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Владикавказский математический журнал Апрель-июнь, 2005, Том 7, Выпуск 2
UDC 517.9
BERNSTEIN-NIKOLSKII TYPE INEQUALITY IN LORENTZ SPACES AND RELATED TOPICS
H. H. Bang, N. M. Cong
Dedicated to academician S. M. Nikolskii on the occasion of his 100th-birthday
In this paper we study the Bernstein-Nikolskii type inequality, the inverse Bernstein theorem and some properties of functions and their spectrum in Lorentz spaces Lp,q(Rn).
1. Introduction
The study of properties of functions in the connection with their spectrum has been implemented by many authors (see, for example, [1-16] and their references). Some geometrical properties of spectrums of functions and relations with the sequence of norms of derivatives (in Orlicz spaces and -spaces) were studied in [1-9]. In this paper we give some results on the Bernstein-Nikolskii type inequality, the inverse Bernstein theorem and some properties of functions and their spectrum in Lorentz spaces Lp'q (rn).
Let us recall some notations. If f £ S' then the spectrum of f is defined to be the support of its Fourier transform f (see [14, 15]). Denote sp(f) = suppf and |E| the Lebesgue measure of E. For an arbitrary measurable function f : rn ^ c (or r), one defines (see [17-22])
Af (y):=|{x £ rn : |f(x)| >y}|, y> 0, f *(t) := inf{y > 0: Af (y) < t}, t> 0,
i/q
p,q
( / 00 \ 1/q
(p 0 (i1/pZ*(t))qdi) , 0 <p< 0 < q <
0 < p ^ <x, q = ж.
sup t1/p f *(t), t> о
Then the Lorentz spaces Lp'q (on rn) are by definition the collection of all measurable functions f such that ||f | |p,q < The case p = to, 0 <q< to is not considered since
oo dt
S (f*(t^ — < to implies f = 0 a. e. (see [17]). Furthermore, there is an alternative 0 t
representation of
\p,q (see, for example, [17, 20])
i/q
p,q
( / oo / \ 1/q
^q 0 yq-1\y (y)dyj , 0 <p< то, 0 < q <
,
sup yA1/p(y), y>o
0 < p ^ то, q = то.
© 2005 Bang H. H., Cong N. M.
In this paper, for p, q fixed, we always let r such that 0 < r ^ 1, r ^ q, and r < p. There are two useful analogues of f * used in some below proofs: Let (see [17])
1/r
1 r ,,r . \
t > 0.
f **(t) = f **(t,r) := supf^ / |f (x)|rdx^ ,
|E 1E J
E
Then, (f**)* = f**, and
i X 1/r
(fT*(t) = (j /(f*(y))rd^ =: f***(t), t > 0.
0
It is known that f *,f** and f *** are non-negative, non-increasing, and
f * ^ f ** ^ f ***
If f * is replaced by f ** or f *** in the expression of ||f\\pq then one gets by definition or |f ||p*q* respectively. It is well-known that || ■ ||p*q is a norm when 1 < p ^ to, 1 ^ q ^ to (set r = 1 in this case), and moreover, Lp'q can be considered as Banach spaces if and only if p = q = 1 or 1 <p ^ to, 1 ^ q ^ to (see [17]). In particular there is at that an useful relation among || ■ ||p>q, || ■ ||p*q and || ■ ||p*q* (see [17])
< ||f iiPrq < |f ||P*q* < (p/(p - r))1/r
Henceforth, Q is a compact subset of and
Av = {e G : foI < Vj, j = 1,...,n} where V = (v1, ..., vn), Vj > 0, j = 1,..., n. Denote by
L£q = {f G Lp,q n S' : sp(f) C Q}.
iv, is denol
2. Results
When Q = Av, Lnq is denoted again by Lp'q. Similarly one has Sn or Sv respectively.
First we give some results on the Bernstein—Nikolskii type inequality for Lorentz spaces. Lemma 1. Let 0 < pi < p2 ^ to, 0 < qi,q2 ^ to. Then for each multi-index a, there exists a positive constant c such that for all ^ G Sn
W-D^Wp!,© < c|MU,qi. (1)
< Step 1 ( p2 = q2 = to and a = (0,..., 0)). Let ^ G S such that ?/>(x) = 1 in some neighbourhood of Q. Then for any x G
|p(x)| = * ^(x)| ^J |p(x - y)^(y)|dy ^ p(x -r(i)r(i)dt
0
CO CO
= J <^W(i)dt < IMIl-r| (t1/pi </(t))rt-r/p1 V*(t)dt
0
CO
< IMlCo-rIMIJi,«/t-r/p1 ^*(t)di = II^IU/fe-rylMCy^ypi;c. 01
C
This deduces at once
Pl ;l№llpi/(pi-r) ^
~ ^ I ~—~imipi/(pi-r),i / H^Hpi,^
vpi — r
Step 2 (a = (0,..., 0)). We only have to show that there is a constant c such that
P £ Sn, (2)
where 0 < p1 < p2 < to, 0 < q2 < to.
Indeed, using the alternative representation of || ■ 11p,q , we have
CO llvllro
= q2J yq2-1A®/p2 (y)dy = Q2 J yq2-1A®/p2 (y)dy 0 0
// / \ ~pi 1 52 / f 52(P2 —Pi) i
(yAl/pi (y)J P2 yq2-1-^pidy < q2 lMiP2iPi/p2 J y— 1dy
0
P2 II.-Il?2pi/p2|| ®(p2-pi)/p2 < C P2 ||m|| P2
... pi Mr Mo ^ Nr ||pi ,C
p2 — pi 1 p2 — pi 1
where the last inequality follows from Step 1. Therefore (2) is obtained.
Step 3. We prove (1) when p1 = p2 = p, q1 = q2 = q. If ^ £ Sn then £ Sn for every multi-index a. Denote by the Hardy—Littlewood maximal function of then (see [14, p. 16]) for all x £ Rn
|D>(x)| < ci((M|^|r)(x))1/r,
where ci is a constant depending only on Moreover it is known that for every measurable function f (see, for example, [18, 19])
t
(Mf )*(t) - 1 J f *(s)ds.
0
Hence,
(D»* < ci((M|^|r)1/r)* = ci((M|^|r)*)1/r < C2^***,
and consequently,
||D>||p,q < C2|^|prq* < C3 11^|p,q. (3)
Step 4. The general case follows immediately from (2), (3) and the property || ' ||p,C ^ || ' ||p,q• The proof so has been fulfilled. >
The theorem below is an extension of the Theorems 1.4.1(i) and 1.4.2 in [16]. Theorem 1. Let 0 < p1 < p2 ^ to, 0 < q1, q2 ^ to.
(i) If a is a multi-index, then there exists a constant c such that for all f £ Lp1^1
||Df ||p2,q2 < c||f ||pi,qi.
(ii) Lpq is a quasi-Banach space for arbitrary 0 < p, q ^ to, and the following topological embeddings hold
Sn C Lpi'qi C LP2'q2 C S'.
< (i): Without loss of generality, one can assume that qi = to and 0 < pi,p2, q2 < to (note that the case pi = to and so, pi = p2 = qi = q2 = to, was proved in [16, Theorem 1.4.1]). Let pi < p < to, and let ^ G S such that ^>(0) = 1 and sp(^) C {x : |x| ^ 1}. For each f G Lp1'^ and 0 < 5 < 1, put (x) = ^>(5x)f (x). Then f ^ f on rn and f G Sn1, where
Qi = {y G rn : 3 x G Q such that |x - y| ^ 1}.
Consequently, it follows from Lemma 1 that
||fWP < M||f5Hp < ci Urn||f5Wpi'^ < ^MU"Wpi^
5\0 5\0
where ci is independent of 5 and f. Hence f G LQ. Now the argument in [16, Theorem 1.4.1] implies that Daf,5 —> Daf in L^ (and this show that the conclusion is true if p2 = q2 = to). Lemma 1 therefore deduces again that
W^ |P2,q2 < Hm W-D" | |P2 ,q2 < c Hm ||f5 Wpi'^ < c|MU||f Wpi'~ < c W^U" | |pi ,qi , 5\0 5\0
where c depends only on pi,p2, q2 and Q.
(ii): First, we show that L^'q is a quasi-Banach space for any 0 < p, q ^ to. Let {fj} be any fundamental sequence in L^'q. Then there is a function f G Lp'q such that fj ^ f in Lp'q as j ^ to.
Moreover, part (i) above with a = (0,..., 0) and p2 = q2 = to shows that {fj} is also a fundamental sequence in L^. Then it implies by standard arguments that fj ^ f in L^, and consequently, fj ^ f in S'. Hence fj ^ f in S' and this yields that sp(f) C Q. Therefore f G L^'q and fj ^ f in Lp'q, and it follows that L^'q is a quasi-Banach space.
Part (i) deduces immediately that Lpi'qi C L^2'q2. Moreover, if 0 < 0 < p < k ^ to, then for any q > 0 (see [16, Theorem 1.4.2])
Sn C LQ c Lp,q C LQ c S'. >
It is difficult to get concrete and good constants for Nikolskii inequality for Lorentz spaces L^'q. Following some ideas in [13], we have a version of the Nikolskii inequality for Lorentz spaces.
Theorem 2. (i) If 0 < pi < 2, then for p2 > pi, q2 > 0,
|"< (^)i/pi-i/p2^ f G LT;
(ii) If 0 < pi < to, then for p2 > pi, q2 > 0,
^ (p2—pi )i/q2 ()i/pi-i/p2 |pi,qi ■ f G
where co(Q) denotes the convex hull of Q and p0 is the smallest integer number such that p0 > pi/2.
< (i): Suppose that 0 < pi < 2, 0 < qi ^ to and f G Lpi'qi, then by Theorem 1, f G L2, so it follows from [13, Theorem 3] that
(11/IU \ i/2
J yA/(y)dyj
0
= |Q|i/^ J (yA//pi(y))piyi-pidyy < IQIi/2||fHp^"-p^ ' .
Therefore,
||f <( TT \\J llpi.oo-
V2 — pv
Applying now the argument in Step 2 of the proof of Lemma 1, we can obtain a similar inequality
( p2 \1/q2 ^ ^
P2,q2 < ( ||f ||f ||C .
p2 — pi
Hence,
11/11 _P2 X1/q2 ( M )i/pi-i/p2,
||f|p2,q2 Mpr—VJ
(ii): Since 0 < pi/po < 2, we get immediately
imi =i |fP^| i/po < ( p2/po ) q2 (|co(sp(f p0 ))| ) ir - |fp0||
||f ||p2,q2 = ||f ||p2/p0>q2/p0 < Vp2/p0 — p1/pj V 2 — p1/p^ ||f ||pi/p0>qi/p0
< (¡T)A((f)"-"»/H- < (^)*()"-*|fHp-
The theorem is proved. >
Lemma 2. Let 1 < p < to, 0 < q < to. If f £ Lp'q, then f £ S' and for any g £ L1
||f * g|p,q < c|f ||p,q ||g 11,
where c is a constant depending only on p, q.
< Firstly, we show that f £ S'. Let E C rn such that 0 < |E| < to. Then the Holder inequality implies
|E| |E| |E|
J |f (x)|dx < J f *(t)dt = J (t1/pf*(t))t-i/pdt < ||f ||p,^ t-i/pdt = c(E)||f ||p,c.
p, C
0 0 0
This deduces easily that f £ S'.
Now, we prove the last conclusion. For an arbitrary t > 0, we define
t
f W(t) = !/ f *(y)dy. 0
Then for any E C rn such that t < |E| < to we have by Jensen's inequality
\W\S |f *g(x)|r dy < E\S |f *g(x)|dx < / |g(y){^/ |f (x—y)|d^dy < f«
|
E ' E x E
Hence,
||f * g||p,q < ||f * g||p*q < ||f| |p,q||g|i
i
It now yields from [22, Lemma 3.2] the existence of a constant c such that (in the case p > 1)
||f (*)|p'q < c||f ||p,q, f G Lp'q,
The lemma therefore is proved completely. >
Theorem 3. Let f G Lp'q (1 < p < to, 0 < q ^ to) such that f ^ 0. Then sp(f) contains only points of condensation.
< Let £0 G sp(f) be an arbitrary point, and let V be any neighbourhood of £0. Choose <(£) G C0°°(rn) such that <(£) = 1 in V. Then by Lemma 2, F-i (<f) = < * f G Lp'q. Hence we can assume that sp(f) is bounded, moreover we merely have to show that sp(f) is uncountable.
It deduces from Theorem 1 that there is a positive integer m such that f G Lm(rn). Hence (fmy g C0(rn). Since f ^ 0, there exists a non-void ball B such that
B C sp(fm) = suppf * ■ ■ ■ * f) (m terms) C sp(f) + ■ ■ ■ + sp(f).
Therefore it follows at once that sp(f) is uncountable. >
It is noticeable that Theorem 3 is a corollary of the following theorem which can be proved by the same method used in [4, Theorem 1].
Theorem 4. Let f G Lp'q (1 < p < to, 0 < q ^ to), f ^ 0 and £0 G sp(f) be an arbitrary point. Then the restriction of f on any neighbourhood of £0 cannot concentrate on any finite number of hyperplanes.
It is trivial that A/(y) < to for all y > 0, f G Lp'q if p < to. Then by the argument used in [7, Theorem 3] and Theorem 1, a property of such functions can be formulated as follows. Theorem 5. If f G Lp'q n S' (0 < p < to, 0 < q ^ to) such that sp(f) is bounded, then
lim f(x) = 0.
|x| —
Remark 1. In contrast with hyperplanes, f may concentrate on surfaces (see [4, Remark 2]). In addition, Theorems 3-5 are not true when p = to, i. e., p = q = to (see
[4, 7]).
To obtain more properties of functions with bounded spectrum, we prove an auxiliary result which is interesting in itself.
Theorem 6. If f G Lp'q (0 < p, q < to), then
lim ||f (a.x) - f (x)||p,q = 0, (4)
a—i
where 1 = (1,..., 1) and a.x = (aixi,..., anxn) for all a, x G rn.
< It is known in [17] that the set A of all measurable simple functions with bounded support is dense in Lp'q if 0 < q < to. Therefore, it suffices to show (4) for each f G A. Hence, let f G A and assume on the contrary that there exist {ak} C rn, ak ^ 1, and e > 0 such that
||ffc - f ||p,q > e, k > 1, (5)
where f (x) = f (ak.x). Since f G Lioc(rn), then for each K = [-¿]n, one obtains
J If (x) - f (x)|dx ^ 0, as k ^ to.
So there is a subsequence of {ak}, which is still denoted by {ak}, such that f ^ f a. e. on K. Therefore, there exists a subsequence, denoted again by {ak}, such that f ^ f a. e. on Rn. Consequently,
lim f*(t) ^ f *(t), t> 0.
fc—m
Furthermore, it is easy to verify that
||fk 11 p,q = K ••• a*)"1 ||f |p,q. The Fatou lemma then yields for arbitrary 0 < u < v < to
CO
En J tq/p-i f*q (t)dt = j—in i J tq/p-iffc*q (t)dt -J tq/p-iffc*q (t)dt)
0 ^ 0 u '
CO CO u
< p kLimr ||fk ||p,q - lim / tq/p-ifk*p(t)dt < p ||f ||p,q - [ tq/p-if *q (t)dt = / tq/p-if *q(t)dt,
q fc—<W q J J
u
and similarly,
/C /C
nm f tq/p-ifk*q(t)dt < / tq/p-if*q(t)dt. j J
v v
Hence, if u < v/2 are chosen such that for c = max(2q-i, 1)
J tq/p-if*q(t)dt <5, J tq/p-if*q(t)dt < 5, (6)
0 v/2
where 5 = peq/(3.2q/p.q.c), then there is a positive constant Ni such that for all k > Ni
/u /C
J tq/p-ifk*q(t)dt <5, J tq/p-ifk*q(t)dt < 5. (7)
0 v/2
Therefore, it follows from (6), (7), and the inequality (f + g)*(t) ^ f *(t/2) + g *(t/2), that for all k > Ni
/u /u /u
J tq/p-i (ffe - f) *q(t)dt < c( J tq/p-ifk*q(t/2)dt + J tq/p-if *q(t/2)dtj
0 0 0 (8)
/ u u . (8)
, ^ J (t)dt + J i,/p-ir , ^ < ^
00
Similarly, one obtains for all k > Ni
/C /C /C
J tq/p-i(ffc - f) *q (t)dt < cf J tq/p-ifk*q (t/2)dt + J tq/p-if *q (t/2)dtj
v ^ v v '
(CO CO V
2t,/p_i^(i)di \lt,/p_ir q (i)iT^
v/2 v/2
u
C
u
C
Next, since ak ^ 1 and suppf is bounded, there is a ball B including suppf such that suppf C B, for all k ^ 1. Thus taking account of f ^ f a. e. on it deduces that f ^ f in measure. Then the definition of the non-increasing rearrangement of a measurable function yields for every t > 0 that
(fk - /)*(t) 0, as k ^to. Applying the dominated convergence theorem, one arrives at
v
J tq/p-1(fk - /)*q(t)dt ^ 0, as k ^ to.
u
Consequently, there exists a number N2 > N1 such that for all k > N2
v
j t^-Vk - f r (t)dt< iqeq. (10)
u
Combining (8), (9) and (10), it is evident that for all k > N2
CO
p llfk - f lip,q = / tq/p-1(fk - f )*q(t)dt< 2q/p+1c£ + peq/3 = peq. 0
This contradicts (5). >
Remark 2. It is well-known that Lp'q can be considered as Banach spaces if and only if p = q = 1 or 1 < p ^ to, 1 ^ q ^ to. Using Theorem 1 and the method of [14], one can obtain the Bernstein inequality for Lp'q spaces in these cases: If f £ LV'q, then there is a constant 1 ^ c ^ e1/p such that
l|DQf < CVa|f (11)
holds for any multi-index a. Moreover this inequality still holds when p =1. Indeed, it yields at once from the dominated convergence theorem when p = 1,1 ^ q < to that ||f 11^ ||f 111 as p \ 1, and the claim follows. Therefore we have only to show that this convergence is also true when q = to and imply directly the desired. Suppose that ||f ||P)C ^ ||f ||1,C as p \ 1. Then there is e > 0 and {pn}, pn \ 1, such that:
Case 1. ||f ||pn,C < ||f ||1,C - e, n ^ 1. Thus there exists 0 < u < ||f ||C such that
l1/Pn
sup yAyPn(y) < uAf (u) - e/2, 0<y<||/
and hence, uA^"(u) < uA/(u) — e/2. Let n ^ to, we get a contradiction.
Case 2. ||/||Pn)<» > ||/||i,<» + e, n ^ 1. Then there is a sequence {y„}, 0 < y„ < such that
y^A^" (yn) > y„A/ (yn) + e/2.
It is easy to see from Theorem 5 and the continuity of / that A/ is continuous. Therefore let v be any accumulative point of {yn} and let n ^ to in the last inequality, we also have a contradiction and then the claim is proved.
Furthermore, using the argument in [7, Theorem 6], one can get a stronger result.
Theorem 7. If Vj > 0, j = 1,..., n and 1 ^ p, q < to, then for all f G LV'q
lim v-a||Daf||p,q = 0.
|a|—C
Remark 3. Applying the Bernstein inequality we have v-a||Daf ||p,q ^ V
^ |Df ||p,q if
a ^ P for such above p, q. Moreover, Theorems 6, 7 fail if p = q = to. But we still don't know what happens if p < to, q = to.
Let us recall some notations about the directional derivatives. Suppose that a = (ai,..., an) G rn is an arbitrary real unit vector. Then
Daf(x) = fa(x) := £ aj f (x) j=i j
is the derivative of f at the point x in the direction a, and
Dmf(x) = Dafam-i) = £ aaDf (x)
|a|=m
is the derivative of order m of f at x in the direction a (m = 1, 2,...).
Denote ha(f) = sup |a£|. By an argument similar to the proof of [8, Theorem 2], one £esp(/)
can obtain the corresponding results for directional derivatives cases in certain Lorentz spaces.
Theorem 8. If 1 ^ p, q ^ to, then there is a constant 1 ^ c ^ ei/p such that for all f G Lp'q n S' satisfying ha(f) < to
||Daf||p,q < cha(f)||f||p,q. (12)
Theorem 9. If f G Lp'q n S' (1 ^ p, q < to) is such that ha(f) < to, then
—m (ha(f ))-m|Dmf ||p,q = 0.
m—+C
It is clearly that one can let c =1 in (11) and (12) if || ■ ||p,q is a norm, and let c = ei/p in general case.
Finally, we will show that the Bernstein inequality wholly characterizes the spaces Lp'q in the case they are normable.
Theorem 10. Suppose that p = q = 1 or 1 <p ^ to, 1 ^ q ^ to and f G S'. Then in order that f G LV'q it is necessary and sufficient that there exists a constant c = c(f) such that
||Daf ||p,q < cva, a G z++. (13)
< Only sufficiency hod to be verified. Assume that (13) holds.
Case 1 (1 < p < to, 1 < q < to). If g G Lp'q(rn), then g G Lioc(rn) by the first part of the proof of Lemma 2. It hence deduces from (13) that Daf G L^^r^ for all a ^ 0. Consequently, we can assume that f G Cc(rn) by virtue of Sobolev embedding theorem. Next let u G Cg°(rn) such that ||w||i = 1, and define for each e > 0
fe(x) = f * ue(x),
where we(x) = e-nw(x/e). Then /e(x) ^ / (x) as e j 0, for every x £ rn. Moreover, by the argument at the first step of Lemma 1 (recall that r = 1 in this case), one has for each multi-index a
sup |DQ/e(x)| < 6e||Da/e||p^ < MD/^ < B£ (14)
where Be > 0 is a constant depending only on e. Thus the Taylor series
ro
E a Da/e(0)-za
|a|=0
converges for any point z £ cn and represents f (x) in rn. Hence taking account of (14), we obtain
n
|/e(z)| < B exp(^ Vj |Zj |J , z £ C™,
i. e., /e(z) is an entire function of exponential type V. It therefore follows from the Paley-Wiener-Schwartz theorem that
sp(/e) = supp / C Av. (15)
Therefore, Theorem 1 and Lemma 2 yield that for each e > 0
|/e|p+l ^ C1 ^ c2 = C2 ||/Hp.ro•
The Banach-Alaoglu theorem hence implies that there are a sequence {en} and an / £ Lp+1(rn) such that /£n ^ / weakly in Lp+1 (rn) as e j 0. Then by standard arguments, one has / = / a. e., that is, /£n ^ / weakly in Lp+1(rn). Because S C L(p+1)/p(rn), the dual space of Lp+1(rn), it follows immediately that /£n ^ / in S'. Consequently, /£n ^ / in S' and this deduces at once from (15) that sp(/) C Av.
Case 2 (p = q = 1). This case can be proved by above manner.
Case 3 (p = q = to). Let ^ and /5, 0 < 5 < 1, as in the proof of Theorem 1. Then it yields from the Leibniz formula, the Bernstein inequality for Lro and (13) that for all a £ z+
|Da/5(x)| < ^ (p(5x))||D/(x)| < c ^ 5IyIv^ = c(v + 5)a,
Y+/3=a Y+/3=a
where 5 = (5,..., 5). Thus, as in Case 1, /5(z) is an entire function of exponential type V + 5 for each 0 < 5 < 1, and therefore, sp(/) C Av+. Moreover, it is clear that /5 ^ / in S' as 5 j 0. This implies obviously that sp(/) C Av for any 0 < 9 < 1 and then sp(/) C Av. >
Theorem 11. Ifp = q = 1 or 1 <p ^ to, 1 ^ q ^ to, then a function / £ S' belongs to LV'q if and only if
Tim (V-a||Da/||p,q)1/H < 1. (16)
< It is sufficient to prove «only if» part. Given any e > 0, there is a positive constant Ce > 0 such that for all a ^ 0
||Da/||p,q < Ce(1 + e)|a|Va It hence deduces from Theorem 10 that sp(/) = supp F/ C A(1+e)v. Therefore sp(/) C
f1£>0 A(1+e)v = Av. >
Remark 4. It is noticeable that the root 1/|a| in (16) cannot be replaced by any 1/|a| t(a), where 0 < t(a), lim t(a) = +to.
|a|—ro
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Received by the editors January 13, 2005. Ha Huy Bang, Doktor Fiz.-Mat. Nauk, Professor
Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam E-mail: [email protected]
Nguyen Minh Cong, Ph. D.
Hanoi University of Education, Department of Mathematics, Cau Giay, Hanoi, Vietnam E-mail: [email protected]
