CC BY
101Probl. Anal. Issues Anal. Vol. 7(25), No. 1, 2018, pp. 3-22
DOI: 10.15393/j3.art.2018.4170
3
UDC 517.98
Samira Bissar, Madani MoussAi
POINTWISE MULTIPLICATION IN THE REALIZED HOMOGENEOUS BESOV AND TRIEBEL-LIZORKIN
SPACES
Abstract. For either homogeneous Besov spaces or
homogeneous Triebel - Lizorkin spaces F£q (Rre), with the conditions either s < n/p, or s = n/p and q < 1 in the BSq-case, p < 1 in the F£q-case, we prove some pointwise multiplication assertions in their realized spaces.
Key words: homogeneous Besov space, homogeneous Triebel -Lizorkin space, pointwise multiplication, realization
2010 Mathematical Subject Classification: 46E35
1. Introduction and the main result. Let B^ q(Rn) be the
homogeneous Besov space and let (Rn) be the homogeneous Triebel -Lizorkin space. These spaces are abbreviated by B and F in the following. We will use the notation Asp q(Rn) for either Bsp q (Rn) or F^ q (Rn) when there is no need to distinguish them. Also, as all function spaces occurring below are defined on Euclidean space Rn, we omit Rn in notations.
The space Aspq is defined by distributions modulo polynomials. By means of the "realization" we can consider the version of Asp q in the tempered distribution space S'. Recall that G. Bourdaud [2] introduced the realization, which is a linear continuous mapping a : Aspq ^ S' such that yf E Alp q the equivalence class of a(f) modulo polynomials is precisely f, i.e., [a(f)]p = f, cf., see Definition 3 below ([f]p denotes the equivalence class of a tempered distribution f modulo polynomials). We then obtain
the so-called realized space of Ap q denoted by Ap q (i.e., Bp q in the B-case or Fspq in the F-case), which has a fundamental property that is, under some conditions on n, s,p and q, all elements in Asp q vanish at the infinity
©Petrozavodsk State University, 2018
in the weak sense, see Subsection 2.2 below.
Our aim is to study, in certain cases, the pointwise multiplication in Apq, since now we have the spaces defined in S' and not in distributions modulo polynomials. Moreover, it is important to say that there is not enough literature on this subject, in comparison to the inhomogeneous case. In this direction, we recall that in [10, Theorem 6.2] it has been proved that any bounded function f such that [f ]p G AA^,« acts by (pointwise)
multiplication on Ap, if (n/p — n)+ < s < n/p in the B-case and if (n/min(p, q) — n)+ < s < n/p in the F-case. So, we will show principally
the pointwise multiplication on Asp, for s < n/p, by bounded functions f such that [f]p belong to Aqi, « for some / > 0 and pi > 0. So, our principal contribution of this paper is the following statement.
Theorem 1. Let 0 < p, p1, p2, q < ro (p, p1, p2 < ro in the F-case) be such that 1/p = 1/p1 + 1/p2. Let 0 < s < / < ro. Let in addition
either : (n/p — n) + < s < n/p and 0 < p < ro in the B-case,
(n/ min(p, q) — n) + < s < n/p in the F-case, (1)
or : s = n/p, 0 < q < 1 in the B-case (0 < p < 1 in the F-case), (2)
be satisfied. If f G L« and g G Aspq are such that [f]p G A,« and
[g]p G Ap2, q, then fg G Ap, q. Moreover, there exists a constant c > 0 such that
II[fg]pIA < <\\f II«ll[g]p 11^4- + ll[f]pwa*^\MvWAp-q) (3)
p, q P, q P1 i °° P2 , q
holds, for all such functions f and g.
In Theorem 1 the case s = 0 can be obtained by taking p = ro in (2) with the B-case. Also, we can consider the case / := s, by taking the space B« ^ instead of Ap~q, since B« ^ presents the largest space of B« when q < 1, see Theorem 2 below for more details. On the other hand, the condition on g guarantees the "good" representative, indeed, if
we replace the assumption g G Asp, by [g]p G Alp , , then, it is possible to fall on a wrong choice of representative which yields a contradiction. For instance, if g is any nonzero polynomial on Rn, i.e., ||[g]p||as =
Ap, q
= \\[9]v11= 0, then the left hand-side of (3) becomes \\[fg]p|| s = 0
Jp2.a Jp,a
for all such functions f, but this is an obviously false assertion.
A consequence of Theorem 1 is the following result which concerns the case fx = n/pi and s = n/p, respectively.
Corollary 1. (i) Let 0 < q < to, 0 < p < pl < to and (n/p — n)+ < s < < n/pl (with (n/min(p, q) — n)+ < s < n/pl in the F-case). Then
\[fg]vs < c(\\f \U + \[f]vWjt^ )\[g]vs
pq pq
holds, for all f E Ld such that [f ]p E A^^d and all g E Ap, .
(ii) Let 0 < p < to and 0 < q < 1, in the B-case. Let 0 < q < to and
0 < p < 1 , in the F-case. Then
\\[fg]vW^/P < c(\\f\d + \\[f]pWjt/p)\[g]vW^p
Ap,q JP,q Jp.a
holds, for all f E Ld such that [f ]p E AA^^ and all g E A^q .
Remark 1. The result [10, Theorem 6.2] is now a particular case of Theorem 1, at least for s < n/p we take pl such that p < pl < n/s and apply Corollary 1(i).
For simplicity of the proofs we split them in two parts. First we decompose the product fg in three terms using the Littlewood-Paley's approach, (we refer to, e.g., [12, Chapter 4], [14, 2.8.1]), then we estimate these terms in lq(Z; Lp(Rn)) or Lp(Rn; lq(Z)), which will be given as independent assertions (Propositions 7-10 below). Afterwards, we reduce the proofs to an approximation by smooth functions by considering the case of g being smooth enough. This is principally justified by the Fatou property of the Ap q spaces. Additionally, we will deduce some improvements for the pointwise multiplication on inhomogeneous Besov and Triebel - Lizorkin spaces.
The paper is organized as follows. In Section 2 we collect some necessary facts about Ap, . Sections 3 and 4 are devoted to the proof and extensions, respectively.
Notation. The symbol N denotes the set of natural numbers including 0, Z the integers and R the real numbers. For s E R, [s] denotes the greatest integer less than or equal to s. For a E R we put a+ := max(0, a). The notation A ^ B indicates that A Q B and is continuous. For any function
f, the symbol f (•) means that x ^ f (x). With || • ||p we denote the quasinorm of Lp (0 < p < ro). If p G [1, +ro], we denote by p' the conjugate exponent, i.e., p' := p/(p — 1). For any quasi-normed space E, if 0 < q < ro, then lq(Z; E) is the set of all sequences (aj)jeZ of elements in E such
that || (aj)j-ez\\iq(Z;E) := ( Ejez l\aj IE)1/q < ro; if E = R or C we note lq(Z). For brevity, we use the notation Ep q as Ep q := lq(Z; Lp(Rn)), i.e., W(fj )jezllsp,q := (Ejez Wfj llqp)1/q, in the B-case and Ep ,q := Lp(Rn; lq (Z)),
i-e-> \\(fj )jeÚzp,q := \\(Eje z\fj\q ) q \\p, in the F-case with the usual modification if p = to or q = to. We denote by V ^ the set of all polynomials in Rn. We denote by S^ the set of all functions f in the Schawrtz space S such that (f,u) = 0 (Vu G Voo). The symbol S« denotes the topological dual of Soo. The mapping which takes any [f ]p to the restriction of f to Sco turns out to be an isomorphism from S'/Vo onto S«, for this reason, S« is called the space of distributions modulo polynomials. Sometimes, we will use the Hardy-Littlewood maximal function Mg of a locally integrable function g defined as Mg(x) := sup |Q|_1 Q \g(y)\dy (Vx G Rn), where the supremum is taken with respect to all cubes Q with sides parallel to the axes and containing x. Here \Q\ means the Lebesgue measure of the cube Q. The standard norms in S are given by cm (f) := supH<m supxeK„ (1 + \x\)m\f(x)\ (m G N). If f G Li, then
Ff (i) = f(i) := fRn f (x)e~lx^ dx and F-1f denote the Fourier transform of f and its inverse Fourier transform, respectively. The operators F and F-1 can be extended to the whole S' in the usual way. Throughout the paper we use the parameters s,p, and q as:
s G R, 0 < p,q < to and p < to in the F-case
unless otherwise stated. Finally, constants c,c1,... are strictly positive and depend only on the fixed parameters, e.g., n,s,p, q,..., and probably on auxiliary functions, their values may vary from line to line.
2. Preliminaries. To introduce the homogeneous as well as the inhomogeneous Besov spaces and Triebel - Lizorkin spaces, the Fourier-theoretical approach via the Littlewood-Paley decomposition presents the basic method. This approach has been developed by Bergh and Lofstrom [1], Peetre [11], Triebel [14, 15], ...
We choose, once and for all, a radial C00 function p, such that 0 < p < 1, p(i) = 1 if \i\ < 1 and p(i) = 0 if \i\ > 3/2. We define Y(Í) := P(Í) — P(2Í) for all i G Rn, with a support located in the compact annulus 1/2 < \i\ < 3/2. Then it holds EjezY(2ji) = 1 (Vi = 0) and
p(2-k£) + £j>k Y(2-j0 = 1 (V£ E Rn, Vk E Z). We introduce the convolution operators denoted by Sj and Qj (j E Z), and defined by means of the formulas Sf (£) := p(2-j f) and Qjf (£) := 7(2-j£)/(£), which are defined on S'. The operators Qj are also defined on S^ since Qj f = 0 (Vj E Z) if and only if f is a polynomial; then in the following we say
if f E S^ we set Qjf : = Qjfl for all fl E S' such that [fl]p = f.
All these operators take values in the space of analytical functions of exponential type, indeed, this fact is an immediate consequence of the Paley-Wiener theorem, see, e.g., [13, Theorem 29.2, p. 311]. Then we obtain the Littlewood - Paley decompositions:
(i) For every f E S oo (S ^, respectively), it holds that f = jei Qj f in So (S'o, respectively).
(ii) For every f E S (S', respectively) and every k E Z, it holds that
f = Sk f + E j>k Qjf in S (S', respectively).
The operators Sj and Qj are uniformly bounded in L(Lp), for any p E [1, +to] (the Young inequality). Also, we have the following statement proved in, e.g., [10, Proposition 2.5]:
Lemma 1. For any N E N, there exist constants cl}c2 > 0 and a natural number m such that
(i) \Qj f \\p < ci2-jNZm(f) for all f ES and all j E N.
(ii) \\Qj f \\p + \\Sj f \\p < c2 2jNZm(f) for all f ESo and all j E Z\N.
2.1. Homogeneous Besov and Triebel —Lizorkin spaces. Definition 1. The homogeneous Besov space BS q (or, the homogeneous
3 s
p,q
Triebel - Lizorkin space ) is the set of f E S^ such that \\f \\j
:= \\(2sjQj f )jez\\£pqa < to. For all f E S', we define
:= \\Lfb\\js • (4)
jls •— ^ j p\\jls
p.a p.a
Apq becomes a quasi-Banach space in relation to this quasi-norm and it has the following properties:
• Soo Apq ^ S'qc , which can be obtained easily by Lemma 1 and an estimate of the Nikol'skij representation method type.
• There exist constants cl,c2 > 0 such that
ci\\f \\j p < \n/p-s\\f (A(-))Hjj s < c2\\f \\j p (Vf E Aspq, VX> 0)- (5) p.q p.q p.q
• Bp,min(p, q) ^ FS,q ^ BP ,max(p, q) and the following embeddingS:
Proposition 1. (See [8]) Let s1,s2 G R and 0 < pi < p2 < œ be such that s1 — n/p1 = s2 — n/p2. Let 0 < q,r < œ. Then Bp,1, ^ B«2, ^ B «2-n/p2 F si ^v B s2 and F si ^ F sa
Boo,q , Fpi,q ^ BP2 pi and Fpi, q ^ F pa , r ■
We will need an estimate of the Nikol'skij's type, for which the proof is essentially given, in homogeneous or inhomogeneous case, in [4, Proposition 4], [9, Proposition 3.4], [10, Proposition 2.15], [12, p. 59] and [16].
Proposition 2. Let a, b be real numbers such that 0 < a < b. Let (uj)jez be a sequence in S' such that
• Uj is supported by the compact annulus a2j < |£| < b2j,
• A := || (2jsuj}j-ez|k,„ < œ.
(i) Then the series Euj converges in S« to a limit denoted by u and which satisfies ||u|| as < cA, where c depends only on n, s,p, q, a and b.
(ii) If in addition s > (n/p — n)+ (s > (n/min(p, q) — n)+ in the F-case), the same conclusion holds for a = 0.
We will also need the Fatou property of Ap, q, see, e.g., [4, Proposition 7] and [9, Proposition 3.13].
Proposition 3. Let f E S«. If there exists a bounded sequence (uk )keN in Ap, such that limk— oo uk = f in S«, then f belongs to Ap , and If II As < c liminf k- o lluk I As .
The characterization of B and F spaces by the maximal functions is also useful in what follows. To recall this thing we introduce the maximal operators, thus for f E S«, a > 0 and j E Z, we set
Q*'af (x) := sup (1 + |2jy\a)-1\Qjf (x — y)\ (Vx E Rn). (6)
Proposition 4. (See e.g., [7, p. 45]) Let a > n/p in the B-case (a > > n/min(p, q) in the F-case). Then we have an equivalent quasi-norm in Apq defined by the expression If ||A := ||(2sj Q]'a f jzhp,,.
2.2. Realized spaces. We begin by the following definition:
Definition 2. A distribution f E S' is said to be vanishing at infinity in the weak sense if lim^—o f(^-1(0) = 0 in S'. The set of all such distributions on Rn is denoted by C0.
Here are some examples: (i) The elements of Lp for 1 < p < to. (ii) The derivatives in D' of any element of C0. (iii) The derivatives in D' of a bounded function. ^
We note that P ofC0 = {0} (it is an easy assertion). Another example will be given by the following assertion proved in, e.g., [3, Proposition 4.4].
Lemma 2. If f E Lo and supp f is a compact subset in Rn\{0}, then f E Co.
Now, we give the notion of the realization:
Definition 3. Let E be a vector subspace of S^ endowed with a quasi-seminorm such that E ^ S^ holds. A realization of E in S' is a continuous linear mapping a : E ^ S' such that [a(f )]p = f for all f E E. The image set a(E) is called the realized space of E.
By the Littlewood - Paley series we have an example of realization. Let
either s < n/p , or s = n/p and 0 < q < 1 in the B-case (0 <p < 1 in the F-case), (7)
then for any u E Ap, q the series Qju converges in S', and the linear
mapping a(u) :=^jeZ Qju is a realization on Ap, q satisfying [a(u)]p = u
in S^ and a(u) E C0; more precisely, the element a(u) is the unique representative of u that belongs to S', see [3, Proposition 4.6] and [10, Theorems 1.2, 4.1, Section 4.2]. So, if we take v E S' fl C0 such that [v]p E Apq, we deduce that a([v]p) — v E Vo f C0 = {0}. Then, without referencing to the Littlewood-Paley decomposition, we define the realized homogeneous space of Ap q:
Definition 4. Assume that (7) holds. The realized homogeneous space
of Ap, , denoted by Asp, , is the set of all f E C0 such that [f ]p E Ap, , and endowed with the quasi-norm \\f \\ ~ := \\f \\j s , see (4).
Jp,a p,a
2.3. Inhomogeneous Besov and Triebel —Lizorkin spaces. The
definition of inhomogeneous Besov and Triebel - Lizorkin spaces relies upon Littlewood - Paley theory, similarly to the case of homogeneous spaces.
Definition 5. (i) Let 0 < p < to. The Besov space Bp, is the set of f ES' such that \\f \\Bsq := \\Sof\\p + (j (2sj\\Qj f \\p)q )i/q < to.
(ii) Let 0 < p < m. The Triebel - Lizorkin space Fps is the set of f £ S' such that \\f\\Flq : = ||sof||p + \\(Ej>1(2sj\Qjf\)q)1/qMp <
We denote by Ap q either BS q or FS q, and recall S ^ Apq ^ S' and the following statement proved in, e.g., [15, p. 98]; other properties can be found in, e.g., [1, 11, 14, 15].
Proposition 5. Assume that s > (n/p — n)+. A function f belongs to Ap,q if and only if f £ Lp and [f ]p £ Apq. Moreover, \\f \\p + \\f \\Ais
an equivalent quasi-norm in As
p,q
p, q'
2.4. Necessary conditions for pointwise multiplication. By the
following assertion, where the proof is similar to that given by Franke [6, Proposition 2.5/1] for Asp, q, we can formulate some necessary conditions
such that ||/l f2 \ \ A s < c\\fi\\A3p1 qi ||f2\\a p2 qo for all /1 '/2 £ .
p,q AP1 ,qi P2 >q2
Proposition 6. Let Si £ R, 0 < Pi,Qi < to (pi < to in the F-case) (i = 1, 2). Assume that there exists a constant c > 0 such that, either \\fl f2\\AS < c\\fl\\B°i \\f2 MaA or \\f1f2 < cWhUps! \\f2\\ps2
Ap,q BPi,qi AS2 ,q2 p,q Fsi,qi FS2,q2
for all f1,f2 £ Soo, are satisfied. Then it follows:
(i) either si > s or si = s and qi < q.
(ii) si + s2 > 0.
3. Proof.
3.1. Decomposition and estimate of the product. "We introduce the bilinear maps Ts,(f,g) := {2ksQk({Sk+vf)(Qk+,g)) }keZ and Rs , (f, g) := {2ks J2j>k Qk ((Qj+u f)(Qj+^g)) }kez, defined on S' x S^ and S^ x S^, respectively, where s £ R and (v, fx) £ {—3, —2,..., 3}2. In the case v = fx = 0, we set Ts := Ts,0 , 0 and Rs := Rs,0 , 0.
Proposition 7. Let f £ L^. Let g £ Lp (0 < p < to) be a function
of class C00 such that g has a compact support. Then the product fg is
correctly defined in S', and for all k £ Z we can write 1 3
Qk(fg)= £ Qk{(Sk-3g)(Qk+vf)) + £ Qk{(Sk+if)(Qk+vg)) +
v=-2 v=-2
1
+ £ £ Qk((Qjf)(Qj+vg)) in S^. (8)
V=-3j>k+2
Moreover, if in addition [fg]p £ Apq then it holds
\\fg\\A}iq < c( £ \\Ts,-3,»(g,f)hP,q+
v= — 2
3 1
+ £ \\Ts, 1, V (f,g)\\Ep q q + £ \\Rs, 0, v (f,9)kP q q) . (9)
v=—2 v= — 3
Proof. Step 1. Let p E Soo• By the Holder inequality \(g,p)\ < Wg^p^^^p' if p > 1. In the case 0 < p < 1 we apply the following lemma proved in, e.g., [11, Lemma 1, p. 54] or [14, Remark 1.4.1/4, p. 23]:
Lemma 3. Let 0 <p < q < o. Then it holds \\f\\q < cRn/p—n/q\\f \\p, for all R> 0 and all f satisfying supp f C : \£ \ < R}.
We have \(g,p)\ < \\p\\oo\\g\\1 < c\\p\\oo\\g\\p where c depends only on p and suppg. This gives g E S«. Let now p E S, then \(fg,p)\ < < Wf \0\M\i, and Wgp\1 can be bounded similarly as above. Then fg E es '.
Steep 2: proof of (8) and (9). Since f, g and fg belong to S«, then
Qk(fg) = £ Qk((Qjf)(Qig)) (Vk e z). (10)
j, lez
Denote by Il, j and Ik intervals [(2max(j ' l) - 3 • 2min(j ' l))+, 3(2j + 2l)] and [2k, 3 • 2k], respectively, for all j,h,£ E Z. A careful examination of the intersection of the supports of y(2—k (•)) and F ((Qj f )(Qlg)) allows us to obtain y(2—k£)F((Qj f )(Qlg))(0 = 0 in the following four cases:
• if j > k + 2, 1 > j + 2 then 2l - 3 • 2j > 3 • 2k, (Ik n It j = 0),
• if j > k + 2, 1 < j - 4 then 2j - 3 • 2l > 3 • 2k, (Ik n It j = 0),
• if j < k + 1, 1 > k + 4 then 2l - 3 • 2j > 3 • 2k, (Ik n It ^ = 0),
• if 1 < k - 3, j < k - 3 then 3(2l + 2j) < 2k, (Ik n Ittj = 0). Consequently, in the right-hand side of (10) we have
£•••=£ E + E E + E E ••• =
j, lez j>k+2j—3<l<j+1 j<k+1k—2<l<k+3 k—2<j<k+1l<k — 3
=: Wk + Vk + Uk.
13
We introduce Qk, 1 := £v=—2 Qk+v, Qk,2 := Ev=—2 Qk+v and Qj,3 : =
:= E V=—3 Qj+v then Qk (fg) = Uk + Vk + Wk where Uk ,Vk, and Wk are
defined as we want in (8), i.e., we have Uk := Qk((Sk—3g)(Qk, 1f)), Vk : =
:= Qk((Sk+1f)(Qk,2g)) and Wk := £ j>k+2 Qk ((Qj f )(Qj,3g)). Finally, (9) follows from the definition of \\ • \\as and (8). □
Proposition 8. There exists a constant c > 0 such that the inequality \\Ts{f,g)\\£P,q < c\\f ||oo||glUPq holds, for al1 f e L^ a^d al1 g e Aspq.
Proof. Let us recall that Ts is defined on S' x S^. On the other hand, we have supfceZ \\Skf \\oo < c\\f \\oo. Then
\Qk ((Sk f )(Qk g))(x)| < c\\f \\ ooQl'ag(x) i (1 + \y\a)\F-1 y (y)\dy, (11)
JRn
for all x e Rn and all a > 0 (see (6) for the definition of Qk'a). Now, to apply Proposition 4 we choose a > n/p in the B-case, and a > n/ min(p, q) in the F-case. The desired estimate follows. □
Proposition 9. (i) Let — œ < s < fx < œ. Let 0 < pi,p2 < œ (pi ,p2 < œ in the F-case) be such that 1/p = l/pi + 1/p2. Then there exists a constant c> 0 such that \\Ts(g, f)\\sp q < c\\f \\à' II^Nâ°
all f G A^,^ and all g G S' with [g]p G À-.
(ii) There exists a constant c> 0 such that, for all f G Àp, and all g G S'
"?0 it h°lds NT (g f )\\ _ < c\\ f IK \\g\\
it holds \\TS (g,f)\\£p q q < c
wiïh [g]p G B^ Proof. As in (11), we first have
A s
Ap, q ■
IB0 1 •
OO , 1
\\Ts(g, f )\Epqq < c
M Qk ' a f)Y,Q ' a4k. (V«> 0). (12)
j<k ke£ Epq
Step 1: proof of (i). We will apply both the following elementary estimate
<£nd , 0 <d < 1,Vj > 0, (13)
j j
and the following lemma, which can be proved as in, e.g., [16, Lemma 3.8]:
Lemma 4. For all 0 < u < <x> and all a > 1, there exists a constant c> 0, such that HQ]£{j-k)<o a£{3-k)n3)kez\\iu(z) < c\\(n3 jezlk(Z) (e = ±1) holds, for all (nj)jez € lu(Z).
We begin with the B-case. We choose a > n/p (then a > n/p\ and
a > n/p2), put d := min(1,p) and apply the Holder inequality, then
1/d-
№(g,f)lk) < ci\\{2ks(£ \\(Ql'af)(Q*'agX)
3<k
<
lq (Z)
< C1
< C2
^ . 2k(S-,) \lQk,af I^^ Ij g\\ä^
j<k
<
lq (Z)
keZ j<k
,, b3-r •
pi,^ P2 ,q
Then by Lemma 4 with u := q/d we obtain the bound c\\f \\¿¡i \\g\
P 1 , 00
We now consider the F-case. We apply the Holder inequality in (12), then \\Ts(g,f)\\Lp(lq) is bounded by
sup 2l^Q*£'a f \ \pi \\ ( £ { £ 2(k-j)(—)(2j(—)Q*'ag) }*)
" \1/"
keZ j<k
P2
so, we choose a > n/min(p, q), which implies a > n/min(p1, q) and a > n/min(p2,q), then Lemma 4 with u := q gives the desired bound
cWf \\piitX MF- .
Step 2: proof of (ii). In (12) we choose a > n/min(p, q) > n/p, then
which is bounded
IlTs(gJ)I\£p,q < c\\{2ksQk>af}kezllep,qj IIQ.
by c\\f II .is IIgIBo • The proof is finished. □
Àp,q B CO ,1
jI
Proposition 10. Let ß G R. Let 0 < p1,p2,r < œ (p1 ,p2 < œ in the F-case) be such that 1/p = l/p1 + l/p2. Assume that s > n/p — n in the B-case and s > n/min(p, q) — n in the F-case. Then there exists a constant C > 0 such that
I\Rs(f, g)hp,q < c\\f \IÀ^q IIgIIi Sp-r (Vf G A Pi,", Vg G Ap-r ). In the case p = p1 and p2 = œ, inequality (14) becomes
(14)
\\Rs(f,g)\\£pqq < c\\f \\A2qq\\g\\m (Vf G APq,Vg G ). (15)
Proof. We will use the Marschall pointwise inequality.
Lemma 5. (See e.g., [17, Proposition 6.1]) Let h G C°° and p G D be such that h and p are supported by the balls |£| < AR and |£| < A,
c
respectively, for some A > 0 and R > 1. With 0 < t < 1, it follows \(F"V) * h(x)\< c(ARr/t-ny\\Bn/t (M\h\t(x))1/t,
where c does not depend on p, h, A, R and x.
Step 1: the B-case. Applying Lemma 5 with h : = (Qjf)(Qjg), p : = := y(2-k(•)), A := 3 • 2k-1 and R := 2j-k+1 (j > k), then, for some 0 <t < 1,
\Qk ((Qj f)(Qjg))(x)\ < c2(j-k)(n/t-n)(M\(Qj f)(Qj g)\t (x))1/t, (16)
for all j > k and all x E Rn; here we used the homogeneous property of Bn/, i.e., \\y(2-k(•))\fBn/t < c2-k(n/t-n)\\y\\bn/t where c depends only on
n and t, see (5). We set d := min(1,p). Choosing t such that 0 < t < p (recall that 0 < t < 1), and using both (13) and the Minkowski inequality with respect to Lp/d(Rn; £1 (Z)), we see that Qk ((Qj f )(Qjg)) ||p
is bounded by
c(f iE 2d(j-k)(n/t-n) (M \(Qj f)(Qj g)\t(x))d/t}P/ddx] ^ <
t ||d/A 1/d
< c( £ 2d(j-k)(n/t-n) ||m\(Qj f )(Qjg)\t||d;/t)
j>k
Now, as the maximal function M is bounded on Lv if 1 < v < to (see [5]), we then apply this assertion with v := p/t > 1, and obtain \\M\(Qjf)(Qjg)\tlp/t < c\\(Qjf)(Qjg)\\p for some c > 0 independent of f, g, and j. Therefore, by the Holder inequality the term \\Rs (f,g)\\i (z;l )
is bounded by c(Ekez 2qks{Ej>k 2d(j-k)(n/t-n)\\Qj f \\£ WQjg^}q/d)'1/q, which is bounded by
c(£ (2j(—)\\Qjg\\P2)r) \ jez
x 2dj-k)(n/t-n-s) j\\Qj f \\Pi )d }q/d)1/q.
Now we also choose n/t — n — s < 0, then it suffices to take t such that n/(n + s) <t< min(1,p) (see the assumption on s), then Lemma 4 with u := q/d yields \\Rs (f,g)\\tq (Z;LP) < c\\f Wb^ Mb;-» •
Step 2: the F-case. By (16), \\Rs(f, g)\\Lp(Rn-,eq (Z)) is estimated by
,t )1/t }q ) 1/q
2......M i (Qj j )(Qjg)
j>k
2(k-j)(n-n/t) (Mi(Qjf)(Qjg)f)1/t} )
keZ
(17)
Since 0 < t < 1, then in (17) the term with J2j>k ••• is bounded by
iEj>k 2(k-j)(nt-n)M\(Qj f )(Qjg)\t(x)}1/t for all x e Rn and all k e Z. By inserting this estimate into (17), choosing
t < min(p, q)
and using the fact that \\(Mhk)k\\lv(r«;tw(Z)) < c\\(hk)k\\lv
1 ; iw (Z))
(18) holds
for 1 < v < to and 1 < w < to and for some c > 0 depending on n,v and w (see [16, Theorem 2.2]); and with v := p/t, w := q/t and hk := 2kts YJj>k 2(k-j)(nt-n)\(Qj f )(Qjg)\t, then using the Holder inequality with 1/p = 1/pi + 1/p2, the term \\Rs(f, g)\\Lp(R™;eq(Z)) is bounded by
ci
q/tN 1/q
(X{2fctS£ 2(k-j)(nt-n) i(Qj f )(Qjg)|t| )
keZ j>k
<
< C2
J2(2j(—) iQjgiy)
1/r
jeZ
x
P2
x
t } q/t N1/q
(J2{J22(k-j)(nt+St-n)(2j»iQjfi)7)
keZ j>k
(19)
pi
under the condition n/(n + s) < t; the compatibility of this condition with (18) is guaranteed by assumptions on s,p,q. Then we choose t as n/(n + s) < t < min(p,q, 1), and deduce the bound c\\f \\p^ \\g\\ps. If p = pi, i.e., p2 = to, we estimate (19) by
c\\g\\è^
2(k-j)(nt+st-n)(^iQj f i)t}q/t )1/q
keZ j>k
which yields the correct bound c\\f Wp^ \\g\\ès-M• □
Remark 2. By easy computations, and owing to the translation invariance on k G Z when we take the sum over all k G Z, Propositions 8, 9 and 10 also hold with Ts,v,^ and Rs,v,^ (where (v, f) G {—3, —2,..., 3}2) instead of Ts and Rs, respectively.
p
p
3.2. Proof of Theorem 1 and Corollary 1. Proof of Theorem 1. We recall that indicates a continuous em-
bedding, see the section of Notation in Introduction. Let f E Loo and
g E Ap,q be such that [f]p E A^,^ and [g] P E AV2, q'
Step 1: proof of fg E S' and [fg]p E Ap, . We defined a sequence (gk)keN by gk \j\<k Qjg = Skg - S-k-ig, which has the following
properties:
• gk is supported by the compact annulus 2-k-1 < |£| < 3 • 2k-1,
• [gk]p belongs to Ap q f Ap2 ^ (this follows by Proposition 2) with
\\gk\\ap — c\\g\\Àp and \\gkHyip-A» — c\\g\\Às-q for all k E N,
1S — c\\g\\À s and \\gk\\à — c\\g\\A s
p , q Àp , q ÀP2 > q ÀP2 > q
• gk E C°° f Lp; indeed, if we put d := min(p, 1), by the embedding Ap,q ^ Bsp>^ (see Subsection 2.1) it follows \\gk\\p — (Em<k \\Qjg\\dp)1/d which is bounded by (£2~Sjd)1/d\\g\\ÈP — c(k)\\g\\ÀP .
\J\— p,œ p,q
Using Propositions 7, 8, 9(i), and 10 with functions f and gk (recall that by Proposition 7, fgk E S ' ), it holds that the two terms \\Ts (f, gk ) \E and \\Ts(gk,f)\\sp,q + \\R.s(f,gk)\Ep, q can be estimated by c\\f \\oo\\g\Up q and c\\j Wa^ \\g\\Àp-a , respectively. Then, owing to Remark 2 we get
\\fgk\\a. < c( \\f \\oo\\g\\A- + \\f „Mai-%), (20)
where the constant c> 0 is independent of f, g, and k. Now, we claim lim (gk - g)f = 0 in S' and fg ES'. (21)
k^ oo
Indeed, we prove (21) with respect to the cases s < n/p and s = n/p, separately:
The case s < n/p. (Recall that here 0 < p < <x> in all spaces, see (1)). Since g — Skg = j>k Qjg (see the beginning of Section 2), we have
K(gk — g)f, v)l < \\f \\oo(\\(S-k-ig)^\\i + E \\(QjgMi), Vv E S; (22)
j>k
and we separately estimate each term in the right-hand side of (22). If p > 1, then using the Holder inequality, we obtain
E\\(Qi g)^\\i — \m\p'E WQj g\\p — w^w^b^E 2~sj — j>k j>k j>k
— c2~sk\\p\\p'\\g\\Às (recallthat s> 0); (23)
if 0 < p < 1, we use the Bernstein inequality (see Lemma 3) and obtain
E \\(Qi9Mi E \Qi9h < ciMOE2j(n/p-n)\\Qjg\\P < j>k j>k j>k 2k{n/p-n-s)
< ci 1 _ 2n/p-n-s MooMb.Pi„ < c22k(n/p-n-sS) MooMAs q; (24) again, by the Bernstein inequality we get
\\(S-k-igMi <M\i E \\Qj9\\oo < ciMi E 2jn/p\\Qjg\\p <
j<-k j<-k
< ci 1 _ 2-{nh-a) \M\iiigNBB^ < C22-k(n/p-s)W^iHgH^; (25)
by inserting (23) or (24) (for each case) and (25) into (22), and by taking k ^ to we obtain the convergence of the sequence {(gk _g)f }kEN to 0 in S' since the conditions (^ _ n)+ < s < ^ (in the B-case) or (min(p q) _ n)+ <
< s < n (in the F-case) are at our disposal; noticing that in the estimate (24) we have n/p_n_s < 0 in the F-case since ^ _n < (^^ q) _n)+ < s. Now, for an arbitrary e > 0, there exists a natural number k£ such that for all k > k£, it holds
\{fg,p)\< e + Kfgk,p)\ < to (Vp gS), (26)
and fg G S'.
The case s = n/p and 0 < q < 1 in the B-case (0 < p < 1 in the F-case).
"This case also includes a situation when we consider B-spaces for p = to (and consequently s = 0)." (27)
First, we have
\fgk(x) _ fg(x)\ < \\f Woo E \\Qjg\\oo a.e. on , (Vk G N). (28)
\j\>k
By the embedding Jjp^ ^ B^^, which can be obtained by the fact that Bp,!qp ^ B, q ^ BA if 0 < q < 1 (in the B-case) and F^, /p ^ B«, p ^ B0,i if 0 <p < 1 (in the F-case), we have
E\\Qjg\o = Mb0 1 < ^VgW^/p■
f Ap,q
jEZ
Then the last term of (28) converges to 0 when k ^ to, and consequently fgk converges to fg pointwise (i.e., lim^ oo fgk(x) = fg(x) a.e. on Rn). Second, \\gk||oo < I]\j\<k \\Qjg\\oo < llglls^ implies that \\fgk loo < c\\f\\oo\\g\\A n/P (Vk E N). Then applying the Lebesgue domi-
Ap,q
nated convergence theorem, we deduce limk^ co fgk = fg in S', and (21) follows as in (26) in this case also.
Now the separate treatment of cases s < n/p and s = n/p is complete and the paragraph below concerns both of them. Trivially, we have lim^ oo {fgk + u, p) = {fg + v, p) for all p E So and all (u,v) E VoXVoo, then the sequence ([fgk]p)keN converges to [fg]p in S^ too. Hence, we apply the Fatou property of Apq in (20) (see Proposition 3). Then the inequality (3) holds.
Step 2: proof of fg E C0 ■
Substep 2.1: the case s < n/p. (Here 0 < p < to, see (1)). We use:
Lemma 6. There exists a constant c > 0 and a natural number m E N, such that \{u, p)\ < c(m(p)\\u\\ as , Vp E S, Vu E S' with [u]p E Ap .
Ap,q
Proof of Lemma 6. The bilinear map {■, ■) is separately continuous on Ap q x S, then, as Ap q and S are Frechet spaces, this map is continuous; cf.,' [13, Section 34.2, Corollary, p. 354]. □
We set fx := f (A-1 (■)) and gx := g(A-1 (■)) for all A > 0. By (5) we have \\gx\\As x An/p-s\\g\\As , \\gx\\As-q x An/p2-s+^\g\As-q and
p , q p , q P2 > q P2 > q
\\f^\A^ x An/pi-^\\f \\a^ . By applying, both Lemma 6 with u : = := fx gx, and the estimate (3) with f\ and g\, we get for all p E S,
\{fxgx,p)\< ciCm(^)(\\f IloollgAll^,, + llfxlA^Ma—J <
< C2 \n/p-sCm(v){llf WooMA}« + llf WAçlia> Ma - q ) (29)
for some natural numbers m and some positive constants c1,c2 independent of f,g, p, A, and m. Taking A ^ 0 into (29), we obtain the result. Substep 2.2: the case s = n/p and 0 < q < 1 in the B-case (0 < p < 1 in the F -case).
"This case also includes a situation when we consider B-spaces for p = to (and consequently s = 0)." (30)
We defined a sequence (uk)keN by uk := Y,\j\<k Qj(fg), and
• uk is supported by the compact annulus 2 k 1 < |£| < 3 • 2k 1,
• \Wk||oo < c\\fg\\Ar,/P (Vk > 0), this follows by A^ ^ B^ 1.
These yield Uk G C0, cf., Lemma 2, on the one hand. On the other hand £u>k \\Qj(fg)\\ o < \\fg\\Bf1 < c\\fg\\A;/qr V G N) implies that the first term converges to 0 when k ^ to. Consequently, we obtain limk^ co \\uk — fg\\oo =0. Now, for all p G S and all A > 0, we get
I(fg(A-1(^)),p)I <\\uk — fgWcMh + IU(A-1(^)),p)I,
and for an arbitrary e > 0, there exists a number k£ G N such that the right-hand side of the last inequality is bounded by e + l(uk(A-1 (0),p)| for all k > k£. Thus, by taking A ^ 0 we have lim^0(uk(A-1 (•)), p) = 0 implies lim^0(fg(A-1 (•)), p) = 0. Hence we obtain a desired result, and the proof of Theorem 1 is complete. □
Before we prove Corollary 1 we need to formulate the complement of Theorem 1 when 1 = s as mentioned in the introduction.
Theorem 2. Assume that either (1) or (2) is satisfied. If f G Lco and
g G Apq are such that [f]P G Ap, and [g]P G B, then fg G Apq . Moreover, there exists a constant c> 0 such that for all such functions f and g it holds WfgWAsS < c(\\f HooNA + WfWA* Mb0! ).
p,q p,q pq 0,1
Proof. Using Propositions 7, 8, 9(ii) and the inequality (15) with (1 = s and r = 1) we obtain (20) with \\g\\Bo 1 instead of \\g\\As-M • Then by observations (27) and (30), the proof is similar to that of Theorem 1. □
Proof of Corollary 1. The assertion (i) follows by both, (3) with 1 := n/p1 and Asp, ^ jA>P2n/Vl where p2 := (1/p — 1/p1 )-1. For (ii) we apply Theorem 2 with s = n/p and B^ B 1 (in B-case) and FnP!p ^ B0,p ^ B^, 1 (in F-case). □
4. An extension to inhomogeneous case. The following two corollaries concern the pointwise multiplication in case of the inhomoge-neous spaces Aspq. For brevity, if E C S', we shall write f G E n Apq if f G E and [f]p G Apq, also E C Apq means that for arbitrary f G E we
have [f]p G Ap q q .
Corollary 2. Let 0 < p < p1 < to. Assume that (n/p — n)+ < s < < n/p1 in the B-case ((n/min(p, q) — n)+ < s < n/p1 in the F-case). Let
g E Ap, and let f E Lco be such that [f]p E . Then fg E Ap, .
Moreover, there exists a constant c > 0 such that \\fg\\Ap < c(\\f \\oo +
Ap ,q. We write (Lo n À^) • Ap^ ^ Ap^.
An/pi )WgWAp ■ we write Loo n ApiO
ap i .no p,q
Proof. We first prove that g G C0. Indeed^if 1 < p < to, the assertion follows from the embeddings Àspq C Lp C C0 (see example (i) just after Definition 2); if 0 < p < 1 then s + n — n/p > 0, and again we have Aspq C Al+qn-n/p C Li C Co. We set p2 := (1/p — 1/pi)-1. Then the
assumption s < n/p1 < n/p yields g G Àspq, and Àspq ^ Àsp2 p/pi implies
[g]p G Àp2p/pi. Then there exist c1,c2 > 0 independent of g such that (see Proposition 5)
\\g\\A s-n/Pi < C1 \\gWA s < c2 Map. • (31)
ap2 > q s . q p 'q
Now, by Theorem 1 (with fx := n/p1 ) (see also Corollary 1) it holds
WfgWAPp < c(\\fg\\p + WfgWAs ) which is bounded by c(\\f\\ooWgWp +
p >q p , q
+ Wf \\oo\\g\\A s + Wf \\A n/pi \\gW A s-n/pi ). Thus, the desired result follows
p.q aPI ,co aP2 .q
by (31), see again Proposition 5. □
Corollary 3. Let p, q be such that 0 < p < to and 0 < q < 1 in the B-case, 0 < p < 1 and 0 < q < to in the F-case. Let g G À^fê and let f G Lco be such that [f]p G . Then fg G Àp/p . Moreover, there exists a constant c> 0 such that \\fg\l An/p < c(\\f\\ o + Wf W a n/p )\\g\\ Wp •
Ap, q Ap, q Ap, q
We write (Lo n Àp/p) • À^ ^ À^.
Proof. Using Proposition 5, Theorem 2, and Corollary 1(ii) the proof is similar to that of Corollary 2. We omit the details. □
The main motivation of Corollaries 2 and 3 is that we have now complements of some previous works given in case of À^. Namely:
Remark 3. (i) Since Àp//p ^ if p1 > p, Corollary 2 implies
(Lo n À^fZ) • Àpq ^ Àpq if (n/p — n)+ < s < n/p (with (n/min(p, q) — —n)+ < s < n/p in the F-case). If in addition 0 < p < 1, we have the result given in [12, Theorem 4.6.2/2(24)-(25), p. 200]. (ii) Because of À1p/qpi ^ Àp/q and Àip/qp2 ^ L^ n Àp/qp, by Corollary 3 we have the following two assertions: Bp• Bp/cpp ^ Bp/p if q : = := max(q1 ,q2) < 1, and Fp!/p • Fp/p ^ Brp/p if 0 < p < 1 and q : =
: = max(q1,q2), given in [12, Theorem 4.6.1/2(20), p. 192] and [12, Theorem 4.6.1/1(8), p. 190], respectively.
Remark 4. The case p = to in Corollary 3 can be given as the following: (Lo n B0,qq) • Bc,q ^ B^,q if 0 <q < 1. (32)
Indeed, assume that f G Lo n B0, q and g G B0 q. By Theorem 2,
fg G S', and as So(fg) = J2j<o Qj (fg) then WSo(fg)Woo < cWfg\\B;0caq. Also, since WfgWB°oa < c(\So(fg) \\o + Wfg\\Bc ), Theorem 2 again yields
00 , q co , q
WfgWBc < c(Wf W oo + Wf Wbc )\\g\\Bc . Now, with respect to
,q ,q ,q
B^q1 • B0, q ^ Bq if 0 <p1 < TO and 0 <q < 1, (33)
see [12, Remark 4.4.4/6, p. 180], we have B^q g Lo n B^q1 C Lo n Hi?0, q. However, (33) fails in case p1 = to (see again [12]), then an interesting problem is to obtain (32) with B0, q instead of Loo H B0, q.
Remark 5. A complement of Remarks 3-4, that it would be interesting to extend to homogeneous spaces the results given in, e.g., [14], [12, 4.6.14.6.2, pp. 190-207] for inhomogeneous ones.
Acknowledgment. We would like to thank the referees for their valuable comments and suggestions.
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Received November 7, 2017.
In revised form, February 9, 2018.
Accepted February 15, 2018.
Published online March 14, 2018.
Laboratory of Functional Analysis and Geometry of Spaces
Mohamed Boudiaf University of M'Sila
28000 M'Sila, Algeria
E-mail: [email protected], [email protected]
