OPTIMAL FUNCTION SPACES FOR THE LAPLACE TRANSFORM. f(s)e ts ds,

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1 OPTIMAL FUNCTION SPACES FOR THE LAPLACE TRANSFORM EVA BURIÁNKOVÁ, DAVID E. EDMUNDS AND LUBOŠ PICK Absrac. We sudy he acion of he Laplace ransform L on rearrangemen-invarian funcion spaces. We focus on he opimaliy of he range and he domain spaces.. Inroducion The Laplace ransform L is a well-known classical linear inegral operaor defined for every appropriae funcion f on (, ), by Lf() := f(s)e s ds, (, ). I is a he same ime one of he mos imporan inegral operaors wih a wide range of applicaions hroughou analysis and oher pars of mahemaics. I is paricularly handy in he heory of ordinary differenial equaions and probabiliy heory and i is useful for he invesigaion of specral properies of pseudo-differenial operaors (see, for insance, [3] and he references given herein) and for he sudy of Fredholm inegral equaions ([2]). While he boundedness of he Laplace ransform on funcion spaces has been sudied (cf. []), lile aenion has been paid o he sharpness of such resuls, perhaps wih he noable excepion of he recen paper [8] in which opimal domains among order-coninuous Banach funcion spaces are sudied for he Laplace ransform, resriced o he case when he given range space is a weighed Lebesgue space. Opimal domains in a more general seing are furher invesigaed in [3]. The Lebesgue spaces naurally play a primary role in analysis and is applicaions, bu here are oher spaces which are also of ineres, and for a saisfacory descripion of a paricular siuaion i ofen urns ou ha he scale of Lebesgue spaces is no rich enough. A basic disinguished insance of such a siuaion surfaces when we ask abou he acion of he Laplace ransform on he Lebesgue space L p wih p > 2. Then here is no Lebesgue space L q ha would allow L : L p L q. So, resriced o Lebesgue spaces, we would end up empy-handed. However, if we are willing o sele for finer scales of funcion spaces, we can sill remain in business. For example, we can show ha L : L p L p,p, where L p,p is he wo-parameer Lorenz space (for definiions see below) and p is he Lebesgue conjugae index, given by p = p p when p (, ) and p = when p =. Moreover, we can prove ha, in a cerain broad sense, his resul is he bes possible. In his paper, we focus on problems illusraed by his example. In paricular, we invesigae he opimaliy of rearrangemen-invarian funcion spaces on which he Laplace ransform acs Dae: May 5, Mahemaics Subjec Classificaion. 46E3, 26D2, 47B38, 46B7. Key words and phrases. Laplace ransform, rearrangemen-invarian spaces, opimal domain, opimal range, Lorenz spaces. This research was suppored by he gran P S of he Gran Agency of he Czech Republic.

2 2 EVA BURIÁNKOVÁ, DAVID E. EDMUNDS AND LUBOŠ PICK boundedly. The approach is based in par on echniques ha have been developed recenly in connecion wih invesigaion of opimal funcion spaces in Sobolev embeddings (cf. e.g. [5, 6, ]). We shall also make use of mehods of real inerpolaion heory and cerain appropriae weighed inequaliies. I is worh noicing ha he inegral in he definiion of Lf is convergen for every funcion f (L +L ), since, for any fixed (, ), he funcion s e s belongs boh o L and L. Moreover, L + L is he larges of all rearrangemen-invarian spaces (in he se-heoreical sense), hence L is well defined on any such space. The principal resuls of he paper are Theorems 3.4 and 3.5 in which he opimal rearrangemeninvarian space on eiher side of L : X Y is characerized when he oher space is given. For example, if X is fixed, hen Y is explicily consruced in Theorem 3.4 such ha i is coninuously embedded ino every oher admissible rearrangemen-invarian candidae for a arge space. An analogous consrucion of X when Y is fixed is given in Theorem 3.5. The main ool in proofs of hese resuls is Theorem 3. in which a Calderón-ype esimae is given for he non-increasing rearrangemen of he image of he Laplace ransform. The resuls are hen applied o he characerizaion of opimal acion of he Laplace ransform on Lorenz spaces. The paper is srucured as follows. We inroduce rearrangemen-invarian funcion spaces and collec all he necessary background maerial in he nex secion. In he las secion we presen all he resuls and proofs. 2. Preliminaries In his secion we recall he definiion and some basic properies of rearrangemen-invarian spaces, and also some useful informaion from inerpolaion heory. The sandard general reference is [2], where a comprehensive exposiion of his opic can be found. We se M(, ) = {f : (, ) [, ]: f is Lebesgue-measurable in (, )}, and M + (, ) = {f M(, ) : f }. The non-increasing rearrangemen f : (, ) [, ] of a funcion f M(, ) is defined as f () = inf{λ (, ) : {s (, ) : f(s) > λ} } for (, ). The operaion f f is monoone in he sense ha f g a.e. in (, ) implies f g. I is however no subaddiive. Insead, for every (, ) and f, g M(, ), we have (2.) (f + g) (s) ds f (s) ds + g (s) ds. We recall ha for every f M + (, ) and every (, ), one has (2.2) {s (, ) : f(s) > f ()}. A basic propery of rearrangemens is he Hardy-Lilewood inequaliy, which assers ha, if f, g M(, ), hen (2.3) f()g() d f ()g ()d. We say ha a funcional ϱ : M + (, ) [, ] is a funcion norm, if, for all f, g and {f j } j N in M + (, ), and every λ, he following properies hold: (P) ϱ(f) = if and only if f = ; ϱ(λf) = λϱ(f); ϱ(f + g) ϱ(f) + ϱ(g); (P2) f g a.e. implies ϱ(f) ϱ(g); (P3) f j f a.e. implies ϱ(f j ) ϱ(f);

3 OPTIMAL FUNCTION SPACES FOR THE LAPLACE TRANSFORM 3 (P4) ϱ(χ G ) < for every G (, ) of finie measure; (P5) G f() d C Gϱ(f) for some C G and every G (, ) of finie measure. If, in addiion, (P6) ϱ(f) = ϱ(g) whenever f = g, hen we say ha ϱ is a rearrangemen-invarian norm. If ϱ is a rearrangemen-invarian norm, hen he collecion X = X(ϱ) = {f M(, ) : ϱ( f ) < } is called a rearrangemen-invarian space. Noe ha he quaniy ϱ( f ) is defined for every f M(, ), and f X ϱ( f ) <. Wih any rearrangemen-invarian norm ϱ is associaed anoher funcional, ϱ, defined for g M + (, ) as { } ϱ (g) = sup f()g() d : f M + (, ), ϱ(f). I urns ou ha ϱ is also a rearrangemen-invarian norm, which is called he associae norm of ϱ. Furhermore, one always has (ϱ ) = ϱ. If ϱ is a rearrangemen-invarian norm, X = X(ϱ) is he rearrangemen-invarian space deermined by ϱ, and ϱ is he associae norm of ϱ, hen he funcion space X(ϱ ) deermined by ϱ is called he associae space of X and is denoed by X. Furhermore, he Hölder inequaliy f()g() d f X g X holds for every f, g M(, ). We define he fundamenal funcion, ϕ X, of a given rearrangemen-invarian space X by ϕ() = χ (,) X(, ), (, ). If X and Y are rearrangemen-invarian spaces, hen we denoe by X Y he coninuous embedding of X ino Y and by T : X Y he boundedness of an operaor T from X o Y. If T : X Y and T is anoher operaor defined a leas on Y wih values in M(, ) and such ha he ideniy T f()g() d = g()t f() d holds for every f Y and every g X, hen i is easy o verify ha T : Y X. In paricular, X Y if and only if Y X wih he same embedding norms, cf. [2, Chaper, Proposiion 2.]. Le us recall ha for any fixed s >, he dilaion operaor E s, defined a every g M(, ) by E s g() = g( s ) for (, ), is bounded on every rearrangemen-invarian space wih norm depending on s and on he space. Among basic examples of funcion norms are hose associaed wih he sandard Lebesgue spaces L p. For p (, ], we define he funcional ϱ p by ϱ p (f) = { ( f() p d ) p if < p <, ess sup (, ) f() if p = for f M + (, ). If p [, ], hen ϱ p is a rearrangemen-invarian norm. If < p, q, we define he funcional ϱ p,q by ϱ p,q (f) = s p q f (s) q

4 4 EVA BURIÁNKOVÁ, DAVID E. EDMUNDS AND LUBOŠ PICK for f M + (, ). The se L p,q defined as he collecion of all f M(, ) saisfying f p,q = ϱ p,q ( f ) < is called a Lorenz space. If eiher < p < and q or p = q = or p = q =, hen ϱ p,q is equivalen o a rearrangemen-invarian norm in he sense ha here exiss a rearrangemen-invarian norm σ and a posiive consan C depending on p, q such ha C σ(f) ϱ p,q (f) Cσ(f) for every f M + (, ), and herefore L p,q is a rearrangemen-invarian space for hese cases. If eiher < p < or p = and q >, hen L p,q is a quasi-normed space. If p = and q <, hen L p,q = {}. For every p [, ], we have L p,p = L p. Furhermore, if p, q, r (, ] and q r, hen he embedding (2.4) L p,q L p,r holds. We shall need some basic informaion from inerpolaion heory. Le X and X be quasinormed spaces, compaible in he sense ha hey are embedded in some common Hausdorff opological vecor space. By X + X we denoe he se of all funcions f M(, ) for which here exiss a decomposiion f = g + h such ha g X and h X. The space X + X is equipped wih he quasinorm f X +X = inf ( g X + h X ), f=g+h where he infimum is exended over all such decomposiions. Suppose ha f X + X. The Peere K-funcional is defined by K(, f; X, X ) := inf f=g+h ( g X + h X ) for >. The funcion K (considered wih respec o he variable ) is increasing and concave on (, ). Moreover, he funcion K(, f; X, X ) is non-increasing on (, ). We recall ([2, Chaper 2, Theorem 6.2]) ha in he case when X = L and X = L, an exac formula for he K funcional is known, namely, (2.5) K(f, ; L, L ) = f (s) ds for (, ) and f (L + L ). Furhermore, for X = L, and X = L, one has (cf. [7, Remark 3]) (2.6) sup sf (s) K(f, ; L,, L ) 2 sup sf (s) <s <s for (, ) and f (L, + L ). We recall ha L, is only a quasi-normed space and ha i is known no o be equivalenly normable. 3. Main resuls We begin by esablishing a poinwise esimae for he non-increasing rearrangemen of he image of a funcion under he Laplace ransform. Theorem 3.. Le f (L + L ) and (, ). Then (3.) (Lf) () f (s) ds. Proof. For every f (L + L ) and every (, ), one has Lf() f(s) e s ds Taking he supremum over (, ), we obain (3.2) Lf f. f(s) ds,

5 OPTIMAL FUNCTION SPACES FOR THE LAPLACE TRANSFORM 5 The funcion e s is decreasing boh in he variable (for every fixed s) and in he variable s (for every fixed ). Using his fac and also he Hardy Lilewood inequaliy (2.3), we ge, for every funcion f (L + L ) and every (, ), (3.3) (Lf) () = ( Lf ) () (L( f )) () = L( f )() = f (s)e s ds = L(f )(). Le f be a bounded nonnegaive funcion and (, ). Then ha is, (Lf) () = Lf() f Taking he supremum over (, ), we ge (Lf) () f. (3.4) Lf, f. e s ds = f, f(s) e s ds By (3.2), (3.4) and he definiion of he K-funcional, we ge he poinwise esimae (3.5) K(Lf, ; L, L, ) K(f, ; L, L ) for every f (L + L ) and (, ). By [2, Chaper 5, Proposiion.2], (3.6) K(Lf, ; L, L, ) = K(Lf, ; L,, L ). Applying he lef inequaliy in (2.6) o in place of and hen muliplying by, we ge (3.7) sup s(lf) (s) K(Lf, ; L,, L ). <s Combining (3.7), (3.6) and (3.5), we obain ha is, by (2.5), Using he rivial esimae we arrive a sup s(lf) (s) K(f, ; L, L ), <s sup s(lf) (s) <s f (s) ds. (Lf) ( ) sup s(lf) (s), <s (Lf) ( ) f (s) ds. Because he las esimae holds for any old (, ), a simple change of variables yields (3.). The proof is complee. Remark 3.2. I will be useful o noice ha (3.) can be reversed up o some exen, a leas for non-negaive funcions. Indeed, if f M + (, ) and (, ), hen Lf() = e s f(s) ds e s f(s) ds e f(s) ds.

6 6 EVA BURIÁNKOVÁ, DAVID E. EDMUNDS AND LUBOŠ PICK Proposiion 3.3. Le Y be a rearrangemen invarian space such ha he condiion (3.8) min{, } Y holds. Then he funcional ϱ, defined by ϱ(f) = is a rearrangemen-invarian norm. f (s) ds Y, f M + (, ), Proof. We have o verify he axioms (P)-(P6) for he funcional ϱ. The axiom (P6) is rivially saisfied, and (P2) and (P3) follow immediaely from he monoone convergence heorem. All he asserions in (P) are rivial excep perhaps he riangle inequaliy, which however follows easily from (2.) and he axiom (P2) for Y. As for (P4), we firs noe ha ϱ(χ (,) ) <. Indeed, his follows immediaely from (3.8) and he ideniy χ (,) (s) ds = min{, } for (, ). Since he dilaion operaor is bounded on every rearrangemen-invarian space, hence in paricular on Y, his implies ha ϱ(χ (,a) ) < for any a (, ). Now le G (, ) be a se of finie non-zero measure. Then χ G = χ (, G ), hence ϱ(χ G ) = ϱ(χ G) = ϱ(χ (, G ) ) <. This esablishes (P4). I remains o verify (P5). Once again, le G (, ) be a se of finie non-zero measure. Then, for f M + (, ), one has ϱ(f) = f (s) ds Y χ (, G )() f (s) ds Y χ (, G ) Y By a special case of he Hardy Lilewood inequaliy, we have G f() d f () d. G G f (s) ds. The combinaion of he las wo esimaes esablishes (P5) wih C G, where C G = χ (, ) Y. G The proof is complee. We shall now characerize he opimal rearrangemen-invarian range space for he Laplace ransform when he domain space is fixed. Theorem 3.4. Le X be a rearrangemen-invarian space such ha (3.9) min{, } X. Define he funcional ϱ by ϱ(g) = g (s) ds X for g M + (, ). Then ϱ is a rearrangemen-invarian norm. We denoe by Y he rearrangemen-invarian space deermined by ϱ, he associae norm of ϱ. Then L : X Y. Moreover, Y is he opimal (smalles) such rearrangemen-invarian space. If he condiion (3.9) is no saisfied, hen here does no exis any rearrangemen-invarian space Y such ha L : X Y. Proof. We know ha ϱ is a rearrangemen-invarian norm hanks o Proposiion 3.3. Fix f X. By he rearrangemen invariance of Y and (3.3), we have Lf Y = (Lf) Y L(f ) Y.

7 OPTIMAL FUNCTION SPACES FOR THE LAPLACE TRANSFORM 7 Since f is non-negaive, L(f ) is non-increasing on (, ). Therefore, he Hardy Lilewood inequaliy implies L(f ) Y = By he Fubini heorem, sup g Y sup g Y g() L(f )() d g ()L(f )() d = sup g Y sup g Y g ()L(f )() d. L(g )()f () d, which, alogeher, combined wih (3.) and he Hölder inequaliy, yields Lf Y sup g Y g (s) dsf () d = sup g Y f X = f X, g Y sup g Y g (s) ds X f X proving L : X Y. We shall now show he opimaliy of Y. Assume ha Z is a rearrangemen-invarian space such ha L : X Z. Since L is a self-adjoin operaor, we have also L : Z X. Consequenly, we obain from Remark 3.2 and he rearrangemen invariance of he space Z, for some C > and all g M(, ), g Y = g (s) ds X e L(g ) X C g Z = C g Z. In oher words, Z Y, ha is, Y Z, showing he opimaliy of Y. Finally, assume ha (3.9) is no saisfied and suppose ha Y is a rearrangemen-invarian space such ha L : X Y. Then Lf Y C f X for some C > and every f X. Since L is a self-adjoin operaor, his implies In paricular, for some posiive consan C, χ (,) Y C Lχ (,) X = C Lg X C g Y for every g Y. e s ds X C min{, } X =, a conradicion. Thus, no rearrangemen-invarian space Y exiss which would allow L : X Y. The proof is complee. In he following heorem we shall characerize he opimal domain space for he Laplace ransform in he case when he range space is fixed. Theorem 3.5. Le Y be a rearrangemen-invarian space such ha (3.) min{, } Y. We denoe by ϱ he funcional defined by ϱ(f) = f (s) ds Y, f M + (, ). Then ϱ is a rearrangemen-invarian norm. Le X be he rearrangemen-invarian space deermined by ϱ. Then L : X Y. Moreover, X is he opimal (larges) such rearrangemeninvarian space. If he condiion (3.) is no saisfied, hen here does no exis any rearrangemeninvarian space X such ha L : X Y.

8 8 EVA BURIÁNKOVÁ, DAVID E. EDMUNDS AND LUBOŠ PICK Proof. Again, Proposiion 3.3 and he assumpion (3.) guaranee ha ϱ is a rearrangemeninvarian norm. Fix f X. By he rearrangemen invariance of Y, (3.) and he definiions of ϱ and X, we have Lf Y = (Lf) Y f (s) ds Y = f X, esablishing L : X Y. To show he opimaliy of he space X, assume ha Z is a rearrangemeninvarian space saisfying L : Z Y. Then, by he rearrangemen invariance of Z and Remark 3.2, for some c > and every f Z, we have f Z = f Z c L(f ) Y c e Consequenly, Z X, as desired. The proof is complee. f (s) ds Y = c e f X. Remark 3.6. We noe ha he condiion (3.9) is equivalen o saying ha he quasi-normed space L, L, equipped wih he usual quasinorm, is conained in X (and an analogous observaion can be made abou (3.)). This condiion is saisfied, for insance, for every X = L p wih p <, bu no for X = L. Roughly speaking, if a space X is oo close o L, hen here is no hope for finding a rearrangemen-invarian arge space ino which he Laplace ransform would ac boundedly from X. We know, for insance, ha L : L L,, bu hen L, is no normable. Likewise, if a space Y is oo close o L (in he sense ha (3.) is violaed), hen i does no have a domain parner wihin he caegory of rearrangemen-invarian spaces. We shall now apply he main resuls o he characerizaion of he opimal acion of he Laplace ransform on Lorenz spaces. We shall firs need an auxiliary observaion. Proposiion 3.7. Assume ha p (, ) and q [, ]. Then (3.) f p,q f (s) ds p,q p f p,q for f L p,q. Proof. Fix f L p,q. Assume firs ha q <. Then, by he definiion of p,q, change of variables and he Hardy inequaliy ([5, Theorem 9.6]), we ge Conversely, since we a once have f (s) ds p,q = p q f (s) ds q = p q p p q f () q = p f p,q. f (s) ds f ( ) for (, ), f (s) ds q f (s) ds p,q p q f ( ) q = p q + f () q = f p,q. This esablishes (3.) in he case q <. Now assume ha q = and se M = f p,. If M <, hen he definiion of p, implies ha, for every s (, ), one has f (s) Ms p.

9 OPTIMAL FUNCTION SPACES FOR THE LAPLACE TRANSFORM 9 Thus, f (s) ds p, = sup p f (s) ds M << sup p s p ds = pm, << showing ha f (s) ds p, p f p,. If M =, hen he laer inequaliy holds rivially. Conversely, by he monooniciy of f and change of variables, f (s) ds p, = sup p f (s) ds sup p f ( ) << << This shows (3.) in he case q =. = sup p f () = f p,. << Theorem 3.8. Assume ha p (, ) and q [, ]. Then L : L p,q L p,q. Moreover, boh he domain space and he arge space are he opimal such rearrangemeninvarian spaces (hence hey form an opimal pair). Proof. Denoe X = L p,q. Then i is well known (see e.g. [2, Chaper 4, Theorem 4.7, p. 22]) ha, up o equivalence of norms, X = L p,q. I is easy o check ha (3.9) is saisfied. Define ϱ(f) = f (s) ds X for f M + (, ). From Proposiion 3.7, applied o p, q in place of p, q, i follows ha he norm ϱ is equivalen o p,q, whence ϱ is equivalen o p,q. Thus, by Theorem 3.4 we have L : L p,q L p,q and he arge space is opimal. The verificaion of he opimaliy of he domain space is similar (in fac, even easier). Remark 3.9. I follows from Theorem 3.8 ha L : L p,q L p,q if and only if p = 2. In paricular, he only L p space which L maps boundedly ino iself is L 2. A much sronger resul is known o hold for he Fourier ransform (see [4, Theorem ]), namely ha i is bounded from a rearrangemen-invarian space X o X if and only if X = L 2. An analogous saemen for he Laplace ransform is false since, as noed above, L akes boundedly ino iself every space of he form L 2,q, q [, ]. Remark 3.. An observaion ha generalizes he houghs concerning he acion of L from a rearrangemen-invarian space ino iself conained in he preceding remark can be obained from Theorem 3.4. Indeed, calculaion yields ha for every s, (, ) one has χ (,s) (y) dy = sχ (, s )() + χ ( s, )(). Therefore, if L : X X for a rearrangemen-invarian space X, hen he opimal space Y such ha L: X Y mus saisfy Y X, ha is, X Y. By Theorem 3.4, his means ha f (y) dy X C f X

10 EVA BURIÁNKOVÁ, DAVID E. EDMUNDS AND LUBOŠ PICK for some posiive consan C independen of f. In paricular, insering f = χ (,s) for a fixed s (, ), his and he above calculaion yield sχ (, s )() + χ ( s, )() X C χ (, s ) X. This rivially implies ha s χ (, s ) X χ (, s ) X. Le us denoe by ϕ he fundamenal funcion of X and ϕ ha of X. Consequenly, he above inequaliy reads as s ϕ( s ) C ϕ(s). Since ([2, Chaper 2, Theorem 5.2]) ϕ(s) ϕ(s) = s for every s, his in fac proves he following, quie ineresing saemen: if L: X X, hen sup ϕ(s) ϕ( s ) <. s (, ) Noe ha he fundamenal funcion ϕ() =, (, ), of he spaces L 2,q (in paricular L 2 ), saisfies his condiion. Moreover, i immediaely follows ha L can be bounded on a space whose fundamenal funcion is equivalen o p only if p = 2. Remark 3.. The resuls of he paper can be exended o inegral operaors of he form T f() = K(, s)f(s) ds, where K : (, ) (, ) (, ) is a bounded non-negaive funcion which is decreasing in s, sricly decreasing in, and such ha ψ() := K(, s) ds < for every (, ). Then he asserions of Theorems 3.4 and 3.5 hold unchanged, wih he funcion min{, } replaced by min{, ψ()} and wih he inegral in he definiion of ϱ replaced by ψ() g (s) ds. 2π Remark 3.2. For p 2, an upper esimae of he norm L L p Lp by ( p ) p was proved by G.H. Hardy as a side resul in [9, Theorem ]. Laer on, Hardy noiced his fac and formulaed i more explicily in [, Theorem 9]. He admied in [] ha he had been unable o prove ha he consan is sharp and conjecured ha i was no. A good deal laer, in [4, Theorem 2.2], i was shown ha his predicion was correc by finding a beer upper bound for L L p L, p namely (p(2 p)) 2 p 2p (π(p )) p. The consan however is sharp when p = 2 (in which case, of course, boh he upper bounds coincide and equal π, see [9] and []). By differen mehods (involving a complex variable) i is calculaed for example in [4, Corollary 2.]. Pleny of relaed resuls can be found in he lieraure. For example, i is shown in [3, Proposiion ] ha he operaor norm of L 2 on L 2 is equal o π. Acknowledgmens. We hank he referee for horough and criical reading of he paper, for many valuable commens and for urning our aenion o some relaed lieraure. We hank Aleš Nekvinda and Jan Malý for simulaing discussions abou cerain pars of he paper. References [] K.F. Andersen, On Hardy s inequaliy and Laplace ransforms in weighed rearrangemen invarian spaces, Proc. Amer. Mah. Soc. 39,2 (973), [2] C. Benne and R. Sharpley, Inerpolaion of Operaors, Pure and Applied Mahemaics Vol. 29, Academic Press, Boson 988. [3] A. Boumenir and A. Al-Shuaibi, The inverse Laplace ransform and analyic pseudo-differenial operaors, J. Mah. Anal. Appl. 228 (998), [4] L. Brandolini and L. Colzani, Fourier ransform, oscillaory mulipliers and evoluion equaions in rearrangemen invarian funcion spaces, Colloq. Mah. 7 (996), [5] A. Cianchi, L. Pick and L. Slavíková, Higher-order Sobolev embeddings and isoperimeric inequaliies, Adv. Mah. 273 (25),

11 OPTIMAL FUNCTION SPACES FOR THE LAPLACE TRANSFORM [6] D. E. Edmunds, R. Kerman and L. Pick, Opimal Sobolev imbeddings involving rearrangemen-invarian quasinorms, J. Func. Anal. 7 (2), [7] S. Ericsson, Exac descripions of some K and E funcionals, J. Approx. Theory 9 (997), [8] O. Galdames Bravo, On he opimal domain of he Laplace ransform, Bull. Malays. Mah. Sci. Soc. 4 (27), [9] G.H. Hardy, Remarks in addiion o Dr. Widder s noe on inequaliies, J. London Mah. Soc. 4 (929), [] G.H. Hardy, The consans of cerain inequaliies, J. London Mah. Soc. 8, no. 2 (933), 4 9. [] R. Kerman and L. Pick, Opimal Sobolev imbeddings, Forum Mah. 8, 4 (26), [2] J.G. McWhirer and E.R. Pike, On he numerical inversion of he Laplace ransform and similar Fredholm inegral equaions of he firs kind, J. Phys. A: Mah. Gen., 9 (978), [3] S. Okada, W.J. Ricker and E.A. Sánchez Pérez, Opimal domain and inegral exension of operaors: acing in funcion spaces, Operaor Theory: Advances and Applicaions 8, Birkhäuser Verlag, Basel, 28. [4] E. Seerqvis, Uniary equivalence. A new approach o he Laplace ransform and he Hardy operaor, Maser Thesis 25:329, Deparmen of Mahemaics, Luleå Universiy of Technology, ISSN:42-67, Luleå 25, 26. [5] A. Zygmund, Trigonomeric Series, Cambridge Universiy Press, Cambridge, 22. Eva Buriánková, Deparmen of Mahemaical Analysis, Faculy of Mahemaics and Physics, Charles Universiy, Sokolovská 83, Praha 8, Czech Republic address: Eva.Buriankova@seznam.cz David E. Edmunds, Universiy of Sussex, Falmer, Brighon, BN 9QH, UK address: davideedmunds@aol.com Luboš Pick, Deparmen of Mahemaical Analysis, Faculy of Mahemaics and Physics, Charles Universiy, Sokolovská 83, Praha 8, Czech Republic address: pick@karlin.mff.cuni.cz