Bernstein - Nikolskii type inequality in Lorentz spaces and related topics

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 N Φ -spaces) were studied in [1–9]. In this paper we give some results on the Bernstein–Nikolski˘ı type inequality, the inverse Bernstein theorem and some properties of functions and their spectrum in Lorentz spaces L ^p,q ( R n ) .

Let us recall some notations. If f ∈ S ⁰ then the spectrum of f is defined to be the support of its Fourier transform f (see [14, 15]). Denote sp (f) = supp f and | E | the Lebesgue measure of E . For an arbitrary measurable function f : R n ^ C (or R ), one defines (see [17-22])

^X f ⁽y) ^:= ^{| { x e} R n ^{: | f ( x )|} >y } | , y> 0,

f•(t) := inf{y > 0: Xf (y) 6 t}, t> 0, kf kp,q

^∞                 1 /q

(p / (f /p f • (< f)

sup t1/p f •(t), t>0

0 < p < to, 0 < q < to,

0 < p 6 to, q = to.

Then the Lorentz spaces L ^p,q (on R ⁿ ) are by definition the collection of all measurable functions f such that k f k p, _q <  to . The case p = to , 0 < q <  to is not considered since ∞ _dt

0 (f • (t)) 7 <  to implies f = 0 a. e. (see [17]). Furthermore, there is an alternative representation of k · k p,q (see, for example, [17, 20])

k f k p,q ⁼

(q / yq 1xf/p(y)dy) /q, sup yXfp(y), y>0

0 < p < to, 0 < q < to,

0 < p 6 to, q = to.

(c) 2005 Bang H. H., Cong N. M.

In this paper, for p, q fixed, we always let r such that 0 < r 6 1 , r 6 q , and r < p . There are two useful analogues of f ^∗ used in some below proofs: Let (see [17])

1 /r

A    I f (x) | ^r dx

| E |

E

, t > 0.

f**(t) = f**(t,r) := sup

| E | > t

Then, (f**)* = f**, and t                  1/r

(fT(t) = I 1 /(f*(y))rdy I =: f*"(t), t > 0. 0

It is known that f∗, f∗∗ and f∗∗∗ are non-negative, non-increasing, and f∗ 6 f∗∗ 6 f∗∗∗.

If f ^∗ is replaced by f ^∗∗ or f ^∗∗∗ in the expression of k f k _p,q then one gets by definition k f k ^∗ _p, ^∗ _q or k f k p ** respectively. It is well-known that k • k p * _q is a norm when 1 < p 6 ro , 1 6 q 6 ro (set r = 1 in this case), and moreover, L ^p,q can be considered as Banach spaces if and only if p = q = 1 or 1 < p 6 ro , 1 6 q 6 ro (see [17]). In particular there is at that an useful relation among k · k p,q , k · k ^∗ _p ^∗ _,q and k · k ^∗ _p, ^∗ _q ^∗ (see [17])

k f Ik, 6 k f 1^ , 6 k / BP" 6 ( p/(p — r)) 1 /r k / li p,, •

Henceforth, Q is a compact subset of R n , and

L Q = { f e L ^p’⁴ П S 0 : sp (f) C Q } .

When Q = A _v , L p,⁴ is denoted again by L p’⁴ . Similarly one has S q or S _v respectively.

• 2.    Results

First we give some results on the Bernstein—Nikolski˘ı type inequality for Lorentz spaces.

Lemma 1. Let 0 < pi < p2 6 ro, 0 < qi,q2 6 ro. Then for each multi-index a, there exists a positive constant c such that for all ϕ ∈ SΩ kDaVkp2’42 6 ck^kpi’41.                                       (1)

C Step 1 ( p2 = q 2 = ro and a = (0,..., 0) ). Let ^ e S such that -0(x) = 1 in some neighbourhood of Q . Then for any x e R ⁿ

И^{х )|} = | ^ * W^x)|

⁶ у | ^⁽x R ⁿ

^- y⁾^⁽y^)|dy ⁶

∞

j ^(x — •)*(t)^*(t)dt

∞∞

= yV(tNWt 6 k v kV j (t ^{1 /p} 1 ^*(t) ) r t - ^r/p 1 ^ * (t)dt 00

∞

• 6 k p k^ r k v k p,„ / t ^{- r/p} 1 *"ft)dt = pp —- k * k p , / ( p , - r ) ’ i k p k i- ^r k p k p i ’ ^ 0

  This deduces at once

  
  

k ϕ k ∞ 6 p ₁ p _- ¹ r k ψ k p 1 / ( p 1 - r ) , 1 ^{1 /}r k ^ϕ k p 1 , ∞ ^.

Step 2 ( a = (0,..., 0) ). We only have to show that there is a constant c such that

Il^k p 2 ,q 2 6 c k ^ k p i , ^ ,      ^ ^E S Q ,                                 ⁽²⁾

where 0 < p ₁ < p ₂ <  to , 0 < q ₂ <  to .

Indeed, using the alternative representation of k · k _p,q , we have

∞                       k ϕ k ∞

IMIPI^ = q2 / yq2-1X^p2 (y)dy = q2 I yq2-1X^p2 (y)dy kϕk∞                                                    kϕk∞ q2 p1 - - q2                                     q2(p2-p1) -

(y^ ^/p 1 (y)) ^p2 y      p ² p dy 6 q 2 ll^k p^/p ² J y    p ²      dy

=   1-2k^ISp/p2k^k®(p2-p-)/p2 6 c p2„, p2 - p1    p1,∞     ∞             p2 - p1    p1,∞ where the last inequality follows from Step 1. Therefore (2) is obtained.

Step 3 . We prove (1) when p ₁ = p ₂ = p , q ₁ = q ₂ = q . If ^ G S q then D ^a E E S q for every multi-index α . Denote by M ϕ the Hardy—Littlewood maximal function of ϕ , then (see [14, p. 16]) for all x ∈ R ⁿ

№(х)1 6 C1((MMr)(x))1/r, where ci is a constant depending only on Q. Moreover it is known that for every measurable function f (see, for example, [18, 19])

t

( M f ) * (t) - -J f * (s)ds.

Hence,

(D“^)* 6 ci((M|^|r)1/r)* = ci((M|^|rГ)1/г 6 C2^***, and consequently,

Il D “ v IU 6 C 2 k ^ k p : q * 6 C 3 k ^ k p,q .                               (3)

Step 4. The general case follows immediately from (2), (3) and the property k · k _{p, ∞} 6 k · k _p,q . The proof so has been fulfilled. B

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 6 to, 0 < q1, q2 6 to.

• (i)    If α is a multi-index, then there exists a constant c such that for all f ∈ LpΩ1,q1

k D ^α f k _{p 2 ,q 2} 6 c k f k _{p 1 ,q 1} .

• (ii)    L ^p _Ω ^,q is a quasi-Banach space for arbitrary 0 < p, q 6 to , and the following topological embeddings hold

S Ω ⊂ L ^p _Ω ^{1 ,q 1} ⊂ L ^p _Ω ^{2 ,q 2} ⊂ S ⁰ .

C (i): Without loss of generality, one can assume that q i = ro and 0 < p i ,p2, q2 <  ro (note that the case p i = ro and so, p i = p 2 = q i = q 2 = ro , was proved in [16, Theorem 1.4.1]). Let p i < p <  ro , and let ^ G S such that ^(0) = 1 and sp (^) C { x : | x | 6 1 } . For each f G L f’ ” and 0 < 5 < 1 , put f g (x) = ^(6x)f (x) . Then f g ^ f on R n and f g G S q _i , where

Q i = |y G R n : 3 x G Q such that | x — y | 6 1}.

where c depends only on p i ,p 2 , q 2 and Q .

(ii): First, we show that L ^ q is a quasi-Banach space for any 0 < p,q 6 ro . Let {f j} be any fundamental sequence in L ^p _Ω ^,q . Then there is a function f ∈ L ^p,q such that f j → f in L ^p,q as j → ∞ .

Moreover, part (i) above with a = (0,..., 0) and p 2 = q 2 = ro shows that { f j } is also a fundamental sequence in L ^∞ . Then it implies by standard arguments that f j → f in L ^∞ , and consequently, f j ^ f in S 0 . Hence f j ^ f in S 0 and this yields that sp (f) C Q . Therefore f ∈ L ^p _Ω ^,q and f j → f in L ^p,q , and it follows that L ^p _Ω ^,q is a quasi-Banach space.

Part (i) deduces immediately that L p i ’^{q i} C Lp'⁴'¹ . Moreover, if 0 <  9 <  p <  к 6 ro , then for any q> 0 (see [16, Theorem 1.4.2])

S Ω ⊂ L ^θ Ω ⊂ L ^p _Ω ^,q ⊂ L ^κ Ω ⊂ S ⁰ . B

It is difficult to get concrete and good constants for Nikolski˘ı inequality for Lorentz spaces L ^p _Ω ^,q . Following some ideas in [13], we have a version of the Nikolski˘ı inequality for Lorentz spaces.

Theorem 2. (i) If 0 < pi < 2, then for p2 > pi,q2 > 0, kf kP2,q2 6 / p2

p 2

-

.

pi)     V2

| Q |

-

p 1

1/p -1/p kfkp,’„,  f G L^'";

(ii) If 0 < pi < ro, then for p2 > pi , q2 > 0, kf kP2,q2 6 / p2

p 2

-

, ■ / p 2 ^{| co (Q)|} V^/pi- ^{i /p2}_|m|              ppqn

й)    I 2p o — pi )          ^kf^{k pi}’^{q i} , f ^G ,

where co(Q) denotes the convex hull of Q and po is the smallest integer number such that po > pi/2.

C (i): Suppose that 0 < pi < 2, 0 < qi 6 ro and f G Lp'qi, then by Theorem 1, f G L2, so it follows from [13, Theorem 3] that kfk∞               1/2

j yXf(y)dyj kfk∞                                1/2

_{k f k} 2 _∞ - p 1

2 — p i

1 / 2 _.

j ( y>. ⁱ/^{p i} (y) ) ^{p i} y ^{i - p i} dy)     6| Q | ^{i / 2} k f k p i / il

Therefore,

kfIk 6 ^

| Q |

-

p 1

¹ k f k p 1 ∞ .

Applying now the argument in Step 2 of the proof of Lemma 1, we can obtain a similar inequality

kf kP2,92 6 Z p2

p 2

-

_{1 /} q ₂     p1         1-p1

² k f k p ^p1² ∞ k f k ∞ ^p2 . p 1

Hence,

kf kP2,92 6 Z p2

p 2

-

^ 1 /q ² pj   V2

| Q |

-

p 1

1 /p 1 - 1 /p 2

k f k p 1 q 1 .

(ii): Since 0 < pi/po < 2, we get immediately kf kp„2 = kf" k^/p,, 6

(

p 2 /p 0 p 2 /p 0 - p 1 /p 0

\ q2 z ^{| co(}sp ^{( f p o} )) | \ pl - p2

) I 2 - p i /p o J        ^kf k -i/Poqi/Po

6 p 2

p 2

-

A q2 Z po|co(sp(f ))| A pi - P2 и „и / Z kf kp1 q1 6 pi/   \ 2 -pi/po /                   pP2

p 2

-

A 4 2 ( ^p 0 ^{| co (}^^)| A P1 P2 II Л1 k f k p q ^. p i       2 p o - p i              ^{1 1}

The theorem is proved. B

Lemma 2. Let 1 < p 6 ro, 0 < q 6 ro. If f E Lp,q, then f E S0 and for any g E L1

k f ∗ g k p,q 6 c k f k p,q k g k 1 ,

where c is a constant depending only on p, q.

C Firstly, we show that f E S 0 . Let E C R ⁿ such that 0 <  | E | <  ro . Then the Holder inequality implies

| E |

| E |

| E |

j I f (x) | dx 6 j f * (t)dt = у ( t ^{i /p} f - (ty^h-dt 6 k f k p„y t ^{- i /p} dt = c(E) k f k „, „ .

E

This deduces easily that f ∈ S ⁰ .

Now, we prove the last conclusion. For an arbitrary t >  0 , we define

f M(t) = 11 f *(y)dy.

Then for any E ⊂ R ⁿ such that t 6 | E | <  ∞ we have by Jensen’s inequality

(/^|f * g^{( x )| r dx}l ⁶      /^|f * g^(x)W ⁶ / Пущ т4г / ^{| f ( x} - y^{)| dx}A dy 6 ^{f (} * ⁾( ^{t )} k g k i .

| E |                                    | E |                                             | E |

E                   E            R ⁿ           E

Hence,

k f ∗ g k p,q 6 k f ∗ g k ^∗ p, ^∗ q 6 k f ^{( ∗ )} k p,q k g k 1 .

It now yields from [22, Lemma 3.2] the existence of a constant c such that (in the case p> 1 )

k f ^{( ∗ )} k p,q 6 c k f k p,q ,      f ∈ L ^p,q ,

The lemma therefore is proved completely. B

Theorem 3. Let f E Lp'q (1 < p < to, 0 < q 6 to) such that f ^ 0. Then sp(f) contains only points of condensation.

C Let ^ o E sp (f) be an arbitrary point, and let V be any neighbourhood of £ q . Choose 9?(^) E C “ ( R ⁿ ) such that ф(^) = 1 in V . Then by Lemma 2, F -1 (ф f) = ^ * f E L p, . 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 E L ^m ( R n ) . Hence (f m у e C ₀ ( R ⁿ ) . Since f ^ 0 , there exists a non-void ball B such that

B C sp (f ^m ) = supp (f * • • • * f) (m terms ) C sp (f) ++ sp (f).

Therefore it follows at once that sp (f) is uncountable. B

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 E Lp’q (1 < p < to, 0 < q 6 to), f ^ 0 and (0 E sp(f) be an arbitrary point. Then the restriction of f on any neighbourhood of ξ0 cannot concentrate on any finite number of hyperplanes.

It is trivial that X f (y) <  to for all y >  0 , f E L^pq 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 E Lp,q П S' (0 < p < to, 0 < q 6 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 E Lp,q (0 < p,q < to), then

lim kkf (a.x) - f (x)kp,q = 0,                                   (4)

a → 1

where 1 = (1,..., 1) and a.x = (aiXi,..., anxn) for all a,x E Rn.

C 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 E A. Hence, let f E A and assume on the contrary that there exist {ak} C Rn, ak ^ 1, and e > 0 such that kfk - f kp,q >E, k > 1,                                  (5)

where fk(x) = f (ak.x). Since f E L11oc(Rn), then for each K' = [—',']n, one obtains j |fk(x) — f (x)|dx ^ 0, as k ^ to.

K '

So there is a subsequence of { a ^k } , which is still denoted by { a ^k } , such that f k → f a. e. on K _` . Therefore, there exists a subsequence, denoted again by { a ^k } , such that f _k → f a. e. on R ⁿ . Consequently,

lim fk(t) > f*(t), t> 0.

k →∞

Furthermore, it is easy to verify that kfkII™ =(ak ••• aS)-1kfkp,q.

The Fatou lemma then yields for arbitrary 0 < u < v <  ro

u

lim [ tq/p-i fk (t)dt = lim k→∞              k→∞

∞

6 p lim k f k k p,q q k →∞

-

∞

lim f/p-ff-t^dt 6 p kf kp„ k→∞                q

u

∞

and similarly,

∞

-

∞

∞

[ f (t)dt\

u

u

j t ^{q/p -} 1 f * ^q (t)dt = у t ^{q/p - 1} f * ^q (t)dt,

u

lim [ tq/p—fq (t)dt 6 [ tq/p-1f*q (t)dt. k→∞

v

v

Hence, if u <  v/2 are chosen such that for c = max(2 ^q 1 ,1)

u у tq/p-1f*q(t)dt < 5,

∞

У t ^{q/p - 1} f * ^q (t)dt < 5, v/ 2

where 5 = pe ^q /(3.2 ^q/p .q.c) , then there is a positive constant N i such that for all k > N i

u ∞

j tq/p-fq (t)dt <5,     j tq/p-fq (t)dt < 5.                   (7)

0                               v/ 2

Therefore, it follows from (6), (7), and the inequality (f + g) * (t) 6 f * (t/2) + g * (t/2) , that for all k > N 1

u j tq/p-i (fk — f Г(t)dt 6 c

u

u

u

У t ^{q/p - i} f ^q ( (t/2)dt )

u

j t ^{q/p -i} f k _k ^q (t)dt + у t ^{q/p - i} f * q (t)dt

< 2q/pc5.

Similarly, one obtains for all k > N 1

∞ j tq/p-1(fk — f )*q(t)dt 6 c v

∞

v

∞

v/ 2

∞

fkq (t/2')dt + /‘tq/p-1f*q (t/2)dt]

v

∞ tq/p-ifkkq(t)dt + у tq/p-1f*q(t)dtj < 2q/pc5.

v/ 2

Next, since a ^k ^ 1 and supp f is bounded, there is a ball B including supp f such that supp f k C B , for all k >  1 . Thus taking account of f k ^ f a. e. on R ⁿ , it deduces that f k ^ f in measure. Then the definition of the non-increasing rearrangement of a measurable function yields for every t >  0 that

(fk - f ) * (t) — > 0, as k ^ to .

Applying the dominated convergence theorem, one arrives at

v j tq/p-1(fk - f Г(t)dt ^ 0, as k ^ to.

u

Consequently, there exists a number N 2 > N 1 such that for all k > N 2

v p εq.

3q

j t ^{q/p -} 1 (f k - f ) * ^q (t)dt<

u

Combining (8), (9) and (10), it is evident that for all k > N 2

∞

p f - f kP,q = j tq/p-1(fk - f Г (t)dt < 2q/p+1c5 + pEq /3 = pEq. 0

This contradicts (5). B

Remark 2. It is well-known that L ^p,q can be considered as Banach spaces if and only if p = q = 1 or 1 < p 6 to , 1 6 q 6 to . Using Theorem 1 and the method of [14], one can obtain the Bernstein inequality for L ^p,q spaces in these cases: If f ∈ L ^pν^,q , then there is a constant 1 6 c 6 e ^{1 /p} such that

| p,q 6 CV ^a\\ f | p,q                                     (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 6 q <  to that \\ f \ _p , _q ^ \\ f ||1 , _q 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, ^ ^ \\ f ||i, ^ as p &  1 . Then there is e > 0 and { p n } , p n &  1 , such that:

Case 1. k f k p _n , ^ < Ilf ||i, ^ — e , n >  1 . Thus there exists 0 < u < ||f ||^ such that

sup   yXf/pn (y) < uXf(u) - e/2,

0

Case 2. ||f ||pn,^ > ||f ||i,^ + e, n > 1. Then there is a sequence {yn}, 0 < yn < ||f |^ such that ynXf/pn(yn) > ynXf (yn) + e/2.

It is easy to see from Theorem 5 and the continuity of f that λf is continuous. Therefore let v be any accumulative point of {y_n} and let n → ∞ 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 6 p,q < ro, then for all f E Lp’q

lim v-akDafkp,9 = 0.

|α|→∞

Remark 3. Applying the Bernstein inequality we have ν^-αkD^αf kp_,q 6 ν^-βkD^βfkp,q if a > в for such above p, q. Moreover, Theorems 6, 7 fail if p = q = ro. But we still don’t know what happens if p < ro, q = ro.

Let us recall some notations about the directional derivatives. Suppose that a = (ai,..., an) E Rn is an arbitrary real unit vector. Then n

∂f dxj(x)

Daf(x) = fa(x) := X aj j=1

is the derivative of f at the point x in the direction a, and

Daf(x) = Dafam-1) = X aaDaf(x)

|a|=m is the derivative of order m of f at x in the direction a (m = 1, 2,...).

Denote h_a(f) = sup |a£|. By an argument similar to the proof of [8, Theorem 2], one can €Gsp(^f)

obtain the corresponding results for directional derivatives cases in certain Lorentz spaces.

Theorem 8. If 1 6 p,q 6 ro, then there is a constant 1 6 c 6 e1/p such that for all f E Lp’q П S' satisfying ha(f) < ro kDaf kp,q 6 cha(f )|f kp,q •                                  (12)

Theorem 9. If f E Lp’q П S' (1 6 p,q < ro) is such that ha(f) < ro, then

lim (Mf ))"m«Dm/|U = 0.

m^+ra

It is clearly that one can let c = 1 in (11) and (12) if k • k_Pjq is a norm, and let c = e^1/pin general case.

Finally, we will show that the Bernstein inequality wholly characterizes the spaces L^pν^,q in the case they are normable.

Theorem 10. Suppose that p = q = 1 or 1 < p 6 ro, 1 6 q 6 ro and f E S'. Then in order that f E Lp’q it is necessary and sufficient that there exists a constant c = c(f) such that kDaf IU 6 cva, a E Z+.                           (13)

C Only sufficiency hod to be verified. Assume that (13) holds.

Case 1 (1 < p < ro, 1 6 q 6 ro). If g E Lp’q(Rn), then g E Ljoc(Rn) by the first part of the proof of Lemma 2. It hence deduces from (13) that D^af E Lloc(Rⁿ') for all a > 0. Consequently, we can assume that f E C^ra(Rⁿ) by virtue of Sobolev embedding theorem.

Next let ш E C^Rn) such that ||^ki = 1, and define for each e > 0

fe(x) = f * we(x), where we(x) = е-пш(х/е). Then fe(x) ^ f (x) as E ^ 0, for every x G 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 α sup |Dafe(x)| 6 bekDafekp,^ 6 bekDafekp,q 6 Be va,                (14)

x∈Rn where Be > 0 is a constant depending only on e. Thus the Taylor series

∞

£ -D^af_e(0).Z^a

a!

|a|=0

converges for any point z G Cⁿand represents f_e(x) in Rn. Hence taking account of (14), we obtain

n

|fe(z)| 6 Be expl £vj |zj | I, z G Cn, j=i i. e., fe(z) is an entire function of exponential type v. It therefore follows from the Paley-Wiener–Schwartz theorem that sp(fe) = supp f С Av•                              (15)

Therefore, Theorem 1 and Lemma 2 yield that for each e > 0

kfekp+l 6 cl ||^fekp,ro 6 ^c2 k^eklkfkp,^ = ^c2 k^f kp,^•

The Banach–Alaoglu theorem hence implies that there are a sequence {ε_n} and an f ∈ L^p+1(Rⁿ) such that f_en ^ f weakly in L^p+1(Rⁿ) as e ^ 0. Then by standard arguments, one has f = fa. e., that is, f_en ^ f weakly in L^p+1(Rⁿ). Because S С L^(p+1)/p(Rⁿ), the dual space of L^p+1(Rⁿ), it follows immediately that f_en ^ f in S'. Consequently, f_en ^ f in S' and this deduces at once from (15) that sp(f) С A_v.

Case 2 (p = q = 1). This case can be proved by above manner.

Case 3 (p = q = to). Let ^ and fg, 0 < 5 < 1, as in the proof of Theorem 1. Then it yields from the Leibniz formula, the Bernstein inequality for L^ and (13) that for all a G Z+

|Dafg(x)l 6 X |DY(^(5x))| |Def (x)|6 c X 5|Y|ve = c(v + 5)a, Y+e=a                         Y+e=a where 5 = (5,..., 5). Thus, as in Case 1, fg(z) is an entire function of exponential type v + 5 for each 0 < 5 < 1, and therefore, sp(fg) С Av+g. Moreover, it is clear that fg ^ f in S' as 5 ^ 0. This implies obviously that sp(f) С Av+q for any 0 < 9 < 1 and then sp(f) С Av. B

Theorem 11. If p = q = 1 or 1 < p 6 to, 1 6 q 6 to, then a function f G S' belongs to Lpν,q if and only if l^ (v—a|Daf |p,q)1/|a| 6 1.                                (16)

|α|→∞

C It is sufficient to prove «only if» part. Given any e > 0, there is a positive constant C_e > 0 such that for all a > 0

kD^af kp,q 6 Ce(1 + E)^|a|V^a.

It hence deduces from Theorem 10 that sp(f) = supp Ff С A(1_+e)_v. Therefore sp(f) С Ae>0 ^A(1+e)v = ^Av• B

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.

|α|→∞