Z. Anal. Anwendungen, 19(3), 781-800, 2000
On more general Lipschitz spaces Dorothee D. Haroske
Summary
The present paper deals with (logarithmic) Lipschitz spaces of type Lip(1p;;?q ) , 1 p 1, 0 < q 1, > 1=q . We study their properties and derive some (sharp) embedding results. In that sense this paper can be regarded as some continuation and extension of our papers [9], [8], but there are also connections with some recent work of Triebel concerning Hardy inequalities and sharp embeddings. Recall that the nowadays almost `classical' forerunner of investigations of this type is the Brezis-Wainger result [6] about the `almost' Lipschitz continuity of elements of the Sobolev spaces Hp1+n=p (Rn ) when 1 < p < 1. key words : limiting embeddings, Lipschitz spaces, function spaces 1991 Math. Subject Classi cation : 26A16, 46E35, 26A15
Introduction The present paper arose in connection with our recent papers [9], [8] as well as after some discussion with H. Triebel about this subject. Concerning our joint papers [9], [8] with D.E. Edmunds from the University of Sussex, we were mainly led by two di erent questions to study certain spaces of Lipschitz type and related embeddings. On the one hand, compact embeddings of type
id : Bps11;q1 ( ) ?! Bps22 ;q2 ( )
(1)
have been investigated for a long time already. Here Rn is a bounded C 1 domain, and Bpsii ;qi , i = 1; 2, are the usual Besov spaces with 0 < pi ; qi 1, si 2 R. The embedding (1) is compact if 1 1 0 < p ; p 1; s ? s > n max ? ; 0 ; 0 < q ; q 1: (2) 1 2
1
2
p1
p2
1 2
The question now arises what happens when (2) is replaced by s1 ? n=p1 = s2 ? n=p2, 0 < p1 p2 1, 0 < q1 ; q2 1, the so-called limiting case. Clearly the embedding (1) is no longer compact.
However, modifying the setting slightly, say, enlarging the target space suciently carefully (where the initial space is assumed to be xed now), may lead to compact limiting embeddings. One approach was given by Edmunds and Triebel in [10, Sect. 3.4], changing the integrability p2 of the target space slightly, where now spaces similar to the well-known Lorentz-Zygmund spaces Lp (log L)a ( ), a 2 R, appear. A parallel result, concerning the (weighted) Rn -setting can be found in [12], [13]. Another idea to recover compactness of (1) when s1 ? n=p1 = s2 ? n=p2, is to decrease the smoothness of the target space in such a way, that the embedding becomes compact again, but the smoothness s2 is preserved; i.e. we stick at the limiting situation. In that way one quite naturally arrives at the introduction of new spaces with additional `logarithmic smoothness'. As an example one may consider the case s2 = 1 and p2 = 1. It turns out that in case of the B -spaces there is an interplay between the (usually neglected) q-parameters and the additional logarithmic smoothness. This result is somewhat surprising in our opinion, though similar results were obtained before; cf. [11]. The second reason to deal with spaces of `logarithmic smoothness' in more detail, is the well-known and celebrated result of Brezis and Wainger [6] in which it was shown that every function u in Hp1+n=p (Rn ) is `almost' Lipschitz-continuous, in the sense that for all x; y 2 Rn , 0 < jx ? yj < 1=2,
1=p0
ju(x) ? u(y)j c jx ? yj log jx ? yj
kujHp1+n=p (Rn )k :
(3)
Here c is a constant independent of x; y and u, and 1=p0 +1=p = 1. Our aim in [9] was to investigate how `sharp' this result is (concerning the exponent of the log-term), as well as to look for possible extensions to the wider scale of F -spaces and parallel results for B -spaces. We found that the
2
1. Preliminaries
exponent 1=p0 is sharp in the F -setting, whereas in case of B -spaces the sharp exponent turned out to be 1=q0. As already mentioned above, this important role played by the q-parameter is rather unusual. Moreover, (3) also suggests some de nition of `logarithmic' Lipschitz spaces in the following way : some f 2 C (Rn ) belongs to Lip(1;? ) (Rn ), 0, if
(1;? ) n
supn j(h f )(x) j
f jLip (R ) := kf jL1(Rn )k + 0 > !(f; t)p q dt > > ; q < 1 = < t j log t j t 0 > ! ( f; t ) > p > ; q=1 ; : sup t j log tj 1 2
=
0 n( p ? 1)+, b 2 R, 0 < q 1, where !2 (f; t)p has to be replaced by !r (f; t)p with r > s, r 2 N ,
Z !r (f; t)p q dt !1=q
(s;?b)
0, 2 R, In particular, for p = q = 1 we arrive at spaces of Zygmund type, C (s;? ) = B1 ;1
(f; t) ; kf jC (s;? )k = kf jL1k + sup ts!jrlog tj 0 0. Then 1 ,! Lip(1;? ) Bp;q p; 1
if
q10 :
(17)
Note, that in case of p = 1 one recovers in that way (a weaker version of) the embedding theorem [9, Thm. 2.1 (ii)]. The counterpart of (17) for spaces Lip(1p;;?q ) reads as follows.
Proposition 11 Let 1 p 1, 0 < q; v 1, > 1=v. Then ( 1 ; v = 1; q0 (1 ; ? ) 1 ,! Lip Bp;q if p; v > v1 + q10 ; v < 1 :
(18)
3.1. Embeddings into spaces of Lipschitz type
7
P r o o f : The upper line in (18) is covered by (17), we thus assume v < 1. In view of (5) and (8) it is sucient to show that
Z !(f; t)p v dt !1=v
1
c f jBp;q t j log tj t 1 2
0
if > v1 + q10 . But obviously
Z !(f; t)p v dt !1=v t j log tj t
p sup !(f; t1)=q 0 t j log t j 0 1=q. Then 8 1 ; q = 1; < 1+n=p ,! Lip(1;? ) 1 Bp;q if : 1; q
> max 1; q ; 0 < q < 1:
(19)
Remark 14 Triebel proved in some so far unpublished notes that (19) holds with = 1 when 1 < q 1, 0 < p < 1, using di erent techniques (involving non-increasing rearrangement, Hardy inequalities and atomic decompositions).
8
3. Embeddings
We brie y turn to F -spaces. Recall the following result rst.
Proposition 15 Let 0 < p < 1, 0 < q 1 and 0. Then 1+n=p ,! Lip(1;? ) Fp;q 1; 1
p10 :
if
(20)
We proved this assertion in [9, Thm. 2.1 (i)]. Moreover, the exponent = 1=p0, when 1 < p < 1, is sharp. Note that in case of 1 < p < 1, q = 2, (20) reproduces the famous Brezis-Wainger result [6]. There is also an extension of Proposition 15 to spaces Lip(1p;;?q ), but is is more convenient for us to postpone this result to Corollary 20 below.
3.2 Embeddings of purely Lipschitzian type
Proposition 16 Let 1 p 1, 0 < q; v 1, > 1=q, > 1=v. Then 8 1 < ? v ? q1 ; v q; (1 ; ? ) (1 ; ? ) if, and only if, : Lip p; q ,! Lip p; v ? v1 > ? q1 ; v < q:
(21)
P r o o f : Note that the upper line in (21) is somehow surprising, as it means that some space Lip(1p;;?q ) can be continuously embedded into Lip(1p;;?v ) even if < . At rst glance this seems impossible : having `less' (logarithmic) smoothness (? ) in the original space than in the target one (? ); but it turns out that this fact simply refers back to the in uence of q in De nition 1. The argument to prove it is indeed a tricky one and due to Bennett and Rudnick in [2] (as far as we know) - what they call some `diagonal' result. But we return to this point later in the proof. Step 1. We rst prove the necessity of the assumptions on the parameters in (21). Let f{; 2 Lp be such that ? !(f{; ; t)p t j log tj{ log j log tj (22)
for small t > 0 and {; 2 R. Note that f{;0 2 Lip(1p;;?q ) if, and only if, { < ? 1=q. Thus, assuming ? 1=v < ? 1=q, 0 < q; v 1, we may choose { such that ? 1=v { < ? 1=q and hence f{;0 2 Lip(1p;;?q ) n Lip(1p;;?v ). Now let ? 1=v = ? 1=q =: {, then f{; 2 Lip(1p;;?q ) if, and only if, > 1=q. Consequently f{; 2 Lip(1p;;?q ) n Lip(1p;;?v ) if { = ? 1=v = ? 1=q and 1=q < 1=v, that is, when ? 1=v = ? 1=q and v < q. Step 2. We prove the suciency in (21) in case of v < q. In view of (5) we have to show that
Z !(f; t)p q dt !1=q Z !(f; t)p v dt !1=v c : t j log tj t t j log tj t 0 0 1 2
1 2
By Holder's inequality we get
Z !(f; t)p v dt ! v Z !(f; t)p q dt ! q Z ? ?? dt ! v ? q j log tj v q t c t t 0 t j log tj 0 t j log tj 0 Z !(f; t)p q dt ! q 1 2
1
1
1 2
c0
1 2
0
1 2
1
t j log tj
t
;
1
1
1
1
3.2. Embeddings of purely Lipschitzian type
9
where the last estimate is correct for ? > v1 ? 1q . This gives the lower line in (21). Step 3. It remains to verify the upper line in (21). By the obvious monotonicity argument it is sucient to prove Lip(1;? ) ,! Lip(1;? ) if ? 1 = ? 1 ; v q: (23) p; q
p; v
v
q
As already mentioned we make use of a clever trick which can be found in [2, Thm. 9.5, p.33]. Recall > 1=q. Obviously h ?( q?1)i0 q ? 1 j log tj = t j log tj q ; t > 0; implies Z t d j log tj?( q?1) = c (24) j log j q : 0
Bennett and Rudnick in [2, Thm. 9.5] then gained from the fact that for some function f its rearrangement f (t) is decreasing (by de nition). In our case we may replace this argument in the following way : by [7, Ch. 2, x6, p.41-42] one has that t?1 !(f; t)p is { roughly speaking { decreasing in t > 0 (up to constants), such that (24) (after multiplying both sides with t?1 !(f; t)p q and
involving the above-described monotonicity) results for 0 < t < 1=2 in
Using the decomposition
!(f; t)p c Z t !(f; )p q d 1=q : 0 j log j t j log tj ? q1
!(f; t) v
!(f; t)p v?q f; t)p q = t!j (log tj t j log tj ?1=q we apply (25) to the last term on the right-hand side of (26), recall (23). Thus p t j log tj
0 B@Z !(f; t)p v t j log tj 0 1 2
8 1v 1v?q q 0 > Z Z q q < dt C c !(f; t)p B !(f; )p d C A @ > tA t : 0 j log tj 0 j log j 0 1vvq?q 8 > Z q f; )p d C j : t j log tj 1
1 2
1 2
1 2
1 2
0
=
0
0 Z !(f; t)p q B c @ t j log tj 0 1 2
(25) (26)
9v > dt = t> ; 9v > dt = t> ; 1
1
1q dt C tA
1
This gives (23).
Remark 17 One recognises that our result (21) resembles the outcome [2, Thm. 9.3, Thm. 9.5] by Bennett and Rudnick when in their setting p = 1. We already mentioned the somehow astonishing result that concerning the embedding Lip(1p;;?q ) into Lip(1p;;?v ) one can `compensate' some gain of logarithmic smoothness ? > ? by `paying' with the additional index q, that is, as long as (? ) ? (? ) 1=q ? 1=v, v q. (s;b) , but we This situation is essentially di erent from the related one when dealing with spaces Bp;q postpone a discussion of this phenomenon to Section 4.
10
3. Embeddings
Remark 18 We used in Step 3 of the above proof that t?1 !(f; t)p , 1 p 1, is (more or less) decreasing in t > 0. This fact immediately implies that Lip(1p;;?q ) with 0 < q < 1, 1=q, is a very poor space, see Remark 2; for assuming that there is some c > 0 such that !(f; t)p c > 0
t
for small t > 0, one can estimate
!1=q Z
(1;? )
d t 1 ;
f jLip p; q C 0 j log tj q t 1 2
but the right-hand side diverges for 1=q. Thus, conversely, for a function f 2 Lip(1p;;?q ), 0 < q < 1, 1=q, we have to assume !(f; t)p = 0 for small t > 0 { and the only constant belonging to Lp is the null function. 1 ,! Lip(1;? ) fails for < 1=q 0 + 1=v . For conRemark 19 We show that the embedding Bp;q p; v venience, let p = 1; but an adapted argument should work for p < 1, too. Assume that 1 ,! Lip(1;? ) for some < 1=q 0 + 1=v . Then by (21) we may continue, B1 ;q 1; v
1;q ,!
B1
Lip(1;? )
1; v ,!
Lip(1;? ) ; 1; 1
= ? v1 < q10 + v1 ? v1 = q10 :
1 ,! However, this contradicts the sharpness assertion in [9, Thm. 2.1(ii)] stating that B1 ;q 1 ,! Lip(1;? ) implies 1=q 0 + 1=v , at least for Lip(11;?; 1 ) if, and only if, 1=q0. Hence Bp;q p; v p = 1. In particular, when 1 q = v 1, we necessarily have 1, see (19).
We now give the promised extension of Proposition 15. Obviously Proposition 16 and (20) imply the following.
Corollary 20 Let 0 < p < 1, 0 < q; v 1, > 1=v. Then ( 1 p0 1+ n=p (1 ; ? ) if Fp;q ,! Lip 1; v 1
; v = 1; > v + p10 ; v < 1 :
In particular,
F 1+n=p p;q
,!
Lip(1;? ) 1; p
if
1
> max 1; p :
(27)
Remark 21 Parallel to Remark 14 we mention that Triebel obtained instead of (27) a sharper assertion with = 1, when 1 < p < 1. Note that by a similar argument, i.e. combination of (17) and (21), we get an alternative proof of Proposition 11.
3.3 Embeddings between `logarithmically smooth' spaces
(s;b) and of type Lip(1;? ) in some detail. Thus In Section 4 we intend to compare spaces of type Bp;q p; q it is of particular interest to derive a few more, rather elementary embeddings between both scales of spaces. We studied this question for p = q = 1 in [9, Prop. 4.2] and obtained the following.
Proposition 22 Let 0. Then (1;? ) ,! Lip(1;? ) ,! B (1;? ) : B1 1; 1 1;1 ;1
3.3. Embeddings between `logarithmically smooth' spaces
11
Moreover, we proved in [9, Prop. 4.4] that
(1;? ) ,! Lip(1;? ) B1 ;q 1; 1
if, and only if, 0 < q 1: In view of characterisation (15) and Marchaud's inequality we may extend Proposition 22 to spaces Lip(1p;;?q ). Corollary 23 Let 1 p 1, 0 < q 1, > 1=q. (i) Then ( < ? 1 ; 0 < q < 1; q (1 ; ? ) (1 ; ? ) (28) Bp;1 ,! Lip p; q if ; q = 1: Moreover, ;?( ? q1 )) Bp;(1min( ,! Lip(1p;;?q ) : (29) q;1) (ii) Then
Lip(1p;;?q ) ,! Bp;(11;?( ? q )) : 1
(30)
P r o o f : Step 1. We deal with (28). Note that for q 1 (28) is a consequence of (29), but we will prove both assertions separately. In view of (15) and (5) it is sucient to show that
9 0 1q 8 > > Z Z q < B@ !(f; t)p dt CA c kf jLpk + !2(f; t)p dt = ; > t j log tj t t j log tj t > : ; 0 0 1
1 2
1 2
(31)
where < ? 1q and 0 < q < 1 (the modi cations for q = 1 will be clear from the argument below). Furthermore, without restriction of generality we can assume that 0 < ? q1 , the rest (s;b) spaces. We apply Marchaud's inequality, see [3, Ch. 5, is done by the monotonicity of Bp;q (4.11), p. 334] or [7, Ch. 2, Thm. 8.1, p. 47], which states the following : let f 2 Lp , 1 p 1, t > 0, and k 2 N ; then Z1 !k+1(f; u)p du k k !k (f; t)p log 2 t (32) uk u: t
In particular, assuming k = 1, then (32) implies that there is some c > 0 such that
Z1 !2(f; u)p du !(f; t)p c t u u; t
for all f 2 Lp and t > 0. We thus may conclude that
0 B@Z !(f; t)p q t j log tj 0 1 2
1q 0 1q Z q 1 !(f; t)p dt C dt C = B t A @ j log tj q t tA 0 8 3q 1q 9 2 0 > > Z Z = < d u d t 1 ! ( f; u ) 7 C 6 B 2 p : c >kf jLpk + @ j log tj q 4 u u 5 t A> ; : t 0 1
1
1 2
1 2
1
1 2
By the monotonicity of the log-function we have
Z !2(f; u)p du Z !2(f; u)p du Z !2(f; u)p du c j log t j c j log t j u u u j log uj u u j log uj u t t 0 1 2
1 2
1 2
(33)
12
3. Embeddings
for any t with 0 < t < 21 ; recall 0. Thus (33) implies
0 Z f; t)p q B @ t!j (log tj 0 1 2
8 1q 1q 2 39 0 > > Z Z < 1 dt C c kf jL k + B dt C 6 !2 (f; u)p du 7= p @ j log tj( ? )q t A 4 u j log uj u 5> > tA ; : 0 0
c0
f jBp;(11;? )
; 1
1
1 2
1 2
where the last inequality is correct for < ? q1 . In case of q = 1 one has to modify the above argument in an obvious manner. Hence (28) is shown. Step 2. We prove (29). Let rst 0 < q 1, then (31) has to be replaced by
1q 8 0 " 0 > #q Z q < B@ !(f; t)p dt CA c kf jLpk + B@Z !2(f; t)p > t j log tj t t j log tj ? q : 0 0 1
1 2
1 2
1
1q 9 > dt C = : t A> ; 1
(34)
Recall 0 < q 1, thus (33) and Fubini's theorem yield
8 1q 0 1q 9 0 > > Z Z Z 1 !(f; t)p q dt C c u u t A> ; : t 0 0 8 0 1q 9 > > u Z Z q < = 1 d u d t ! ( f; u ) B C 2 p = c >kf jLpk + @ u j log tj q t u A > : ; 0 0 8 1q 9 0 " > # q Z !2(f; u)p du C > < = B 0 c >kf jLpk + @ ; A ? q u > u j log u j : ; 0 1 2
1
1 2
1
1 2
1
1 2
1
1 2
1
where we also used > 1=q. This gives (34), i.e. (29) for 0 < q 1. Assume now 1 q 1. Put := 1 + ( ? 1q ) > 1, i.e. ? q1 = ? 1. The just proved result (29) with q = 1 implies (35) Bp;(11;?( ? q )) = Bp;(11;?( ?1)) ,! Lip(1p;;?1 ) : Moreover, (21) provides Lip(1p;;?1 ) ,! Lip(1p;;?q ) because q 1 and ? 1 = ? 1q . Thus we can continue (35) to the desired result (29), now for 1 q 1. 1
Step 3. It remains to show (30). We gain from Proposition 16 in the following way: by (21) we 1 have Lip(1p;;?q ) ,! Lip(1p;;?1( ? q )) and can thus reduce (30) to the veri cation of Lip(1p;;?1 ) ,! Bp;(11;? ) for some > 0 and 1 p 1. But this is obvious by (15), (5) and !2 (f; t)p c !(f; t)p .
Remark 24 Note that for p = q = 1 assertions (28), (29) and (30) coincide with [9, Prop. 4.2]. (1;? ) for > 1=v ) in Moreover, by Step 3 of the above proof (in particular, Lip(1p;;?v ) ,! Bp;v connection with Proposition 16 we immediately obtain the following extension of (30).
4. Some discussion
13
Corollary 25 Let 1 p 1, 0 < q; v 1, > 1=q, > 1=v. Then 8 1 < ? v ? 1q ; v q ; (1 ; ? ) (1 ; ? ) if Lip p; q ,! Bp;v : ? v1 > ? 1q ; v < q :
(36)
On the other hand, we may also complement (29) by a similar assertion.
Corollary 26 Let 1 p 1, 1 q 1, > 1. Then (1;?( ?1)) ,! Lip(1;? ) : Bp;q p; q
(37)
P r o o f : We proceed similarly to the proof of Corollary 23. Now (31) has to be replaced by
0 1q 8 0 > Z q < B@ !(f; t)p dt CA c kf jLpk + B@Z !2(f; t)p q > t j log tj t t j log tj ?1 : 0 0 1
1 2
1 2
1q 9 > dt C = : t A> ; 1
(38)
Recall q 1, thus (33) together with Holder's inequality and Fubini's theorem imply
8 0 0 1q 1q 9 > > Z Z Z q q < 1 !2 (f; u)p du dt C = B@ !(f; t)p dt CA c kf jLpk + B@ q( ? q0 ) > t j log tj t u u t A> ; : t 0 0 j log tj 8 1q 9 0 > > u Z !2(f; u)p q Z < = 1 d u d t C B = c >kf jLpk + @ A q( ? q0 ) t u > u : ; 0 j log tj 0 8 1q 9 0 > Z q du > = < ! ( f; u ) B 2 p c0 >kf jLpk + @ u j log uj ?1 u C A>; ; : 0 1 2
1
1
1 2
1 2
1
1
1 2
1
1 2
1
because > 1. Hence (38) is veri ed. Recall the notation for spaces C (1;? ) , 0, see (16) with s = 1, r = 2. We proved in [8, Prop. 2.7] that Lip(11;?; 1 ) ,! C (1;? ) ,! Lip(11;?; 1 ) (39) if, and only if, ; and + 1: Hence (37) coincides with the right-hand embedding in (39) when p = q = 1.
4 Some discussion
(s;b) , introduced by Leopold in [16], and We compare `logarithmically smooth' Besov spaces Bp;q `logarithmic' Lipschitz spaces Lip(1p;;?q ). From the point of dealing with these spaces in view of (s;b) , arise by a atomic decompositions etc. it is essential that the logarithmic B -spaces, that is Bp;q s ), see (6), whereas the logarithmic Lipschitz Fourier-analytical approach (like the usual spaces Bp;q
14
4. Some discussion
spaces Lip(1p;;?q ) , de ned via rst di erences, see (5), remain as `Fourier-unfriendly' as were their classical forerunners (with p = q = 1, = 0). In fact, the almost inconspicuous modi cation in (5) compared with (15), namely the substitution of !2 (f; t)p by !1 (f; t)p , causes a striking di erence in the features of the corresponding spaces (as it does for = b = 0). (s;b) , obtained by Leopold in [16, We return to Proposition 16. The counterpart for spaces Bp;q Thm. 1], reads as follows : Let s 2 R, b1 ; b2 2 R, 0 < p 1 and 0 < q1 ; q2 1. Then ( b ?b 0 ; q 1 q2 1 2 ( s;b ) ( s;b ) (40) if, and only if, Bp;q11 ,! Bp;q22 1 1
b1 ? b2 > q2 ? q1 ; q1 > q2 :
It is obvious, that { though (21) and (40) appear related somehow { the role played by the parameter q in either case is obviously di erent. The `diagonal argument' (essentially used in Step 3 of the proof of Proposition 16 and borrowed from Bennett and Rudnick) does not apply in that (s;b) and case. In other words, the parallel notation (taking the same parameter q) in both cases Bp;q (1 ; ? ) Lip p; q , respectively, is a dangerous one (though suggestive in either case), possibly pretending at rst glance that the construction involving q might be the same; however, it is not. On the other hand, it is nevertheless surprising that the ` ne index' q in these limiting cases becomes so important. Furthermore, we study the question now `where' the Lipschitz spaces Lip(1p;;?q ) can be found within (s;b) . Let 1 p 1 and 0 < q 1. Concerning the scale of the scale of Besov spaces Bp;q (1 ;b logarithmic Besov spaces Bp;q ) for xed p and q, but arbitrary b 2 R, we may locate the Lipschitz spaces Lip(1p;;?q ) as follows. Denote by q := min(q; 1) and assume > 1=q . Then (1;?( ? q )) (1;? ) ; ,! Lip(1p;;?q ) ,! Bp;q Bp;q 1
(41) see (29), (36) and (37). Insisting, however, on the same (logarithmic) smoothness in both nestling (1;b) , that is, for xed p and b, but varying q , we found spaces of type Bp;q
(1;?( ? q )) (42) Bp;q ,! Lip(1p;;?q ) ,! Bp;(11;?( ? q )) ; recall (29) and (30). Note, nally, that for 1 < q < 1 the respective initial spaces and endpoint 1
1
spaces in (41) and (42) are incomparable in the 1sense that neither1 of them is contained in the (1;?( ? q )) (1;?( ? q )) (1;? ) and corresponding other one; this refers to Bp;q and Bp;q as well as to Bp;q 1 Bp;(11;?( ? q )) , respectively. Obviously they coincide, respectively, when 0 < q 1 (in case of the initial spaces) and when q = 1 (concerning the endpoint spaces). Thus we have the general situation that (1;?( ? q )) B p;q % & 1
(1;?( ? q )) Bp;q 1
&
(1;?( ? q )) Bp;q 1
Lip(1;? ) p; q
%
% &
(1;? ) Bp;q
Bp;(11;?( ? q )) 1
& B (1;? ) : (43) % p;1
(1;?( ? q )) Recall that we have the same diagram with Lip(1p;;?q ) replaced by Bp;q . These spaces, however, are not comparable (in the above sense) when 1 < q < 1. On the one hand, one might strengthen structural arguments to disprove this assumption, but on the other hand it can also be (1;?( ? q1 )) (1 ; ? ) for some q < 1. But functions of type f{;0 seen as follows. Assume that Lip p; q ,! Bp;q (1;?( ? q1 )) (1 ; ? ) choosing { such that ( ? q1 ) ? q1 < { < ( ? 1q ). given by (22) belong to Lip p; q n Bp;q 1 (1;?( ? q )) ,! Lip(1p;;?q ) for some q > 1, then one can easily disprove Conversely, assume that Bp;q this assertion for p = 1 by the following argument : choose u > q, thus (40) and (21) imply that 1
(1;?( ? q )) 1 ,! Lip(11;?; q ) ,! Lip(11;?; 1( ? q )) B1 ;u ,! B1;q 1
1
4. Some discussion
15
(1;?( ? q )) 1 we obtained in [9, for ? q1 > q1 ? u1 . However, studying the embedding B1 ;u ,! Lip 1; 1 Thm. 2.1 (ii)] that the exponent ? 1q = 1 ? u1 is sharp, see also (17); but 1 ? u1 > q1 ? u1 for (1;?( ? q1 )) cannot be contained in Lip(1p;;?q ) for q > 1 (at least when p = 1, but q > 1. Thus Bp;q there should be a similar counter-argument when 1 p < 1). Hence the above diagram (43) looks in general like 1
(1;?( ? q )) & % Bp;q 1
(1;?( ? q1 )) Bp;q
&
(1;?( ? q1 )) Bp;q
%
Lip(1p;;?q ) (1;?( ? q )) Bp;q 1
% &
(1;? ) Bp;q (1;?( ? q ))
Bp;1
1
& B (1;? ) ; % p;1
(44)
where the left part of (44) collapses to one space for 0 < q 1 and likewise the right one when q = 1. There are a lot of further related approaches to spaces of Lipschitz type, recall Remark 3. Let us nally mention only a few, more recent papers : Aksoy and Maligranda, see [1], studied descriptions of spaces of Lipschitz-Orlicz type Lip( ; LM ) and Zyg( ; LM ) in terms of Poisson integrals; Brandolini, [5], introduced generalised Lipschitz spaces, i.e. spaces of type X (Rn ), > 0 and X being either Lp;1 (Rn ) or Lp (Rn ), given by
kf j X (Rn )k = kf jX k + sup !k (f;t )X : >0
0) , the modi cation In other words, for X = Lp (Rn ) and = 1 these are the above spaces Lip(1p;;1 for X = Lp;1 (Rn ) and arbitrary > 0 are clear. Finally, the closest approach we found in the literature so far { really dealing with logarithmic or similar modi cations of the usual Lipschitz spaces { is given in the paper [4] by Bloom and De Sousa. They concentrated on weighted Lipschitz spaces of type Lip %, where % : [0; 2] ! [0; 1) is a nondecreasing weight function with %(0) = 0. With a slight modi cation we may regard % (t) tj log tj , t > 0 small, as such a weight, belonging to their classes bp for p > 1, but % 2= b1, see [4, Def. 1]. Here some weight % is in the class bp , p 1, if
Z2 %(t)
R
h
%(h) tp+1 dt C hp :
Moreover, % is Dini, as 0h %(tt) dt c %(h) for each h > 0. In their notation we obtain that Lip % = Lip(11;?; 1 ) and for the Zygmund spaces (% ) = C (1;? ). Recall that similar spaces of type C 0; (t) ( ), Rn , were introduced by Kufner, John and Fuck in [15, Def. 7.2.12, p. 361], see Remark 3.
References [1] A.G. Aksoy and L. Maligranda. Lipschitz-Orlicz spaces and the Laplace equation. Math. Nachr., 178:81{101, 1996. [2] C. Bennett and K. Rudnick. On Lorentz-Zygmund spaces. Dissertationes Math., 175:1{72, 1980. [3] C. Bennett and R. Sharpley. Interpolation of operators. Academic Press, Boston, 1988. [4] S. Bloom and G.S. De Souza. Weighted Lipschitz spaces and their analytic characterizations. Constr. Approx., 10:339{376, 1994.
16
4. Some discussion
[5] L. Brandolini. Imbedding theorems for Lipschitz spaces generated by the weak-Lp metric. J. Approx. Theory, 94:173{190, 1998. [6] H. Brezis and S. Wainger. A note on limiting cases of Sobolev embeddings and convolution inequalities. Comm. Partial Di erential Equations, 5:773{789, 1980. [7] R. DeVore and G.G. Lorentz. Constructive Approximation. Springer, Berlin, 1993. [8] D.E. Edmunds and D.D. Haroske. Embeddings in spaces of Lipschitz type, entropy and approximation numbers, and applications. Forschungsergebnisse Math/Inf/98/31, Universitat Jena, Germany, 1998. [9] D.E. Edmunds and D.D. Haroske. Spaces of Lipschitz type, embeddings and entropy numbers. Dissertationes Math., 380:1{43, 1999. [10] D.E. Edmunds and H. Triebel. Function Spaces, Entropy Numbers, Di erential Operators. Cambridge Univ. Press, Cambridge, 1996. [11] W.D. Evans, B. Opic, and L. Pick. Interpolation of operators on scales of generalized LorentzZygmund spaces. Math. Nachr., 182:127{181, 1996. [12] D. Haroske. Some logarithmic function spaces, entropy numbers, applications to spectral theory. Dissertationes Math., 373:1{59, 1998. [13] D.D. Haroske. Logarithmic Sobolev spaces on Rn ; entropy numbers, and some application. Forum Math., to appear. [14] M. Krbec and H.-J. Schmeier. Imbeddings of Brezis-Wainger type. The case of missing derivatives. Preprint 130, Math. Inst. Czech Acad. Sci., Prague, 1998. [15] A. Kufner, O. John, and S. Fucik. Function spaces. Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977. [16] H.-G. Leopold. Limiting embeddings and entropy numbers. Forschungsergebnisse Math/Inf/98/05, Universitat Jena, Germany, 1998. [17] P.-L. Lions. Mathematical Topics in Fluid Mechanics. Vol. 2 : Compressible Models. Oxford Lecture Series in Mathematics and its Applications, 10. Clarendon Press, Oxford, 1998. [18] H. Triebel. Theory of Function Spaces. Birkhauser, Basel, 1983. [19] H. Triebel. Theory of Function Spaces II. Birkhauser, Basel, 1992. [20] M. Vishik. Hydrodynamics in Besov spaces. Arch. Rational Mech. Anal., 145:197{214, 1998.
Dorothee D. Haroske Mathematical Institute Friedrich-Schiller-University Jena D-07740 Jena Germany
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