4.33 Mixed-norm LP estimates of the type considered in Paragraphs 2.48-2.51 and used in the proof of Gagliardo's averaging lemma 4.23 can contribute to gen- eralizations of Sobolev's inequality. We examine briefly two such generalizations:
(a) anisotropic Sobolev inequalities, in which different L p norms are used for different partial derivatives on the right side of (19), and
(b) reduced Sobolev inequalities, in which the seminorm Iq~lm,p,R, on the fight side of (19) is replaced with a similar seminorm involving only a subset of the partial derivatives of order m of 4).
Questions of this sort are discussed in [BIN1 ] and [BIN2]. We follow the treatment in [A3] and [A4] and most of the details will be omitted here.
4.34 (A First-Order Anisotropic Sobolev Inequality) If pj ~ 1 for each j with 1 _< j _< n and 4) ~ C ~ (I~ n ), then an inequality of the form
[[~[Iq ~ K ~ IIDj II
j = l
(22)
is a (first-order) anisotropic Sobolev inequality because different L p norms are used to estimate the derivatives of cp in different coordinate directions. A dilation argument involving ~b0~lXl . . . . , XnXn) for 0 < ~j < c~, 1 < j < n shows that no such anisotropic inequality is possible for finite q unless
n 1 1 1 @ , 1 1
j ~ l > 1 and - = - Z ~
.= pj
qnj=l P j n
If these conditions are satisfied, then (22) does hold. The proof is a generaliza- tion of that of Theorem 4.31 and uses the mixed-norm H61der and permutation inequalities. (See [A3] for the details.)
4.35 (Higher-Order Anisotropic Sobolev Inequalities) The generalization of (22) to an ruth order inequality by induction on m is somewhat more problematic.
Variations of Sobolev's Inequality 105 The mth order isotropic inequality (19) follows f r o m its special case m -- 1 by simple induction. We can also obtain
II~bllq ~ K ~ IID~q~llp~, ]oe[=m
where
1 = 1 ~ m ( m ) 1 m ( m ) _ m!
q n m . _ _ ot poe lit ol ol 1 !o121 . . . ol n I 9 ,
by induction f r o m (22) under suitable restrictions on the exponents poe, but the restriction
1 ~ ( m ) 1 m
tim Ot Poe /It
I =
will not suffice in general for the induction even though Y~loel=m (m) = nm. The conditions mpoe < n for each ot with lot] - m will suffice, but are stronger than necessary.
For any multi-index/3 and 1 _< j < n, let
f l [ j ] -- (ill . . . ~ - 1 , flj + 1, ~ + 1 . . . ]~n)-
Evidently, [fl[j][ = 1/31 + 1 and it can be verified that if the n u m b e r s poe are defined for all ot with ]ol[ -- m, then
]fl]--m-1 /~ j--1 P f [ J ] ]oel--m c~ poe
This provides the induction step necessary to verify the following theorem, for which the details can again be found in [A3].
4.36 T H E O R E M Let poe >_ 1 for all o~ with Joel -- m. Suppose that for every fl with 1/31 = m - 1 we have
n 1
> m .
T h e n there exists a constant K such that the inequality ]l~bl[q < g ~ I[Doe~b[lp~
[oel=m holds for all 4~ 6 C ~ (I~ n ), where
1 l m
t i m ~--7"-,,. Ol poe ti I ~,.
106 The Sobolev Imbedding Theorem
4.37 ( R e d u c e d Sobolev Inequalities) Another variation of Sobolev's inequal- ity addresses the question of whether the number of derivatives estimated in the seminorm on the right side of (19) (or, equivalently, (18)) can be reduced with- out jeopardizing the validity of the inequality for all ~b ~ C ~ (I~ n). If m >_ 2, the answer is yes; only those partial derivatives of order m that are "completely mixed" (in the sense that all m differentiations are taken with respect to different variables) need be included in the seminorm. Specifically, if we denote
,M - . M ( n , m ) - {or 9
I~1-
m, c~j = 0 o r o t j = 1 for 1 < j < n, then the reduced Sobolev inequalityII~llq
~ K Z
IID~q~IIp otr.h/[holds for all q~ 6 C~(]~n), provided m p < n and q = n p / ( n - m p ) . Again the proof depends on mixed-norm estimates; it can be found in [A4] where the possibility of further reductions in the number of derivatives estimated on the right side of Sobolev's inequality is also considered. See also Section 13 in [BIN1 ].
Wm,P(/~) as a Banach Algebra
4.38 Given u and v in W m,p (~"2), where f2 is a domain in I~ ~, one cannot in general expect that their pointwise product u v will belong to w m ' P ( ~ " 2 ) . The imbedding theorem, however, shows that this is the case provided m p > n and f2 satisfies the cone condition. (See [Sr] and [Mz2].)
4.39 T H E O R E M Let f2 be a domain in I~ n satisfying the cone condition.
If m p > n or p = 1 and m > n, then there exists a constant K* depending on m, p, n, and the cone C determining the cone condition for f2, such that for
U, 1) E W m ' p (~'2) the product uv, defined pointwise a.e. in f2, satisfies
[luvllm,p,a <_ K* Ilullm,p,~ Ilvllm,p,a. (23)
In particular, equipped with the equivalent norm II'll*m,p,~ defined by Ilull~,,p,~ - K* Ilullm,p,~,
W m'p (~"2) is a commutative Banach algebra with respect to pointwise multiplication in that
Iluvll,~p,~ < Ilull* , - - m , p , ~ Ilpll~, p,~ , 9
Proof. We assume m p > n; the case p -- 1, m - n is simpler. In order to establish (23) it is sufficient to show that if Iotl _< m, then
f l D ~ [ u ( x ) v ( x ) ] l p < g ~ Ilullm,p,a IlVllm,p,~,
W m'p (~"2) as a Banach Algebra 107
w h e r e K~ -- K~(m, p , n , C). L e t u s a s s u m e for the m o m e n t t h a t u E C ~ ( f 2 ) . By the Leibniz rule for distributional derivatives, that is,
it is sufficient to show that for any fi < or, loll < m, we have f l D ~ u ( x ) D ~ - ~ v ( x ) l p dx <_% K~,~ IlullPm,p,~ IlvllPm,p,~,
w h e r e K~,~ -- K~,~(m, p, n, C). By the i m b e d d i n g t h e o r e m there exists, for a n y / 3 with 1/31 < m, a constant K(/3) -- K(fi, m, p , n , C) such that for any
W G_. W m ' p ( ~ ) ,
f a lD~w(x)] r dx < g ( f i ) Ilwllr p , ~ , (24) p r o v i d e d (m - ] / 3 1 ) p ~ n and p < r < n p / ( n - [m - J i l l ] P ) [or p ~ r < oc if
(m - I f i l ) p = n], or alternatively
ID~w(x)l ~ K(fi) IlWllm,p,S~ a.e. in S2 p r o v i d e d (m - Ifi I)P > n.
Let k be the largest integer such that (m - k ) p > n. Since mp > n we have k > 0.
If I/~1 _< k, then (m - I f i l ) P > n, so
L I D ~ u ( x ) D ~ - ~ v ( x ) l p dx < g ( ~ ) p IlullPm p ~ [[O~-~vl[ p
- - , , O , p , S 2
< K ( f l ) p IlullPm,p,a IlvllPm,p,a 9 Similarly, if lot - / 3 1 < k, then
f lD~u(x)D~-~v(x)] p dx < K(oe - [3) p -- Ilull p m , p , ~ Ilvll m,p,f2 " P
N o w if Ifil > k and lot - fil > k, then, in fact, 1r ~ k + 1 and lot - fll > k + 1 so that n >_ (m - ] f l l ) p and n >_ (m - l o t - fil)p. M o r e o v e r ,
n - ( m - I f i ] ) P n - ( m - l o t -
fil)P
( 2 m - I c c l ) p m p+ = 2 - < 2 - - - < 1 .
/1 g/ /'/ g/
H e n c e there exist positive n u m b e r s r and r' with ( 1 / r ) + ( 1 / r ' ) - 1 such that
np np
p < rp < , p < r ' p < .
- n - ( m - [ f i ] ) p - n - ( m - ] o r -
fi[)p
108
The Sobolev Imbedding TheoremThus by H61der's inequality and (24) we have
(ff2 )l/r (ff2 )l/r'
s2 l D ~ u ( x ) D ~ p d x < [ D ~ u ( x ) [ rp d x [ D ~ - ~ v ( x ) [ r'p d x
< (K (fl))1/r (K (Ol fl))1/r'
- Ilull p P-- m,p,f2
Ilvllm,p,a.This completes the proof of (23) for u E C ~ (f2), v ~ W m'p
(~"l).
If u