On interpolation inequalities in the space H™,P(Q ) with mixed norm

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXV (1985)

M a g d a len a J a r o sz e w sk a (Poznan)

On interpolation inequalities in the space H™,P(Q ) with mixed norm

1. In this paper we prove some interpolation inequalities for functions / from the space Щ р(й) with mixed norm. At first, we shall give some lemmas for the functions from H m,p(Q). The results contain some inequalities from [4] for n = 1 and also from [3], [10] for = 0.

2. The index i runs through 1, n unless otherwise stated. Det R be the set of real numbers and /с, > 0 an integer, 1 < p, < go , Я; ^ 0. In the following we shall use vector notations, i.e., x = ( x lf x„), p = (pl5 . .. , p„) etc. In this paper we assume that Q, is a cube, because the interpolation theorem, proved for a cube, can be proven also for open, bounded Qt having the restricted cone property^). We can observe this fact for mixed norm studying, for instance, the proof of Theorem 1, [6]. So let Qt denote an open,

k- _

bounded cube, subset of the real Euclidean space R '. Let Q = P Üh Q

i = 1

n " к П

= P Üh R n = P R k\ У ki — N. The measure means always Lebesgue

«=1 i= l « = 1

measure. To simplify the notation we shall write for example:

$ \ f(x )\ d x = J ... J |/(x)|dx! . . . d x n,

Q Qn Q i

J \ f W d x = l l / C (ei = S [■■■ S (S

bQ ^{Qn Q i Q} l

J I ^{\f\’ d x =} j [ . . . ( ) 1 \ М х Г Ч х 1р 1рЧ х 2 . . У ’'1’’- Ч Х ' .

bQ i = 1 Qn Q i i=1

1 P tJP n - 1

Let

HI = I HI. HI = I l D 'f(x)

i = 1

a1'1/) * )

a«i,}... гх1%

i1) For definitions of the cone property and the restricted cone property see among others

Ш, [6].

if |/| > 0 , Dlf (x) = / (x ) if / = 0. Let & m denote the set of polynomials P (x) of degree ^ m.

We shall denote by C m(Q) the space of functions defined in Q which have continuous partial derivatives of order up to m ^ 0.

We shall denote by H m p(Q), p ^ 1, m integer ^ 1 the space of functions which is the completion of C °°(0) with respect to the norm

( 1 ) ! X V r / C J 11"--

П We denote by f Q the mean integral value of the function / on Q = P

i= 1

/ e = p ц - ' т i f(x)dx.

i ~

Q

We denote by L p£ (Q), x; ^ 0, the subspace of U (Q ) of functions / for which

(2) Ш/НкР-^я, = SUP f [ 1 ц т Г 1‘Р""“Р‘ inf I \ f-P \ pd x }llp" < CO,

Qi 1 Pz& m b Q

where Q( is the subcube of Ц .

The space L P,*(Q) is a Banach space under the norm (3) II/II l £ a(«) = II/II lp №) + III/III l £a№)-

This space was studied in [2], [6], [7] and else. We know that

R e m a r k 1. The space L $ X(Q) is isomorphic to M P,X(Q) if 0 ^ Я,- < k t and to C 0,a(Q) if < Я , k, -f pf, af = (Я, — /c^/p,-, for definition of M P,X(Q) and C°'a(Q) see [9].

We shall denote by Щ ,р(й) the space of functions which is the completion of С °°(0) with respect to the norm

(5) \\Л\н™’р(а> = \\f\\H m’P{ + I lll^ / IIL g -W To simplify the notation we set

\jyf\ _LP(Q) { I ll£>7ïl

i*i =j

Pn LP(Q)

VPn

W f l L M « ) = I W f \ \ L P A 0 V

In this paper we use the definition of the space with mixed norm from [2].

The notations and some parts of the proofs are analogous as in [3 ] ; for the sake of completeness we outline these results here.

3. We know from (17) [6 ]: If f e H 1 ,p{Q), then

< j \f\” d x } ' " ’" « 0 { J X I D ' j f l pd x } ‘ ,p’ , j = 1...

bQ '

( 6 ) n,

where r-} is the length of edge of Qjt j — 1, n, and hence j f |/|'’d x}1,p” « c 1(p)cJ. £ ! J | D » / | •’ d x }'1"".

bQ M =1 bQ

Multiplying the above inequalities for j = 1, n side by side and applying for f - f Q we get

( 7 ) ! J \ f - f Qi " d x } llPn < c(n, p ) П г/'" I ! j \D' f\’ d x } ' " ’\

bQ *'=i l«l = i bQ

L emma 1. Let Q{ be a cube o f the edge rt and let / е Я ш,р( 0 ; then fo r every system o f numbers [aa}|a| = m there exists a polynomial T (x)e # (x ) such that

(8) j \ f - 3 r ( x ) - U - ^ m Q \ P d x ^ c 2(n,p) П r", J ’ £ f m - a l i ’ dx.

bQ j = 1 |s| = m bQ

P ro o f. Lemma 1 holds for m = 1. Fixing N of numbers a lt . . . , a N, it is sufficient to assume

(9) ^ { x ) = Y , a i x i .

i= 1 From (8), really, we have

J \ f - ^ ( x ) - U ~ . r ( x n Q\ " d x ^ c 2 (n, p ) П ^{A Jn} L I J I D J - a i ’ dx.

bQ J=1 1=1 bQ

If m > 1, then the thesis follows by induction. Let us assume that the lemma holds for f e H m~i,p(Q), fixed numbers [aa}|a| = m. Then there exist N of polynomials ^~i (x )e i such that

(10) £ J ID J - F M - l D J - r ^ d x

« = i bQ

« с 2(и,р) n L f

j = 1 |s| = m bQ

N xi

For ,T (x ) = £ [ J ^ i ( x u . . . , t, xf + 1, . .. , x J d t + lD J - J T fx ^ Q - X i]

i= 1 0 by (7) we have

(11) f | / - ^ ( х ) - [ / - ^ ( х ) ] в | 'Л с

f \ D , U - ^ m pdx.

J=1 i=i bQ

From (10) and (11) we get (8).

R e m a r k 2. We observe, from the above, that ^~(x) is of the form

^ ~ (x )= £ as xs + R (x), R ( x ) e ^ m- 1.

|s| = m

T heorem 1. I f Qi is a cube o f the edge rh then fo r every f e H 2,p{Q) (12) i IIA / -(A / W l1P(ei ^ c3 \D2f\]ym ||/-/e ||{&,,

i = 1

where c3 does not depend on r{.

P ro o f. Let v e H 2,P(Q); then we know from Theorem 1, [6],

l ° ‘ v\m Q ) « { l » 2 < 4 > l l < 4 , + £ " 1 IW I

® } :

c4 does not depend on r; .

Studying the proof of the generalization of Theorem 3.2, [1], for the scalar p and studying the proof of Theorem 1, [6], we write

а з ) in 1 »\Lm> « <5 n < 4 1+ '-~ ‘ 'LP(Q)

where r = m axri5 i = 1, . .. , n.

Let = Y, aiXi + a о be a polynomial such that i = 1

(14) J If - P l \p d x = inf J \ f-P \ pdx.

bQ Pe* i bQ

Let us write (13) for v = / —P i, taking into account that r ■ЧП ri lln

i = 1

( I 5)

I m - a A \ ^ c 6 № 2/ ii/ 4 ) n / - p 1iii/ 4 ) + n '■r1/- w f - P A r J -

i « i =1 i=i

Applying Lemma 1 and Remark 2 we get

П n 11" р)\°гл]!г1 ^{LP(Q) ‘}

(16)

J= 1 By 1.10, [3], we know

WV ~ VÀ\

P(Q)

< 2in f lle - c ll^ e ,, taf ll»--P|l1, (e) « inf ll»-c|lu,B) « ll«’- ‘’o l U e,- From (15), (16) and the above two inequalities there holds (12).

T heorem 2. L et Q{ be a cube o f the edge r(; then fo r every f e H 2,p(Q) we

have

(17) £ inf | 1 А '/ -Л и е ) « с 7 £ inf|U>,Dj/-cH$ inf Н/-Л1[йв)-

i = l P e & i i,j- 1

where c7 does not depend on r,.

P ro o f. Let P 2 be a polynomial from such that I I / - E l l i n a = “ rf HZ-f’ lU g , and P 2 (x) = £ üij xf Xj + R (x), R (x) e ^ .

»j= i

By (13) for v = f —P l , we conclude (18) X iia / - d , p 2|

i= 1 ^LP(Q)

r f ‘'”1 1 /- P A w + {( I IIB. D jf -null L PQ )m I I /- A ll^ g ,} '

i = 1 i j = 1

Let be a complex number such that

Il A D J - b,i\ = inf II D, D j f - c|| i . i = 1... ».

ceC

We know by Lemma 1 that there exists a polynomial ^ " ( x ) e ^ 2 W such that (19) \ \ f - ^ ^ K m < c 9 W f ln E I I A V - b y l

f = l i,j= 1 LP(Q)-

By Remark 1 , 3 T { X ) = bij X,- Xj + R (x), R (x) e ^ . Applying Lemma 1, [7],

»'j=i

to the polynomial P 2 {x) — ^ { x ) , we get for every i, j

(20) _{f = l} П r r VP' Ifly-iyl « c 10 П '•rl',l| I A -^ W I I lP(01. _{t= 1 t- 1} i ^{HI = 2.}

By (19), (20) and from the fact that ]^[ r,-2/n) ij ^ 1, we have

i = 1

( 21 )

Let us write

N

п ^ к{1р' к - ь и1 ^ с п x i i A z y - а д

t=i •J=l LP(Q) •

(22) { £ I I A V - a y l l ^ e . ) U=1

< c 12 {( I m D J - h j W ^ J 11+ ( I Iny-byl П

ij — 1 ij= l < — 1

By (21) and (22) we get

(23) ! I \ \ D , D j f - a , j \ \ L P W ) } 112 i>j= 1

< Ci3( Z IID iD jf-b ,

i j = 1

I - Г '

'LP(Q)f

Moreover, we have

(24) П ^T ш r - ^2llt p(CI < П ^{' Г I I / - p}

i= 1 i = 1 Ii& )

Inequalities (19) and (24) yield

(25) П ' T 1," l l / - P 2 l l l p(ai < c 9 | | /-P

i= 1 11,2 ( I I I A V - ^ L o , )

1/2

'rP(Q) U=i From (18), (22), (25) we get (17).

Following Campanato we rewrite from [3] the next two lemmas:

L emma 2. I f B 0, B lf B 2 and a are non-negative numbers, then the next inequalities are equivalent:

(26) B 1 ^ c l0 (eB2 + £ aB 0 + B 0), £ > 0 , (27) B x ^ с 11(вВ2 + Е~а B 0), 0 < e < l , (28) B x ^ c 12( B f 1+a)B j /(1+a> + B 0),

where c 10, c n , c 12 are positive constants which do not depend on B 0, B\, B 2, £.

L emma 3. I f B 0, B x, B m(m > 1) are non-negative numbers and fo r every j, 1 < j ^ m — 1 satisfy the inequality

(29) B j ^ c ^ d U s B j ^ + e - ' B j ^ ) , 0 < e < 1, then there exists a constant c lA( m ,j) such that

(30) Bj < c i4 (m, j)(£m~j B m + £~j B 0), 0 < e < 1.

T heorem 3. F or every / e H f ,p((2) there exists a constant c(n, p) such that

(31) I I I I A / l l l i f . W M x ll|D,VIII*.fî1«i.),,2|ll/llli^<ffl-

I = 1 i,j = 1

P r o o f. For every cube Qt <= we write inequality (17). Dividing both

sides of this inequality by ]~J Lu (Qd] A,/k'P' we get i = 1

X 1 № , ) У ^ кт inf \\DJ-P\\m o .

i= 1 Pei?!

« C l ! x Ы < Ш ~ 4 " ‘m inf II Д 0 / -с | | ^ в)! ‘ '2 X

i , J = 1 C6C

x l l î [ р ( Ш Г 2,'"“'’' inf Н/-Л11Л0)) 1/2-

i = 1 Pe;3»2

Taking into account (2) we get (31).

T heorem 4. F or every function f e H j,p(Q) n L$,P(Q) the follow ing inequali­

ty holds:

(32) £ III A / lllif« ,« « c7 (n, p) ( £ III A Djf\\\L^ ( a ) ^1/2

i= 1 i.j= 1

1

2

L% Ш)’

where & = ( i i + Jui)/2.

П

P ro o f. Analogously as before we divide (17) by Yl [^(6/)] (2, + k,)/2fc,p‘

i — 1 and by (2), we get (32).

T heorem 5. F or every function f e H f,p(Q) the following inequality holds:

(33) X HA/ILg^m, «

1= 1

P ro o f. We get (33) repeating the proof of Theorem 1, applying (12) instead of (17) and taking into account (3) and (4).

T

6 . F or every function / е Я ? , р ( 0 ) , 0 ^ л , < /с; + р ь there exists a constant c 15(n, p) such that

(34) \ D l f \ L P 0>\n) ^ c l5 (n, P ) {\ D 2f \ l^ \ a ) II/ llLg’A№) + l l / l l L g ’W -

P ro o f. We take the thesis of Theorem 1, [6], for the function f - P , then

Л

we divide both sides of this inequality by П r fiPn/ki and for every e > 0 we i = 1

get

Z IIIA/III lp . a №) ^ c l6 (n, p) [e т о / \ \ ш Ш)+ е- 1 lli/llli

i = 1 i , j = 1 р2Лп)\-

We have also from Theorem 1, [6], that for every a > 0

ID'f\LP(m « P) !s l£,2/lLP,a + e" 1 + ll/lU i » } ’ Summing the two inequalities side to side we have

(35) \Dl f\Lp,\m «S c I8 {£\D2f\L^ {a + r l W f W ^ + W fW ^ }- Inequality (34) follows by (35) and by three next relations (a) IIf\\LP(n) < \\Л\ьрЛп) ЬУ definition,

(b) Ц л (й) and L f A(f2) are isomorphic to Lg,A(0 ) for 0 ^ c & j + p, (see Theorem 1, [9]),

(c) (26) and (28) are equivalent.

Applying the definition of C m,lx(Q) from [8 ] and suitable the theorems from [8] and [9] we can prove analogous theorem as Campanato for p scalar in [3].

T heorem 7. F or every function f e C 2,1 (Ü)

(36) iDlf l L{ ’"+1(n) ^ c 1 9 {\^2f\c 0,l(t2) II/II l L "+ l(fi) + II/II l J’" +1(fi)} •

C orollary 1. I f f e H ^ p(Q), m ^ 2, 0 ^ A{ < kt + pit then fo r every 0 < j < m there holds the inequality

( 37 ) \ D > f \ ^ « c 20 { | / Г / | 1 '?Д(п) Ц/ l l i ^ â + Ц/П^Двд}.

P r o o f. By Theorem 6 and by the equivalence of (26) and (28) we have for every s e ( 0, 1] and 1 < j < m

l^/kg-Vo, « c21 {6|iy+I/ l L ^ ( » + ^ 1 Hence by Lemma 3 we get for every £e(0, 1]

« C22 {e“--'|D"/|i f* (0) + a--'ll/llL^ffl}.

From the above inequalities and from Lemma 2, (37) follows.

C orollary 2. I f feH ™ ’p(Q), m > 2, 0 ^ < /c.-fft, then fo r every e > 0 and 1 < j < m there holds the inequality

(38) 11/11я{’р(Я) ^ C23 {£m J \\Л\н™’РШ) + £ L$X(f 2 )}- P r o o f. From (37) we get for £ > 0,

|O '/ lig -A(0> < ^c2i { e " _ J |D “ / li.g ^ (i))+ £ ” ■' ll/llr g '^ c i + ll/llrg '^ m i

and then analogously as in [3]

\ & Л ь р 0 ’\п) < c 25 {em-J II/II h ^>p(^ + e_J II/II lp .^ o )} • At the same time from Theorem [6] we have for e > 0,

Wf\\Hj,P{n) < c 26 {e J \\f\\Hm,P{Q) + £ J' l l / l l L p ( f i ) } . Summing side to- side the two above inequalities we get (38).

References

[1] S. Agm on, Lectures on elliptic boundary value problems, d. Van Nostrand Company, Canada, 1965.

[2] A. B en ed ek , R. P an zo n e , The spaces U with mixed norm, Duke Math. J. 28 (1961), 301-324.

[3] S. C a m p a n a to , Maggiorazioni in terp o la to r negli spazi Hmp(Q), Ann. Mat. Рига Appl.

75 (1967), 261-276.

[4] —, Proprieta di una fam iglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa 18 (1964), 137-160.

[5] E. G a g lia rd o , Proprieta di alcune classi di funzioni in piu variabili, Ricerche Mat. 7 (1958), 102-137.

[6] M. Ja ro s z e w s k a , On interpolation inequalities with mixed norm, Comment. Math., this fasc., 227-237.

[7] —, On interpolation in the IT,4>-spaces with mixed norms, Functiones Approx. 8 (1980), 119-128.

[8] —, On the spaces (Q) with mixed norms. I, Comment. Math, (to appear).

[9] —, On the spaces L%l (Q) with mixed norms. II, Demonstratio Math. 15 (1982), 87-100.

[10] L. N iren b e rg , Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8 (1955), 648-674.

INSTYTUT MATEMATYKI UNIWERSYTETU im. A. MICKIEWICZA, POZNAN