^{1/2}
\right]_{u=x_n^{\gamma_i}}
x_n^{\gamma_j-1}\,dR^{n-1}
=
]
[
-k\int_{R^{n-1}}\sum_{j=1}^{n-1}\gamma_j
\left[
\varphi(t_1,\ldots,t_j,\ldots,t_{n-1})
-
\varphi(t_1,\ldots,x_j,\ldots,t_{n-1})
\right]\times
]
[
\times
\prod_{i=1}^{n-1}{}'
\frac{x_n^{\gamma_i}}{(t_i-x_i)^2+x_n^{2\gamma_i}}\,
\frac{\partial^2}{\partial t_j^2}
\left[
\ln\left[
\frac{1}{(t_j-x_j)^2+x_n^{2\gamma_i}}
\right]^{1/2}
\right]
x_n^{\gamma_j-1}\,dR^{n-1};
]
[
\left|\frac{\partial f}{\partial x_n}\right|
\leq
c\sum_{j=1}^{n-1}\int_{R^{n-1}}
|\Delta_{j,t_j}\varphi|
\prod_{i=1}^{n-1}{}'
\frac{x_n^{\gamma_i}}{(t_i-x_i)^2+x_n^{2\gamma_i}}\,
\frac{x_n^{\gamma_j-1}}{t_j^2+x_n^{2\gamma_j}}\,dR^{n-1};
]
[
\int_{R_0^n}x_n^{\alpha_n}
\left|\frac{\partial f}{\partial x_n}\right|^p\,dR_0^n
\leq
]
[
\leq
c\int_{R_0^n}x_n^{\alpha_n}
\sum_{j=1}^{n-1}
\left[
\int_{R^{n-1}}
|\Delta_{j,t_j}\varphi|
\prod_{i=1}^{n-1}{}'
\frac{x_n^{\gamma_i}}{(t_i-x_i)^2+x_n^{2\gamma_i}}\,
\frac{x_n^{\gamma_j-1}}{t_j^2+x_n^{2\gamma_j}}\,dR^{n-1}
\right]^p dR_0^n
\leq
]
[
\leq
c\sum_{j=1}^{n-1}\int_{R_0^n}x_n^{\alpha_n}
\int_{R^{n-1}}
|\Delta_{j,t_j}\varphi|^p
\prod_{i=1}^{n-1}{}'
\frac{x_n^{p\gamma_i}}{(t_i-x_i)^2+x_n^{2\gamma_i}}\,
\frac{x_n^{p(\gamma_j-1)}}{[t_j^2+x_n^{2\gamma_j}]^{\varepsilon p/2}}
\,dR^{n-1}\times
]
[
\times
\left[
\int_{R^{n-1}}
\prod_{i=1}^{n-1}{}'
\frac{1}{(t_i-x_i)^2+x_n^{2\gamma_i}}\,
\frac{1}{[t_j^2+x_n^{2\gamma_j}]^{(2-\varepsilon)q/2}}
\,dR^{n-1}
\right]^{p/q}dR_0^n
\leq
]
[
\left(\frac{1}{p}+\beta_j<\varepsilon<1+\frac{1}{p}\right)
]
[
\leq
c_1\sum_{j=1}^{n-1}\int_{-\infty}^{\infty}
\int_{R^{n-1}}|\Delta_{j,t_j}\varphi|^p
\int_0^{\infty}
\frac{x_n^{\alpha_n-\gamma_j-p+p\gamma_j\varepsilon}}
{(t_j^2+x_n^{2\gamma_j})^{\varepsilon p/2}}
\,dx_n\,dR^{n-1}\,dt_j
\leq
]
[
\left(\text{substitution }\frac{x_n^{\gamma_j}}{t_j}=v\right)
]
[
\leq
c_2\sum_{j=1}^{n-1}\int_{-\infty}^{\infty}
\int_{R^{n-1}}
\frac{|\Delta_{j,t_j}\varphi|^p}{|t_j|^{1+p\beta_j}}
\int_0^{\infty}
\frac{v^\theta}{[1+v^2]^{\varepsilon p/2}}\,dv\,dR^{n-1}\,dt_j
\leq
]
[
\left(p\varepsilon>1+p\beta_j,\ \theta=\frac{\alpha_n}{\gamma_j}-1-\frac{p}{\gamma_j}+p\varepsilon+\frac{1}{\gamma_j}-1>-1\right)
]
[
\leq c_2 \sum_{j=1}^{n-1}\int_0^\infty \int_{R^{n-1}}
\frac{|\Delta_j h^\varphi|^p}{h^{1+p\beta_j}}\,dR^{n-1}\,dh.
]
Estimates for the derivatives (\partial f/\partial x_i) ((i=1,\ldots,n-1)) are obtained similarly.
Theorems 1 and 2 also carry over to bounded domains with sufficiently smooth boundaries.
Institute of Mathematics with Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR
Received
19 I 1961
REFERENCES CITED
- S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
- L. N. Slobodetskii, DAN, 120, No. 3 (1958).
- A. A. Vasharin, DAN, 117, No. 5 (1957).
- P. I. Lizorkin, DAN, 126, No. 4 (1959).
- L. D. Kudryavtsev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 55 (1960).
- G. G. Hardy, D. E. Littlewood, G. P. Pólya, Inequalities, M., 1948.