Integrali Singolari e Questioni Connesse 3642109160, 9783642109164, 9783642109188 [PDF]

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G. Fichera x E. Magenes (Eds.)

Integrali singolari e questioni connesse Lectures given at the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, June 10-19, 1957

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-10916-4 e-ISBN: 978-3-642-10918-8 DOI:10.1007/978-3-642-10918-8 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 ed. C.I.M.E., Ed. Cremonese, Roma 1958 With kind permission of C.I.M.E.

Printed on acid-free paper

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CENTRO INTERNATIONALE MATEMATICO ESTIVO (C.I.M.E)

Reprint of 1st ed.- Varenna, Italy, June 10-19, 1957

INTEGRALI SINGOLARI E QUESTIONI CONNESSE

E. Magenes:

Il problema della derivata obliqua regolare per le equazioni lineari ellittico-paraboliche del secondo ordine in m variabili ..........................................

1

G. Stampacchia:

Completamenti funzionali ed applicazione alla teoria dei potenziali di dominio ...................................... 53

A. Zygmund:

On singular integrals .............................................................. 69

S. Faedo:

Applicazione ai problemi di derivata obliqua di un principio esistenziale e di una legge di dualità fra le formule di maggiorazione ............................ 109

G. Fichera:

Una introduzione alla teoria delle equazioni integrali singolari ................................................................... 127

MAGENES, ENRICO 1957 Rendiconti di Matematica (3-4), Vol. 16, pp. 363-414

II problema della derivata obliqua regolare per Ie equazioni lineari ellittico-paraboliche del secondo ordine in m variabili (*) di ENRICO llIAGENES (Genova)

n. 1 Esempi intt·oduttivi. - Uno dei pitt importanti e prIml problemi cbe banIlo portato allo studio degli integrali singolari e delle equazioni singolari e it cosiddetto problema della derivata obliqua nella teoria del potenziale ordinario (v. ad es. [9aJ (i)). Esso cOIlsiste nel deterlllinare una fnnzione armollica u in Ull campo A dello spazio a 3 dimensioni quando sia assegnato sulla fl'ontiera .s di A it val ore della derivata secondo una direzione prefissata l, in generale variabile da punto a punto, cioe:

au = aI

(1)

h su 2

III ipotesi di opportulla regolal'ita sui dati del problema, cite per ora non e il caso di pl'ecisal'e, quando si cercbi la soluzione di (1) sotto la forma di potenziale di semplice strato

(*) Qllesta Memoria ripl'odnce IJrevemente Ie lezioni svolte dall' A. nel 2 0 cicIo di corsi del Centro Internaziona.le Matematico Estivo (CIME) tenuto a Val'enna dal 10 al 19 Gillgno 1957 sn «Integrali Bingo lad e qlteBtioni conneBBe ». (1) I llllmeri tra [] si riferisoono aHa bilJliografia finale della presente relazione.

1

2

[364]

ENRICO MAGENES

si e COlHlotti allo studio della derivata obliqua del potellziale (2) ed e facile, se cp e sufficientemellte l'egolare, trovare per questa derivata la formula limite: I,

(3)

1m

cos (n; , 1;)

. 0 1t (x) _

+

-----;;z; - -

2

(1:) cp s

+ J*cp ()y -0-40 (t , ,I}) d 0" 8

4

X~o (811 ttl; )

dove no e Passe normale a:4 nel punto ~ rivolto verso l'illterno di A e Pintegl'ale con asterisco e da intendersi q uale illtegrale prilleipale alla Canel.y, cioe come

, J

(4)

lun ,'0

,a 8 (~,

(p (y)

y)

a ll;

4-4.

Ii

0"

essendo :4 e la porzione di :4 clle si proietta suI piaHo tangen te a :4 nel punto ~ nel cercbio di centro ~ e raggio e (e> 0) (2). II nucleo 0 8

(~ ,

o 11;

Y) , salvo nel caso 1

non e sommabile su 2 esso

e nell'illtOl'110

di

~

=n

su 2 (problema di Neumann),

(come fUllziolle di y pel' ogni

~

fissato):

del tipo 0 ( 1 ) (3), ma tuttavia esiste il ~y2

limite (4). Si arri va COS1 per (1) allo studio dell'equazione integrale singolare - cos (nl; , 11;) cp (~) 2

(5)

+ J* cP (y) 0 s (~ , y) d Oy = h (~) 04

4

nell'incognita cp (~) • Un problema analogo a (1) puo porsi (ed e Y3

si trovano (come si vedra pili esattamente nel n. 11), se cp e sufficientemente regolare, la formula limite analoga alIa (3)

(8)

I ,lIn

+) x~g(su v
ko:

Quilldi la snccessiolle estratta Uk) COIl verge nil ifol'Ulemell te fllori di 00

(E t:1a)

U Bk k~J

per ogni scelta di k o ' D'altra parte si ha, perla 4):

59

58

[420J

GUIDO STAMPACCHIA

e quindi per la 4):

Di qui si deduce cite la snccessiolle estratta converge Illlifol'memellt.e ill tutti i pUllti - oi Tao eccezione di quelli di un insieme I di [, per cui # (I) e illfel'iore ad un llumero positivo pl'efissato. Consideriamo inune l'insieme: E

=

= n 00

lim sup. Bk

00

U Rk

8=1 k = s

ed osserviamo elle, essendo, qualullqlle sia I'inoiee

8:

00

e sussistendo la (3), E e Ull insieme di [,. La successiolle estl'atta cOllverge in tutti i pUllti di T clle non appartengono ad E e cio completa la dimostl'azione del teol'ema (8). II teol'ema precedente permette di risolvere it problema del completamento fUllzionale come fOl'mulato all'inizio; infatti ad ogni successiolle di CaucllY di S U;,l possiamo associare almeno una sllccessiolle estratta (lnkl elle con verge, come detto nell'enunciato precedente, ad uua fllllziolle- defiuita ill T a meno di illsiemi della classe eccezionale [,. Quindi ad ogni PUllto di S* velliamo ad associare almeno una funzione definita a menD di insiemi di [,. Siano ora I' ed j" due fllllzioni associate ad uno stesso punto di S* secondo il criterio precedente; segue facilmente clle # (E

[I' (x) =1= j" (x)j) =

0

(8) La dimostrazione del teorema e analoga a quella del ricordato teorema di Weil- Riesz relati vo alla oonvergenza in lllisura di una sottosuocessione convel'gente fort ellente in LV. Che la dimostrazione di questa teorema lion sfrutta l'additivita della misul'a, rna solo la Bubadditivita e stato osservato da diversi Autori: Cartan, Deny, Stampacohia, Deny - Lions, per particolari tipi di funzioni d'insieme suubaditive. Una prima formlllazione astratta oi questa teorema e stata data da F. Cafiero [2J; successivamente Aronszajn e Smith hanno dato ]a fOl'mulazione rip or taLa nel testo, ohe l'innncia rispetto a quella di F. Catiero - all ' ipotesi ehe la fnnzione f-' (I) sia definita in nna famiglia cOlDpletamente additiva.

60

[ 421]

COlllpletalllen ti funzionali ed applicazione alla teoria ecc.

59

cioe

E [f' (x) =1= f" (x)l E S . Ma di pin, possiamo dire che Ie funzioni di S sono «quasi continue» se Ie fUllziolli della classe S sono continne (9). Uua consegllenza immediata di quanto dimostrato e Ia seguente: Supposto che le funzioni di S sianu continne e che in et si abbia:

c (B)"? m

>0

pet" ogni

B=I=0

si 1IUO concludel'e che le fUllzioni iii S sono anche ('sse cOlttinue. Infatti in tal caso il teorelll3 preceden I.e assicum che Ia COllvergenza delle successioni di Oauchy di S unifol'llle e quindi 10 spazio S e costituito da funzioni continue.

e

5. Diamo ora un semplice esempio - a titolo illustmtivo di quanto detto precedentemente. Sia S costituito da funzioni di una variabile f(x) contiuue con Ie derivate prime Ilell'illtel'vallo 0 n Ie funzioni di S Sono differellziabili secondo Stolz (14); se illvece p n Ie funziolli f sono differenziabili asiutoticamellte di ordine l' secondo Ulla nozione illtrodotta indipelldelltemente da Rado e da Caccioppoli e Scorza Dragoni e generalizzata da Tibaldo e poi da Pezzaua [l1J. Una funzione f(P) di 8 gode infatti per quasi tutti i puuti di T della seguente proprieta:


0

11. (x) I.

In view of Theorem 1 of § 1, the function ;: is finite almost everywhere for each f E LP, 1 l. We may consider a generalization of the transform (7) to spaces of higher dimensions. IJet e1 , e2 , ••• ,en be a system of n mutually orthogonal non zero veetors in En. Consider the set of all lattice points h generated by this system. Hence the h are of the form fl1 e1 fl2 e2 fln en, where fl1' ,fln are arbitrary integers. Since the set of all lattice points is denumerable, we may arrange them into a single infinite sequence I hq I = (h o : hi , .•. ) . Given any infinite sequence of numbers I Xr 1= (XO , Xi , ••• ) we may consider the transformation

+

"'2 , ...

+ ... +

~

(9)

X'l

=

+00

2' K(h q

hr)xr ,

-

(q

=

0 , 1 , 2 , ... )

r~O

which obviously generalizes (7), and for which we have the following analogue of l\L Riesz"s theorem: 8. If the kernel K (x) = Q (x')1 Ix In satisfies the conditions (1), then the transformation (9) is from lP into lP, for p strictly THEOiml\I

greater than 1 . .iJ1ore precisely, if X = l;;q 1, then I ~Yllp < Ap II X lip· We shall now consider singular integrals for periodic functions. We recall that parallel to the theory of Hilbert transforms (8) we have a theory of conjugate functions

(10) -n;

The function f here is initially defined in (- n ,n) and then extended to all X by the. condition of periodicity. The properties of conjugate functions are close to those of Hilbert transforms (8),

79

On singular integrals

[477]

77

and this is not surprising since using the formula

(11)

1 (1 -cot -1 x = 1- + +00 ~ - -1-) , 2 2 x n~-oo X 2 11: n 2 :n n

+

we can put (10) in the form l(x) =

~

r

+00

f

11:.

-00

(y) d y •

x-y

We can apply the same procedure to the general kernel K (x) • Suppose for simplicity that the orthogonal vectors e 1 , e2 , ••• ,en are all of length 2 11: and set (12)

K* (.-r)

=

K (x)

+ +00 ~' jK (x + h

q) -

K (h q )\

q~l

assuming, for simplicity of notation, that ho = 0 . Using the fact that Q E Lip IX, it is not difficult to see that the series (12) converges uniformly over any finite portion of Ii] provided we drop the first few terms of the series which have singularities there; the function K* (x) is of period 2 11: in each of the components ~ j of x . Let Qn be the hypercube in En with center at the orig'in and half-dimensions 11:. Consider the function

n,

(13)

f(x) = ff(y) K* (x -

y) d y.

Qn

Since K - K* is continuous in Qn, Theorem 1 implies that the integral (13) exists almost everywhere in Q", and from Theorem 2 we can deduce without difficulty the following resuit: THEOREM

9. If f is in LP on Qn, where p

~ inLP, and llfllp 1,

stands for

then

f

is also

(f If(x) Ip dx )l /P. Q"

We conclude this section with a few words about singnlar int.egrals on curves.

80

'is

[478]

A. ZYGMUND

C be a rectifiable curve in the complex plane, and let f (C) be and integrable function on C (that is, f an integrable function of the arc length). Consider the integral I~et

j

(14)

f(C)dC

z-C '

a

where z is also on C. By the pt'incipal value of this integral we mean the limit, as e -. 0, of

j z-C1 ' f(C)d

as (0)

where C. (z) is the part of the curve C which is outside the circle with center z and radius e. One would expect that under these conditions the integral (14) would exist at almost all points of C; whether this is so we do not know, and the best result so far obtained is the following THEOREM 10. If the curve C has bounded curVlttUt'e and. x and f is integrable on C, then the integral (14) exists, in the sense of principal value, at almost all points of C. D~finitions of singular integrals can be extended from Euclidean spaces to curved varieties, but very little is known in the general case, and we do not discuss the topic here.

§ 3. Given a function j (x) =

Fourier transform of

f

(I; 1

, •••

,I;n), we denote by

/'0-

f the

f,

f(x) = (2 n) -

+ff n

(y)

(}-i(xy)

dy ,

+ ... +

where (x y) stands for the scalar product /;1 'f/l /;n 'f/n of the vectors x = (1;1"'" I;n) and y = ('f/t , .•• ,.'f/n)' We take for granted /'0elementary properties of the Fourier transform f, in particular the

f

facts that if fE L2, then exists (in the metric LZ), that = II f 112, and that we have the inversion formula

(1)

f(y)

=

(2

n)-+nj.i(X) ei(xy) r7.1:, 81

IIill2 =

On singular integrals

[479]

79

Moreover we shall need the fact that if we define the convolution h of functions ! and g by the formula

h (x) = (2 n)-

+f! n

(y) g (x -

y) d y ,

and if one of the functions!, g is in L and the other in L2, then

h=!

(2)

g.

/'..

Since a singular integral! is a convolution of two functions ! and g, we may expect (2) to be valid in this case in some sense, and we are led to study the Fourier transform of the kernel K (x) = = Q (x') / I x In. The kernel being not integrable near the origin, we must first define the Fourier transform of K. Let K e ,'} (x) be the function coinciding with K for IJ < I x I < 'YJ and equal to 0 otherwise. "Ve define the Fourier transform of K by the formula K (xl

(3)

=

lim lim K e,') (x) , e---+O

't}-HXJ

/'..

and we first study the behavior of K e ,'}. We snppose temporarily that Q is merely bounded and satisfies the usual condition

r

Q (x') d x' = 0 .

I

I 1=

I I=

We introduce polar coordinates and set x r, y (2 , (x y) = r e cos cp. We have d y = e"-l d e d a, where d a stands for the element of area of ~, and (2 n)n/2

(4)

Jr.,') (x) =

J

da

I

'}

e- 1 Q (y') e-irecoAip (7 e =

:E

=

r

. :E

Q (y') il e

/7~-iecos'l'

. er

e

de.

Let g (e) be the function eq nal to 1 for e < 1 and to 0 for e

>1.

Then, in view of the condition

I

:E

82

Q (y') d y' = (),

the right

80

[480]

A. ZYGMUND

hand side of (4) is

J J 1Jr

de·

e

er

:E

g (e)

e-iecosrp -

Q (y')

Denote the inner integral by I. vVe will show that (.5)

1II

< log 1cos1 ffJ 1+ 0,

where 0 is an absolute constant. Consider first the case 13 r < 1

R~l

1

1=

e-iecosrp -

1

e

er

say. The inequality

1 eit -

d

----;::- d

'Jr

( e-iecosrp

.

e+

-e- d e = + 1 II

.

1

1

1

1

(! / •

"

1

We have

J

J

R e-ie

I

sup

01 =

r; 1", and set


1 the situation is similar. In the former, using the same estimate as for It above, we obtain 1 II < 1; and in the latter

+

111=/ J

J

I --glle e- ie ~e-(lel
0

11.(x) I

(thus /; is always non-negative, possibly infinite). It is known that the one-dimensional inequality (6) can be strengthened to (p> 1)

(8)

88

86

[486]

A. ZYGMUND

an inequality which is equivalent to saying that for any measurable step function e (x) we have II h e(x ) (x) lip < Ap II h (x) lip,

(p> 1)

where Ap is a constant depending on p, but not the choice of e (x) • Suppose now that the e in (2) is a function of x, say a step function (by a step function in En we mean a function constant in a finite number of non-overlapping n-dimensional rectangles and 0 elsewhere). We have Ih(X) (x) I::::;:

~

J

IQ

(t) ge(x) (x , t)

Id t
O

and so also

~ (x) :::;;: 2, 1 j~

I' I Q

(t)

I g* (x , t) d t ,

:?:

from which, by the previous argument, we deduce

I IJ~ (x) lip
1).

1.

This inequality implies that < 00 almost everywhere, and in particular that at almost all points x the integral h (x) remains bounded as e ~ O. But at also implies easily that for almost aU x the limit l(x) = lim h (x) necessarily exists. To see this we observe that exists everywhere if f is continuously differentiable and, say vanishes outside a sufficiently large sphere. The class of such functions f is dense in every LP . Now, if we set

1

() (x)

= () (x, f) =

lim suph (x) -

._+0

lim infh (x), e~+O

then (9) implies that (10)

II () (x ,f) lip < 2 Ap II Q 111 IIflip·

Now () (x ,f) = a(x, f- g) for any g such that g (xl = lim g. (x) exists and, selecting g such that Ilf - gl lp is arbitrarily small, we deduce

89

On singular integrals

[487]

that

1

87

110 (x , f) lip =

0 , (} (x , f) = 0 almost everywhere, so that = lim]; exists almost everywhere. Applying this to (7) we see that

(11)

Let us summarize the results obtained so far in this section. Taking for granted results for the one-dimensional Hilbert transform, and assuming that the function Q is odd and integrable over ~ (this, of course, implies that the integral of Q over ~ is 0) we deduced that 1(x) = lim]; (x) eX'ists almost everywhere, and that l zed by a function in LP; in particular

is rnajori-

111-];llp = o.

lim

The method we used, which we may call the method of rotation, may be applied to more general kernels, already considered in § 2, provided the kernels are odd. Oonsider the integral (12)

-ff (- ) I I

f~• (x ) -

y Q (xyn, y') d y,

x

1!l1;;;;':·

where f is in L p , p > 1, and Q is a function of the variable y' E ~ and of the parameter x E En. We assume that Q is odd in y', and that there exists a function Q* (y') integrable on ~ and such that

IQ (x , y') I ~ Q* (y')

(13)

for all x. If g. (x , t) has the same meaning as above, then arguing as before we have

~ (x) f.

IJ

="2

Q (x , t) g. (x , t) d t ,

I

11. (x) I< ~ JIQ* (t) II g. (x , t) I d t , I

from which it follows that

II]; (x) lip
1).

88

[488]

A. ZYGMUND

The last inequality can be extended to

h, (x) =

where

=

sup

_>0

I]; (x) I,

which, as before, implies that l(x)

=

lim.f. (x) exists almost everywhere.

§ 5. In this section we cousider the limiting case p = 1, and we show that if Theorem 1 is valid for p = 2, then it is also valid for p = 1. Thus the result is of conditional nature, but is of interest since in the preceding section we showed the validity of Theorem 1 for odd kernels and any p > 1. The method we use is itself of interest and has wider application. We begin with the proof of the following lemma. LEMMA I. Let P be a bounded perfect set in En, and A a sphere containing P. Let b (x) be the distance of x from P. Then for any A. 0 and almost all points x in P we have

>

(1)

I (x) =

f

0" (y) y In+l d y

!x _

0)

with v = If I and w = I log 1/1' I, and it is easy to refine the boundedness of tt to continuity. The result fails to hold (tt may be everywhere unbounded) if we replace the condition f E L* by a weaker one. If we differentiate the integral defining u formally with respect to x or y we obtain a convolution off with the kernel x/(x 2 y2) or y /(x 2 y2), as the case may be. Both kernels are locally integrable, and the differentiated integrals converge absolutely almost. everywhere to functions which are locally integrable. In particular u", and u y exist almost everywhere and u is an absolutely continuous function of y for almost every x, and vice versa. This is true under the sole hypothesis that f is integrable. We now pass to the second derivatives of u. If f is continuous and satisfies a Lipschitz condition of positive order, there exist classical formulas expressing u"''''' ttwy ,U yy in terms of Hilbert transforms of f, with kernels

+

+

It can be shown these formulas hold almost everywhere if f is in L*. vVhether these formulas hold for f E L, is still an open problem,

and it is conceivable that in this case the second derivatives need not exist in the classical sense. That certain results may fail to hold if we pass from L* to L may be seen from the following fact. Suppose thatfE L*. vVe have

106

104

[504]

A. ZYGJlIUND

just mentioned that in this case u has almost everywhere the derivatives ttxx , ttxy , U yy • But another result holds in this case: if f EL"', then u has almost everywhere a second Peano differential, that is

+ h , y + u (x , y) = A h + B k + + ~ C h + D h 7c + ~ E + (h + 7c

u (x

Te) -

2

Te2

0

2

2 ).

This, of course, implies that u is bounded in the neighborhood of almost all points, and also indicates that the integrability of f log+ If I cannot be weakened here siuce otherwise tt may be everywhere unbounded. If we set k = 0 in the last equation, we obtain u (x

+ h , y) -

tt (x , y)

=

A h

+ 21

C h2

+

0

(h 2 )

•

This indicates that the second derivative of tt with respect to x in the Peano sense exists almost everywhere. The result is weaker than the existence almost everywhere of the classical second derivative, but in is not impossible that in this form the result is extensible to functions f of the class L. We conclude with a few words about the Newtonian potential in the space En with n > 3 :

L-i-

f E +e, then it is a simple consequence of Holder's inequality that u exists almost everywhere and is continuous; if f is only in

If

n

L 2, U may be everywhere unbounded. Suppose now that f E L*. Then, as in the case of the logarithmic potential, u has almost everywhere all second derivatives, and these derivatives are given by classical formulas. The second differential, however, need not exist at a single point since already in the case when f E Ln/2 the potential tt may be everywhere unbounded. If however in u (x) = U (~1' ~2, ... ,~n) we fix ~1' ~2 , ••• , ~n-2, then for almost all choices of (~1 , ••• , ~n-2) in En-2, tt has almost everywhere in E2 a second differential with respect to ;n-1 and ;".

107

[505]

105

On singular integrals

BIBLIOGRAPHICAL

NOTE

Tho ono-dimensional Hilbert transform is a classical topic_ Hilbert transforms in higher dimensions seem to have ben first considered by Tricollli (for 11 = 2) and Girand, bnt the introduction of the Lebesgue integral and the modern theory of operators seems to be due to Michlin. Miohlin's main work is discussed in his expository article Singular integral equations, Uspekhi Matemati~eskich Nallk, vol. 3 (1948), No.3, pp. 29-112 (there is an English translation in Allleri"an Math. Soc. translations, No. 24 (1950». It also contains a discussion of, and references to, the earlier work of Tricomi and Giraud. The developments (8) and (9) of § 6 will be found there (proved by a totally dift'erent argument). The series 13 of § 6 oocurs already in the note of Giraud Sur une class gelllJrale d'equations integrales p1'incipales, C. R. de l' Acad. :"ci. Paris, vol. 202 (1936), 2124-2125, but no Fonrier integral is menLionell, no proofs are giv6u, and it remains a mystery how Giraud arrived at the development, though he explicity mentions the fact that to the composition of integrals corresponds multiplication of the devel(lpments, an obvious hint nowadays to Fourier integrals. That the development is actually the Fourier ntegral of K is implicitly contained in the paper of Bochner, Theta j'elatiolls with spltel'ical ha1'11!onics, Proc. of the Nat. Acad. USA, 37 (1951\,804-808, which contains the formnla (11) of II 6; see also Michlin, On the theOl'y oj nwltidimensional sing'llar integral equations, Vestnik Leningrad University, Series of Math., Astronomy and Mechanics, Vol. 1 (1956), No. I, p. 1-24. Theorems 5 and 6 about iterated Hilbert transforms are proved by M. Cotlar, Some generaliziations oj the HardyLittlewood maximal theorem, Hevista Matelllatica Cuyanil, vol. I, fase. 2, pp. 85-104. A. P. Calder6n's and the author's work is contained in the foll()wing papers: (i) On the existence oj certain integrals, Acta Mat. 88 (1952), 85-139; (ii) On a problem oj Michlin, Trans. American Math. Soc. 78 (1955), 209-224 (Addenda, Ibid. 84 (19~7), 559-560); (iii) Singular integrals and periodic jnnctions, Studia Math. 14 (1954), 249-271; (iv) On singulal' integrals, American J. of Math. 78 (1956),289-309; (v) Algebras oj certain singulm' operatOl's, Ibid. p. 310-320; (vi) Singular integral operators and differential equations, Ibid. 79 (1957), PI). 801-821.

a

[Entrata in RedaziQne il 29 novembrc 1957]

109

FAEDO, SANDRO 1957 Remliconti eli Matelllatica (3-4), Vol. 16, pp. 515-532

Applicazione ai problemi di derivata obliqua di un principio esistenziale e di una legge di dualita fra Ie formule di maggiorazione (*) di SANDRO FAEDO (Pisa)

1. - Nel convegno iuteruazionale snlle equazioni aIle derivate parziali tellllto a Trieste nell'estate del 1954 G. FiclJera lJa comuIlicato Ull principio generale di esistenza nell' Allalisi lineare, clJe gli lJa permesso di dare una tmttazione unitaria a llumel'osi problemi esistenziali. Siano V un insieme astratto, lilleal'e rispetto al corpo reale [complessoJ e Bi e R~ due spazi di Banach l'eali [colllplessiJ. Siano definite in V due trasforrnazioni lineal'i MI (v) e M2 (v), aventi codominio rispettivamente ill Bl e B 2 . Sia assegnato Illl funzionale lineare e continuo if! (Wi)} definito in Bl e si considel'i l'equazione

1) nell'incognita IfF (w 2 ), essendo IfF (w 2 ) un funzionale lilleare e continuo defiuito in B 2 • II principio esistenziale di Fichera si pub ('oSI ellllnciare: «Condizione necessaria e sllfficiente affinche esista la solnzione dell'equlIzione 1), dato comllnque if! , e che esista una costante K, tale clJe sia per ogni v c V 2)

II

M{ (v)

II