SOME INTEGRAL INEQUALITIES WITH APPLICATIONS TO … · SOME INTEGRAL INEQUALITIES WITH APPLICATIONS TO THE ... via certain integral inequalities which generalize inequalities due

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

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

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


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)


" 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


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


(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., ê 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



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.



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


Since rkiy) < 1 on ilk j it follows that \ûiy)\ .< Xûix)]. 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\


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,


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^'-pÂir,*)dr

Documents

SOME INTEGRAL INEQUALITIES WITH APPLICATIONS TO … · SOME INTEGRAL INEQUALITIES WITH APPLICATIONS TO THE ... via certain integral inequalities which generalize inequalities due