IRREGULAR AMALGAMS

The amalgam of L p and q consists of those functions for which the sequence of LP-norms over the intervals [n,n+l) is in q. These spaces (L p,q) have been studied in several recent papers. Here we replace the intervals [n,n+l) by a cover {In;neZ of the real line consisting of disjoint half-open intervals I n each of the form [a,b), and investigate which properties of (LP,q) carry over to these irregular amalgams (LP,q) In particular, we study how (LP,q) varies as p, q, and vary and determine conditions under which translation is continuous.


INTRODUCTION.
The amalgam of L p and q is the space (LP,q) of functions f which are locally in L p and satisfy f!
[ [[n+l if(x)iPdx]q/p}I/q < =, (I.I) p,q -n n=that is, the LP-norms over the intervals [n,n+l] form an q-sequence.(If either p or q is infinite, the expression in (I.I) is modified in the usual way.)Special cases of amalgams were first introduced by Wiener [I], [2] and Stepanoff [3] in the 1920's but their first systematic study is due to Holland [4].
Notice that for p q we have (LP,p) L p and indeed some of the properties of Lebesgue spaces can be carried over to amalgams.For instance, if I/p + I/p' I, and the analogues of Holder's inequality and Young's inequality are also valid.
It is not difficult to establish the following inclusion relations between amalgams.
In this paper we propose to replace the cover consisting of .theintervals In n+l) in (I I) by a cover {I ;neZ} of the real line consisting of disjoint half-open intervals I each of the form [ab), whose union is the real llne.
n Throughout the paper we use the term cover to refer to a cover of this type.If 'f!
{" [fl If(x)IPdx]q/p}I/q P,q, n n we define the irregular amal.am(L p q) {f; flp,q,a Such spaces have arisen in the work of Jakimovski and Russell [6] on approxima- tion theory.Given a complex sequence (Yn)nEZ and a linear space S of real-valued functions, they considered the interpolation problem: For a given increasing sequence (=), find a function f in S such that f(a Yn" For certain normed n n sequence spaces E they took S to consist of the functions f with L p an and showed the existence of optimal solutions to the interpolation problem.Notice that if we take E q then S becomes the irregular amalgam (L p q) where n An important special case of an irregular amalgam is the dyadic amalgam (L p,q) where {[n,n+l)} is the cover given by o O, n 2n if n > O, -2 -n If n < O. J.W. Wells proved that if f g LP(--,-), < p 2, then its n Fourier transform f lies in the dyadic amalgam (L p',2). (Kellogg [7] and Williams [8] proved similar results for the circle group and connected groups, respec- tively.) Certain facts about irregular amalgams are straightforward generalizations of kno results about regular amalgams.Thus we state the following theore without proof.
THEOREM I.If p,q < , the irregular aIgam (L p, q) is a Banach space * (L p ,q whose dual space is (L p q) L THEOREM 2. If f e (LP,q) and g e (L p ,q ), where p,q I, then fg e and lfgH |fl; glp, P,q,a ,q , Other aspects of regular amalgams do not always generalize to irregular amalgams.For instance, as stated in [5], although the inclusion (1.2) holds for irregular amalgams, (1.3) does not hold in general.In fact we show in Section 2 that the analogue of (1.3) holds if and only if the lengths of the intervals I are n bounded above.
In Section 3 we discuss inclusion relations between irregular amalgams when p and q are fixed and the sequence varies.We write .B if the intervals I E n intersect boundedly many of the intervals of B and show that if q p and = B, then (LP'q)c (LP'q)B with strictness if =.If q p, the inclusion is (L p Eq) if and only if = B and B .
reversed.It then follows that (L p q) B We write the latter conditions as B and this defines an equivalence relation on the set of all amalgams.
In Section 4 we investigate conditions under which translation is a continuous operator on (Lp,gq) In particular we show that all the translation operators are continuous, with uniform bound on their norms, if and only if = p, where On {[n,n+l)} is the cover that gives the regular amalgam.
Finally, in Section 5, we discuss generalizations to functions on measure spaces and groups. 2.
THE VARIATION OF (Lp,gq) WITlt p q In this section we consider the irregular amalgam (L p,q) defined by a fixed cover a {I and investigate how it varies when p and q vary.n The variation with q is an immediate consequence of Jensen's inequality [9, p. 28]: q I/q2 q I/q [ lanl 2] < [. lanl I] I, 0 < ql < q2 Taking an Ill If(x)IPdx] I/p, we have the following inclusion.When (LP,q) varies with p, the truth of the corresponding inclusion will depend on the lengths llnl of the intervals In E a"

P2 Pl
Then (L ,gq) c (L ,q) if and only if the set THEOREM 4. Let Pl 'q'= n n Pl n n P2 n p2,q, and so where K is an upper bound for the set {II n P2 Pl (L ,q) c (L ,q) q(I/PI-I/P2) nZ}.It follows that Now assume that {llnl;nZ} is not bounded above.Without loss of generality we may assume that llnl lln+ll. (If not, we take a subsequence, reorder if neces- sary, and assume the function constructed below is zero on intervals not in this subsequence.)Observe that if f is constant on I say f(x) c on I then n n n P2 Pl llf c II I/p.In order to construct f E (L q) with f (L q) we shall n p n n choose the c's in such a way that n while 1/P2 ]q To do this we use the following fact due to Stieltjes [I0, p. 41]: If a x 0, then there is a sequence (d n) of positive numbers such that .d diverges while .adn n n converges.If we choose 0 since II -.By the result of Stieltjes there exist numbers d with then a n n n , .a d < and d .Let This shows that (L ,q) is not contained in (L ,gq). 3.

THE VARIATION OF (L p q) WITH
In this section we fix p and q and consider how the irregular amalgam (L p, q) varies when the cover = varies.For that purpose we need the inequalities provided by the following lemmas.l, then N-q/P'( I If q) Ifl q Nq/q' I If q) n=l n p p n=l n p PROOF: (a) Since q/p I, Jensen's inequality gives If lq [I f IP] q/p I If q p n p n p (b) Here q/p and the result again follows from Jensen's inequality.
(c) From Lemma we have q(l/P2-1/P Then (2.2) and (2.3) give Iflq [I If IP]q/P N-q/P'II If )q N-q/P'II Ifnlq p n p n p p The inequality on the right follows from Minkowski's inequality and Lemma I: Ifl q (I Ifnl )q Nq/q'[l If q] p p n p DEFINITION.Let = {Inl and B {Jn} be covers of the real llne of the type described in Section 1.We say that has index N in B and write "N B if each I n [ntersects at most N of the J 's.The notation B will mean that K 8 for m N sore N.
THEOREM 5. Suppose a has index N in B.
(i) If q > p I, then (LP,q)Bc (LP,q)= and P,q, P,q,B c (L p q) and (ii) If q < p, then (L p q) NfH N I/q NfN P,q,B P,q, PROOF: (i) Using parts (c) and (a) of Lemma 2, we have NfNq I Nfll Nq I Nq/q' I Nfll NJ q p,q,= n p n m p n n m Nq/q' I llflJ q N q/q' NfN q m p P,q, m Thus IfN N l'q' I NfN P,q, P,q,B (ii) Using Theorems and 2 we write Nf Np,q, B sup {lffgl; Nglp, ,q, ,B sup {NfN NgN ; NgNp, P,q, P ,q ,q ,B But q' > p' and so, by part (i), slnce q > p, but

Then
We note that part (ii) could also be proved directly (using Lemmas and 2), but the constant N I/q would be replaced by N I/p'.Ifllp,q, sup {fl Igll ; igi N I/q} N llq Ifl P,q, P ,q P',q , Ik are all disjoint.Let l'm.Im.N Jm and define fll' c. n-(I/p+I/q)ll' -I/p m. y. 'flJ ,q .Ifll' ,P]q/P Pq,8 mp m m j=l mj p m m I .cp. I' I] q/p I I m-l-p/q] q/p .m 3 m.m j m Thus, in both cases, (LP,q) s (LP,q) (ii) We consider the same two cases as in (i).In the first case we define -I/q I' -I/p n n n Then 'Ifq I cq I' q/p I p,q,a n n n n whlle 'fliP I cp I'1 I n-P/q < p,q,13 n n n n since q < p.In the second case we take -2/q I' -I/p f I' "f'q I I =P.I' I] q/p= Y I m-2p/q] q/p= re(q/p-2) < p,q, j m.

m j=l m m 3
This completes the proof.By combining Theorems 5 and 6 we obtain the following.
if and only If = .
THEOREM 7. (i) If q > p I, then (L p q)B c (L p q) (ii) If q < p, then (L p,q)C (L p,q) if and only if = K . (iii) If p,q I, p * q, then (L p q) (L p q) if and only if and K Furthermore, the norms are equivalent.
DEFINITION.We write a if a B and < a.
From Theorem 7 (iii) we note that is an equivalence relation on the set of all irregular amalgams.For example, consider the class of amalgams given by (r) {[an,an+i) where the regular amalgam (L p, q) discussed in Section 1. Thus we have infinitely many (InlU "I" Ulnk)tC I N Since k is arbitrary, this contradicts our hypothesis.
Therefore we can write sup lnl M where M is an integer.It follows that any interval In meets at most M + intervals from p, i.e., Suppose now that a. Then there are intervals in 0 which contain arbitrar1y many l.'s.It follows from the pigeon-hole principle that there are interval of R of any given size which contain arbitrarily many l.'s.In s particular, a given interval in a can be translated to cover arbitrarily many lj so a N at" To prove the converse we assume that a .Then sup llnl K < .Suppose there is no N such that a a for all t.This means that we can find an interval N t in a containing arbitrarily many I +t's for some t.But, since II K, it follows n n (as above) that there is a unit interval which intersects arbitrarily many I +t's n and therefore a unit interval [k,k+l) meeting arbitrarily many I 's.This is a n contradiction.

THEOREM I0
The translation operators Tt, t R, are continuous on (L p q) with uniform bound on their norms, if and only if a O.In this case we have sup ;ITt N I/q' if q p sup l;Tt N I/q if q p where N [s the smallest integer such that a N at for all t.
PROOF.This follows from the two preceding theorems together with the proof of Theorem 4.1.
REMARK.Continuity of translation is an essential ingredient in the proof that if f (Lp,q), I p,q 2, then the Fourier transform f (L q ,P [4, Theorem 8].Thus, in view of Theorem I0, it seems unlikely that any such result will hold for amalgams other than those equivalent to the regular amalgam. Translation also arises in the consideration of convolution and Young's inequality.If {In} B {Jm }, and y are covers and a + B {I n + Jm; n,m Z}, we write y ~N(a + B) if y N (a + B) and (a + B) N Y" Then a O by Theorem 9. Similarly B 0 and it follows that y also.

AMALGAMS ON MORE GENERAL SPACES.
In this section we discuss the extent to which our results can be generalized from functions on R to more general functions.The amalgam spaces themselves make sense on any measure space.
Let (X,v) be a measure space and E {E; % J} any covering of X by disjoint measurable sets of finite measure:

X kJ EX' u(Ek) <
In terms of this decomposition E of X we define the amalgam (L p q) to consist of E functions f such that llfH I lflq ]I/q < p q E p J L (E ) Then all of the results of the first three sections extend to this setting In particular, the extension of Theorem follows from a general result [12, p. 359]   the dual of the space q(B%) of nets x (xk), where x B%, each B is concerning a Banach space, and llxN [ llxnUq]I/q < .Taking B--LP(Ek), and using this result, we have (LP'q)* =E q(LP(Ex))* q'(LP'(Ex)) (Lp''q')E Theorems 3, 4, 5, 6, and 7 have verbatim proofs in the general context of measure spaces For translation, of course, we need algebraic structure and so we assume that G is a locally compact abelian group (with Haar measure) and easily see that Theorem 8 s valid for amalgams on G.The question then arises as to an analogue of Theorem 10 for groups Regular amalgams on groups have been defined and studied in [13], [14], and [II].Nonetheless we do not see how to extend Theorem I0 to groups without imposing severe restrictions on the shape of the sets E k, even for groups as simple as G R 2.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Pl
I;nEZ} is bounded above.n PROOF: Let f be the function which agrees with f on I and is 0 elsewhere.n n Suppose that the set {[I l;nZ} is bounded above.It follows from Holder'< I flf Iq II < K If q left inequality follows from Holder's inequality and the right one is just Jensen's inequality (2.1).LEMMA 2. Suppose f f]En, where El, ..., E N are disjoint measurable subsets n

THEOREM 6 .
If B , then the inclusions in Theorem 5 are strict.PROOF: (i) If B , there are two cases.First we consider the case where 's.Let I' there is an interval, say Jl' which intersects infinitely many I Ifl p: I cp li'l 17 n n P q g n If no Jm intersects infinitely many In'S then we may assume there exist Jl' J2' such that J intersects at least m of the I 's, say I .-., I and the intervals m n m. m n . (In these and the following examples we assume that 0 and n (r) o =-.)Then (s whenever 0 < r < s and (I) O, where (L p q) is -n n p

First
Round of Reviews March 1, 2009 With this notation we can state the following version of Young's inequality for irregular amalgams.The proof is similar to that of Theorem 4.2 in [II].