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TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 178, April 1973
SOME INTEGRAL INEQUALITIES WITH APPLICATIONS TO THEIMBEDDING OF SOBOLEV SPACES DEFINED OVER
IRREGULAR DOMAINS
BY
R. A. ADAMS1)
ABSTRACT. This paper examines the possibility of extending the SobolevImbedding Theorem to certain classes of domains which fail to have the "coneproperty" normally required for that theorem. It is shown that no extension ispossible for certain types of domains (e.g. those with exponentially sharp cuspsor which are unbounded and have finite volume), while extensions are obtainedfor other types (domains with less sharp cusps). These results are developedvia certain integral inequalities which generalize inequalities due to Hardy andto Sobolev, and are of some interest in their own right.
The paper is divided into two parts. Part I establishes the integral in-equalities; Part II deals with extensions of the imbedding theorem. Furtherintroductory information may be found in the first section of each part.
PART I. INTEGRAL INEQUALITIES
1.1 Introduction. The inequalities developed in this section generalize cer-
tain well-known integral inequalities of G. H. Hardy and S. L. Sobolev and con-
cern estimates for weighted L?-norms, uniform norms and Holder norms for con-
tinuously differentiable functions defined on open intervals, cones or balls in
terms of weighted L^-norms of the function and its first derivatives. The in-
equalities will be used in Part II to prove imbedding theorems for (unweighted)
Sobolev spaces defined over irregular domains.
The one-dimensional case is treated in 1.2, and the results obtained ex-
tended to (rz + l)-dimensional Euclidean space E , in the remaining sections,
1.3 dealing with bounds for weighted L -norms, and 1.4 with pointwise bounds
and Holder conditions.
Functions u may be assumed complex-valued in general. We shall not be
concerned with the problem of finding the best constants for our inequalities.
1.2 The one-dimensional case. Throughout this section we consider functions
u continuously differentiable on an open interval (0, T) for fixed T > 0. In each
inequality studied it may be assumed that the right-hand side is finite.
Received by the editors May 31, 1972.AMS (MOS) subject classifications (1970). Primary 46E35, 26A84; Secondary 26A87.Key words and phrases. Sobolev space, imbedding theorem, integral inequality.A) Research partially supported by the National Research Council of Canada under
Operating Grant number A-3973. ,. , ,. tr Copyright 1973, /American Mathematical Society
401
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402 R. A. ADAMS
1.2.1 Lemma. If 8 >l and a > 0 then
il) (T\uit)\Sta-Xdt
SOME INTEGRAL INEQUALITIES 4O3
boundary restrictions by the requirement that jQ \uit)\pt dt be included on the
right-hand side.
1.2.3 Lemma. If 8 >1 and a > 0 we have the following pair of inequalities:
(3) sup \uit)\*
404 R. A. ADAMS
sup \uit)\*ta+1-p
SOME INTEGRAL INEQUALITIES 405
|a(r)| < |a(7")| + f0 I a (o")| do. Integtation of r over (0, T) and application of
H'lder's inequality in case p > 1 yields
T\uit)\ < jTQ \uir)\ dr+T /[ \u'ia)\ do
< t{7- j; [- ,.'(rt,.] .**}"*{ .-*-"* r"/Pfrom which (7) follows. If z> = 1, (7) follows from the first inequality above.
Remarks. 1. For a < p - 1 and T < (5) holds for 1 < y < .2. Under the assumptions of Lemma 1.2.5 it can be shown further that
(8) sup Ia(t)-"(r)| < const i r |" + |Wl r-*}l/#
where /i = 1 - (a+ l)/p. We defer the proof of this inequality as it is similar to,
and a special case of, that of Theorem 1.4.3 below.
1.3 The multi-dimensional case Lp estimates. In this section x= (x,,..-,
x ,) will denote a point in (rz + l)-dimensional Euclidean space E j in > 1),
and we shall use the spherical polar coordinate representation x = (p, cp., cp ,
, cp ) = (p, cp) where p > 0, - 77 < cp. < n, 0 <
406 R. A. ADAMS
spanned by the axes x, ,,, x ,, while r ,(x) is just the distance from xr J k + i 77 + 1 ' 77 + 1 'to the origin. In connection with the use of product symbols of the form P =
nm_i, P -, be it agreed hereafter that P = 1 if m < k.J K. J
Throughout this section A shall denote an open, conical domain in F .
specified in polar coordinates by the inequalities
(2) 0
SOME INTEGRAL INEQUALITIES 407
By virtue of the restrictions placed on a, rzz, and k in the statement of the lemma,
(4) and (5) are both special cases of
Jq kl si/'' 0 if /' > z. We prove (6) by backwards in-
duction on i. For z = tz + 1, (6) is obtained by applying Lemma 1.2.1 to u con-
sidered as a function of p and then integrating the remaining variables with the
appropriate weights. Assume therefore that (6) has been proved for i = I + 1 where
1 < / < tz. If . < 77 we have
(7) sin
408 R. A. ADAMS
This completes the induction establishing (6) and hence the lemma.
We now state without proof a special case, suitable for our purposes, of a
well-known combinatorial lemma which is central to one of the standard proofs of
the Sobolev Imbedding Theorem. The proof of this lemma may be found in
Gagliardo [4, p. 117], or Clark [3].
1.3.2 Lemma. Let fl be a domain in E , and let A ., / = 1, 2, , n + 1,be the projection of fl onto the n-dimensional coordinate hyperplane orthogonal
to the jth coordinate axis in E ,. Let F (
SOME INTEGRAL INEQUALITIES
fi and il. ate domains in E . We define functions Fn = FAp ) and F0 ; 72 U 0 ' jF {cp A as follows:
7 ^7
Fo(p*) = Fo(
410 R. A. ADAMS
since \du/d \ < p II" . sin c . Heneei ,i _ r ,=7 + 1 ,
r iF.icb*)]ndpdcp.
(13)
SOME INTEGRAL INEQUALITIES 411
" 8{/o p^ + i'wi*l [rtwim&l "
since |Vzz (y)| = |Va(x)|, u being independent of z.
4. If max (1 - k, p - 72 - 1) < 0.J < a< a2 < o then the constant Q in (9) can
be chosen so as to depend on a, and a. but not on a. This can be seen by re-
viewing the effect of the constants in formulas (1) and (3) of 1.2 on the constant
K , of (3) above, and finally on Q. This fact will be useful later.Theorem 1.3.3 may be generalized in the direction of the corollary to Theorem
1.2.4 as follows.
1.3.4 Theorem. Let p > 1, 1 < k < n + 1, and 0 < s < p. Suppose that a>max (1 k + s s/p, p - n - l). Then
(14) |ja |a(x)|7^(x)]a-xl /7S p - k and generalizes Theorem 1.2.2. If Ta'p Lpiilao) where
^tx. = Kf9' ^: 0 < p < 00, (a/2, 0) e 21 then we obtain, letting a -. in formula(15),
(16) f |z,(x)|"k(x)]a-Vx
412 R. A. ADAMS
for a> p - k, a generalization of Hardy's inequality.
1.2.5 Example. Let p > 1, I < k max (l - k + s is/p), p n - l). Let uix) = p~p and suppose y > y . It is
readily checked that
(17) jJ]^+|VB|] ,;*)//>. it is pos-
sible to choose so that (17) and (18) both hold. This example shows that the
exponent y in (14) (or y in (9)) is the best possible.
1.4 The multi-dimensional caseboundedness and Holder continuity. We now
turn to the case a> 0, a+- n + 1 - p < 0. It is convenient to deal directly with
domains fl C E . more general than those considered in 1.3. fl is said to
have the "cone property" if there exists a finite cone C (the intersection of an
open ball in F x centred at the origin, with a set of the form {Xx: X > 0,
x e E ,, |x - y| < r\ where r > 0 and y is a fixed point in E x with |y| > r)
such that each point x on the boundary
SOME INTEGRAL INEQUALITIES 4jj
1.4.2 Theorem. Let il be a domain with the cone property in E .. Let 1 0 and a + n + I - p < 0 then for all u C iil)
we have
(2) sUP|zz(x)|
414 R. A. ADAMS
It follows that
Jc \Vuix)\dx
SOME INTEGRAL INEQUALITIES 415
a(x)- I u(z) dzp"+l JP
< f dz C |Va(x + Az - x))\ dt- p" Jap J
= f1 t-"~ldt f \Vuiz)\dzp"J \p
< K6pPo r(a+" + l)/pdt lfa\Vuiz)\p[riz)]adzy
416 R. A. ADAMS
PART IL IMBEDDING AND NONIMBEDDING THEOREMS FORSOBOLEV SPACES
2.1. The Sobolev Imbedding Theorem. Let A be an open domain in F . The
Sobolev space Wm'pi) is, for m = I, 2, and p > 1, the space of all (possibly
complex-valued) functions u in LP(A) whose distributional partial derivatives of
orders up to and including m also belong to LP(A). Wm,pil) is a Banach space
with respect to the norm
(1) \u:Wm-*il)\ = f Z \Dsu: Lpi)\p\l/pl0
SOME INTEGRAL INEQUALITIES 417
(i) if mp
418 R. A. ADAMS
2.2 is concerned with unbounded domains which become narrow at infinity.
We show that generally no imbeddings of the desired type are possible.
2.3 is concerned with classes of domains having cusps. We show that if these
cusps have "power sharpness" Theorem 2.1.1 survives but with weakened con-
clusions, establishing imbeddings of all three types for a large, though by no
means exhaustive, class of domains with such cusps. Our results sharpen and
generalize certain similar results obtained by I. Globenko ([5], [6]) by different
methods. Finally we show that no imbeddings of the desired types are possible if
the domain has cusps of "exponential sharpness", i.e. cusps sharper than any
power cusp.
2.2 Unbounded domainsa nonimbedding theorem. An unbounded domain A
in E may have a smooth boundary and still fail to satisfy the cone condition if
it becomes narrow at infinity. For unbounded A let A., denote the set ix e A:
N < \x\ < N + 11. The writer and John Fournier have shown in [l] that if there is
any imbedding of the form
(1) Wm'piQ) -> L"i)
where q > p then either
(a) vol A = 00 and lim^^^vol A^ > 0, or(b) vol A < 00 and lim., ^e vol A., = 0 for any k.
Unbounded domains with the cone property fall under the alternative (a).
Example. The domain A = i(x, y) e E2- x > 0, 0 < y < e~x \ satisfies (b)
above. However, the function zz(x, y) = ex tl is easily seen to belong to Wm,piQ,)
tot 1 < p < a and any m, but not to L9(A).This example leads us to speculate that there are no unbounded domains in
class (b) above for which (1) holds for some q > p. Such a result was proved
for connected A and m = 1 by R. Andersson [2]. We prove it in general.
2.2.1 Theorem. // A is unbounded and has finite volume there exist no im-
beddings of type (1) for any q > p.
Proof. The method of proof is suggested by the example given above. We
construct a function uix) depending only on the distance of x from the origin,
whose growth is rapid enough to prevent membership in Lqi) but still slow
enough to allow membership in W,piQ,).
Let Air) denote the surface area (Lebesgue in - l)-measure) of the inter-
section of A with the spherical surface of radius r centred at the origin. Then
J Air)dr = vol A < 00.0Without loss of generality we may assume that vol A = 1. We define numbers rn
in = 0; 1, 2, ) by
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SOME INTEGRAL INEQUALITIES 419
J~ Air)dr = 2-"=f" Air) drn 72 1
so that clearly rn = 0 and r T as 72 > . Let Ar = r . - r and fix f suchJ U 72 7772+177that 0 < e < imp)' - imq)~ . There must exist an increasing sequence \n .}\
such that Ar > 2 'for otherwise Ar < 2 for all but possibly finitely many
72 whence 2 Ar < , a contradiction. For convenience we assume 72, > 1 so77=0 77 1
that 72. > 7 for all /'. We denote a. = 0, a . = r ., b . = r (j = 1, 2, ), and7 - ' ' 0 ; 77/ + 1' ; 71/ '
note that a . , < b . < a . and a . - b . = Ar > 2~"i.7-17 7 7 7 72/ -
Let / be a fixed, nonnegative, infinitely differentiable function on (- , oo)
with the properties
(i) 0 < fit) < 1 for all t,(ii) /(/) = 0 if / 1,
(iii) f{k)(t) < M fot all t if 1 < k < 772.For x in fi let r = |x| and define a function u in C iii) as follows (taking
720=0)
a(x) = 2"-l/9
uW = 2nHA+(2"/9./rA
for a . . < r < b .J- 1 - - 7
-2 '-1 )/((r-.)/(a.->.)) for . < r < a ..' ill i - - i
Denoting il . = Sx e Q: a _ , < |x| < a S we have
/ \uix)\p dx = if J +f"'\[u(x)]*A(r)dri i ai-1 i>
n , p/q rco , . 72 .p/q -a .~l J A(r)a"r+2 f '(^J y. 1 J b.
-V\2~n'-l(l~P/,q) j'"'1'1'"^] - 7-('-l)(l-p/,q)
Since p
420 R. A. ADAMS
where C = 1 - p/q -ekp>0 since e < l/mp - l/mq. It follows that j \dku/drk\p dx< oo. Finally, we note that
f \uix)\qdx>2"'-1 F' Air) drSI - J a. . 7 7-1
77. , . n . , 1 77. 1= 2 7_1[2 7- -2 ' ] > 1/4
whence j\uix)\g dx = oo. Since u belongs to Wm,pi,) but not to Lqi) the theo-
rem is proved.
Remarks. 1. Following the discussion at the beginning of this section Theo-
rem 2.2.1 has the force of precluding the existence of imbeddings of type (1) for
any a > p whenever A is unbounded and satisfies lim., ^ vol A,, = 0, a condi-
tion obviously much weaker than finite volume.
2. Since the counterexample function u constructed in the proof of the above
theorem is unbounded it serves also to show that there can be no imbedding of
Wm,p(A) into C;(A) for any / (if A is unbounded and has finite volume.)
2.3 Domains with cusps. Let it be assumed from the outset that each domain
A C E considered in this section has boundary dA consisting of in - l)-dimen-
sional surfaces, and that A lies on only one side of f3A. A is said to have a cusp
at x0 e 9A if no finite cone of positive volume contained in A can have vertex
at x~. The failure of a domain A to have any cusps does not, of course, guarantee
that the domain has the cone property.
We begin by considering cusps of power sharpness.
2.3.1 Definition. For 1 < k < n - 1 and > 1 we denote by A . the stand-ard power cusp domain in F specified by the inequalities:
x2 + + x\ < x2. ,,1 k k+i'
(1) x, . > 0, ..., x > 0,k + i 77
lx2. + ...+x2Al^ + xl ,+... + x2< a21 k fe + l 77
where a is the radius of the ball of unit volume in E . Clearly a < 1. A, has77 J K,\
axial plane spanned by the x, ., > xn coordinate axes, and vertical plane
spanned by x 2, , x . If k = n - 1 the origin is the only vertex point of
A, . . The outer boundary surface (as determined by (1)) is taken to be of this
form in order to simplify calculations later. It could be taken to be a sphere or
more general surface bounded away from the origin.
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SOME INTEGRAL INEQUALITIES 421
Example. Let n = 3. H2 2 is the domain in E, specified in cylindrical
polar coordinates (r, 0, z) by
r0, r + z2 0, \x\ +y2 + z2 < [l/4n]2//i.
This domain has a one-dimensional cusp line along the z-axis.
Together with il, . we shall consider the associated standard cone il, .
which is a domain of the type considered in 1.3. il, , is specified in Cartesian
coordinates y.., , y by
2 2^2yi + ...+yk0'---'y,z>0>
y2 + ...+y2n
422 R. A. Adams
x, = zvsin j sin cp"2 sin p - n. It is sufficient
to prove (4) for y = (a + n)p/i
SOME INTEGRAL INEQUALITIES 423
Since rkiy) < 1 on ilk j it follows that \^uiy)\ .< X^uix)]. Hence (4) follows
from (5) in this case. For a. < p - n and any y the proof is similar, and uses the
second remark following Theorem 1.3.3.
In order to show that the constant X in (4) can be chosen to be independent
of k and X provided a= ( - l)k < a. we note that it is sufficient to prove that
there is a constant / such that for any k, X with 1 < k < n - 1, a. < a , and all
v Clk ,)
(6) // j \viyV[rkiy)]adyy * < ] {f ^ [\viy)\p + \Vviy)\p][rk]ady\" \
In fact it is sufficient to prove (6) with / depending on k as we can then maxi-
mize ]ik) over the finitely many allowed values of k. We distinguish three cases.: a p - n. Again it is enough to deal with y = (a + n)p/ia + n - p).
From Theorem 1.3.3
(8) tt i \viy)\V[rkiy)]ady\l * y and / j is independent of a for p - n < aQ 4.We now consider imbeddings into spaces of continuous functions.
2.3.4 Lemma. Let 0 < a < mp n. There exists a constant Q = Qin, p, a )
such that if 1 < k < n - 1 aW A > 1 satisfy a = (A - 1)& < a /erz /or a//a eCmiu .)
ft,
(11) sup |a(x)| pink )\.xilk,X
Proof. First suppose m = 1. For u e C (A, ) we have by Theorem 1.4.2
and via the method of the first part of the proof of Lemma 2.3.2
sup \uix)\ = sup |a(y)|
Up
xak,x y\i
(12) -QAia l\b>\P+\VWttrkiy)]ady\
SOME INTEGRAL INEQUALITIES 427
Then
(13) Wm-HQ) C(A).
More generally, if a. < im j)p n where 0 < ;' < m - 1 then
(14) Wm'PiQ) -^c>i).
Proof. It is sufficient to prove (13). If yj maps G C A onto A, . we havefor a e Cm(A)
sup |a(x)| = sup |" o t/z" (y)|xG y{lk,X
428 R. A. ADAMS
(16) lim Air, il)/rk = 0.70 +
2.3.7 Theorem. // il is a domain in E having an exponential cusp at xn
p, or into cAil)
for any j.
Proof. We consttuct a function u Wm,piil) which fails to belong to Lqiil)
(a > p) or CAil) because it becomes unbounded too rapidly near x_. We make use
of Theorem 2.2.1 in the construction.
Without loss of generality we assume x = 0 so that r= |x|. Let it =
\y = x/\x\ : x il, \x\ < 1 \. It is easily seen that il is unbounded and has finite
volume, and that Air, il*) = r2{n~ l)Ail/r, il). Let t satisfy p < t < q. By Theorem
2.2.1 there exists a function v Cmi0,
SOME INTEGRAL INEQUALITIES 429
if r>l/a then rk~ 2nAir, A*) < rk~ 2pk = r-2. Thus
f~ r(+> + ')p-2"\ZU)ir)\pAir, Q*)dr
= ^\vU)ir)\p/k-2^'-p^Air,*)dr