Estimates of bilinear pseudodifferential operators associated to bilinear Hörmander classes in Besov and Triebel–Lizorkin spaces with variable exponents

In this paper, we give Leibniz-type estimates of bilinear pseudodifferential operators associated to bilinear Hörmander classes in Besov and Triebel–Lizorkin spaces with variable exponents. To obtain the estimate for Triebel–Lizorkin spaces with variable exponents, we present their approximation characterization.


Introduction
The theory of bilinear pseudodifferential operators with symbols in the Hörmander classes has been extensively studied by many authors. Different from their linear counterparts S 0 ρ,δ , 0 ≤ δ ≤ ρ < 1, whose corresponding pseudodifferential operators are bounded on L 2 (R n ), the classes BS 0 ρ,δ (its definition is in Sect. 2) contain symbols for which the corresponding bilinear pseudodifferential operators do not map any product L P 1 (R n ) × L P 2 (R n ), into any L P (R n ) with 1/P = 1/P 1 + 1/P 2 ; see [6]. Moreover, BS 0 1,1 contains symbols for which the corresponding bilinear operators are unbounded from any L P 1 (R n ) × L P 2 (R n ) into any L P (R n ) with 1/P = 1/P 1 + 1/P 2 . Nevertheless, the operators with symbols in BS 0 1,1 are proved to be bounded on products of Sobolev spaces with positive smoothness in [8]. However, the classes BS 0 ρ,δ with 0 ≤ δ < 1, like their linear setting, the corresponding bilinear pseudodiffer ential operators are bilinear Calderón-Zygmund operators. In [7], the properties of symbols, and boundedness properties of bilinear pseudodifferential operators in Lebesgue spaces were given. For pseudodifferential operators with symbols in the bilinear Hörmander classes of sufficiently negative order, their boundedness properties in Lebesgue spaces, weak-type spaces, BMO and Sobolev spaces are established in [6]. In [8], by establishing a symbolic calculus for the transposes of a class of bilinear pseudodifferential operators, Benyi and Torres proved that these operators are bounded on products of Lebesgue spaces. In [24], Herbert and Naibo showed that bilinear pseudodifferential operators with symbols in Besov spaces are bounded on products of Lebesgue spaces. In [36], Miyachi and Tomita determined the order m for which all the bilinear pseudodifferential operators with symbols in the Hörmander class BS m 0,0 are bounded among Lebesgue spaces, local Hardy spaces, and bmo spaces. In [35], Michalowski, Rule and Staubach obtained the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in Lebesgue spaces and the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander classe BS m ρ,δ . In [42], Rodríguez-López and Staubach obtained the boundedness of rough Fourier integral and pseudodifferential operators. As applications, then they considered boundedness results for Hörmander class bilinear pseudodifferential operators, certain classes of bilinear (as well as multilinear) Fourier integral operators, and rough multilinear operators. Recently, in [37] Naibo obtained boundedness properties on the scales of inhomogeneous Triebel-Lizorkin and Besov spaces of positive smoothness for pseudodifferential operators with symbols in certain bilinear Hörmander classes.
The plan of the paper is as follows. In Sect. 2, we shall state notions, preliminary results. In particular, we give the approximation characterizations of Triebel-Lizorkin spaces with variable exponents. In Sect. 3, we present the proofs of the main results.

Preliminaries
We denote by S(R n ) the usual Schwartz space of rapidly decreasing complex-valued functions and S (R n ) the dual space of tempered distributions. As usual, we denote by f or F(f ) the Fourier transform of f ∈ S (R n ). In particular, we use the formula We denote by F -1 (f ) orf the inverse Fourier transform of f . Given a real number r ≥ 0, the homogeneous derivative of order r, D r , is defined by Let m ∈ R and 0 ≤ δ ≤ ρ ≤ 1. A function σ on R 3n , is an element of the bilinear Hörmander class BS m ρ,δ if for all multi-indices γ , α, β ∈ N n 0 there exist some positive constants C γ ,α,β such that for all x, ξ , η ∈ R n , where |γ | denotes the sum of its components, |α| and |β| are similar. The bilinear pseudodifferential operator associated to σ is defined by Then BS m ρ,δ becomes a Fréchet space with the family of norms { σ N,M : N, M ∈ N 0 }. If a ≤ cb and b ≤ ca we will write a ≈ b. C is always a positive constant but it may change from line to line.
For a measurable function p on R n , we denote p -:= ess inf x∈R n p(x) and p + := ess sup x∈R n p(x). We denote by P 0 the subset of measurable functions on R n with values in (0, ∞] such that p -> 0, and by P the subset of measure functions with values in [1, ∞]. For p(·) ∈ P 0 , the function ρ p is defined as follows: The convention 1 ∞ = 0 is adopted in order for ρ p to be left-continuous. The variable exponent modular is defined by The variable exponent Lebesgue space L p(·) consists of measurable functions f : R n → R, with ρ p(·) (λf ) < ∞ for some λ > 0. The Luxemburg (quasi)-norm on this space is defined by the formula f p(·) := inf λ > 0 : ρ p(·) (f /λ) ≤ 1 .
Let p, q ∈ P 0 . For a sequence of L p(·) -functions (f v ) v , we define the modular where we use the convention λ 1 ∞ = 1. Then the norm in the mixed Lebesgue-sequence space q(·) (L p(·) ) is defined by .
Since the above right-hand side expression is much simpler, we use this notation to represent the above left-hand side even when q + = ∞, and that means for the modular. Let (f v ) v∈N 0 be a sequence of measurable functions on R n , then the norm of (f v ) v∈N 0 in the space L p(·) ( q(·) ) is defined by In the development of the variable exponent function spaces, the concept of log-Hölder continuity is the cornerstone, which was introduced in [10,11]. Definition 2.1 Let g be a real function on R n .
(i) The function g is called locally log-Hölder continuous, abbreviated g ∈ C log loc , if there exists C log > 0 such that (ii) The function g is called globally log-Hölder continuous, abbreviated g ∈ C log , if it is locally log-Hölder continuous and there exists g ∞ ∈ R such that , ∀x ∈ R n .
The notation P log is used for those variable exponents p ∈ P with 1 p ∈ C log . The class P log 0 is defined analogously. Let f , g be in L 1 (R n ). Define the convolution f * g by If p ∈ P log , then convolution with a radially decreasing L 1 -function is bounded on L P(·) :
Thus we obtain the Littlewood- We also putψ 0 :=φ 0 +φ 1 andψ k :=φ k-1 +φ k +φ k+1 for k ∈ N. It is easy to see that ϕ kψk =φ k for k ∈ N 0 and For an appropriate function h, h(D) will stand for the multiplier operator givenĥ(D)f = hf for f ∈ S (R n ).
(ii) Let p, q ∈ P log 0 (R n ) and let s ∈ C log loc (R n ).
The key tool will be the Peetre maximal operators, which were introduced by Peetre in [40]. Let a be a positive number and a system ( k ) k∈N 0 in S(R n ). Then the Peetre maximal operators associated to ( k ) k∈N 0 are defined by for each distribution f ∈ S (R n ) * a We start with two given functions φ 0 , φ 1 ∈ S(R n ). We define Moreover, for each j ∈ N 0 , we denote j =φ j . We shall use the following result.

Lemma 2.5 (Lemma A.3 in [13])
Let v 0 , v 1 ≥ 0 and m > n. Then Here the implicit constant depends only on m and n.
Lemma 2.6 (Lemma A.6 in [13]) Let r > 0, v ≥ 0 and m > n. Then there exists c > 0, which depends only on m, n and r, such that, for all g ∈ S (R n ) with suppĝ ⊂ {ξ ∈ R n : |ξ | ≤ 2 v+1 }, holds for every sequence (f v ) v∈N 0 of locally integrable functions and m > n.
In Lemmas 2.7 and 2.8, we required that p -, q -> 1. This restriction can often be overcome by using Lemma 2.6 and the following identity: . Lemma 2.9 (Lemma 6.1 in [13]) If s(·) ∈ C log loc , then there exists t ∈ (n, ∞) such that if m > t, then with c > 0 independent of x, y ∈ R n and v ∈ N 0 . Therefore, Lemma 2.10 (Lemma 9 in [31]) Let p, q ∈ P 0 (R n ) and δ > 0. Let (g k ) k∈Z be a sequence of non-negative measurable functions on R n and define Then there exist constants c 1 , c 2 > 0, depending on p(·), q(·) and δ, such that Lemma 2.11 (Theorem 3.6 in [3]) Let p, q ∈ P. If either 1 p + 1 q ≤ 1 pointwise, or q is a constant, then · q(·) (L p(·) ) is a norm.

Let f ∈ S (R n ). Then f is in B s(·) p(·),q(·) if and only if there exists
Furthermore, where the infimum is taken over all admissible systems ω ∈ p(·) (R n ), is an equivalent quasi- Furthermore, where the infimum is taken over all admissible systems ω ∈ p(·) (R n ), is an equivalent quasinorm in F s(·) p(·),q(·) .
Proof First we show that there is a constant C independent of f such that Let (ϕ j ) j be functions in R n as defined in Definition 2.2, then Thus (ω j ) j ∈ p(·) (R n ) and in S (R n ). Notice that 2 -ks(·) ≤ 2 -ksand that s -> 0 by assumption. If q(x) ∈ [1, ∞], by Minkowski's inequality, we have Thus from (6) and (7), for q ∈ P 0 , we have Taking the L p(·) (R n )-quasi-norm on the above inequality, we obtain (5) since . Now we show the opposite inequality of (5). Let (ω k ) k ∈ p(·) (R n ) such that f = lim k→∞ ω k and f ω < ∞. Then ϕ j * f = ∞ k=-1 ϕ j * (ω k+jω k+j-1 ), j ∈ N 0 (with ω -1 = 0). Since Let r ∈ (0, min{p -, q -, 1}). By the definition of ϕ j , there exists a constant C > 0 such that |ϕ j | ≤ Cη j, 2m r , and using Lemma 2.6, then we conclude that By Minkowski's integral inequality (with exponent 1 r > 1) and Lemma 2.5 we obtain Hence, by Lemma 2.9, Since 2 js(·) ≤ 2 (j+k)s(·) 2 -ks -. So we have where the last inequality is due to Lemma 2.7. Now using |ω k+jω k+j-1 | ≤ |fω k+j | + |fω k+j-1 |, we find that Since the sequence space is invariant with respect to shifts, we arrive at that the left-hand side can be estimated by a constant times f ω . Taking the infimum over ω, we conclude that f F s(·) The following generalized Hölder inequality will often be used in the sequel. It is Theorem 2.3 in [25].

Lemma 2.16
Let p, p 1 , p 2 ∈ P 0 (R n ) with p + 1 , p + 2 < ∞ such that 1 p(x) = 1 . Then there exists a constant C p,p 1 independent of the functions f and g such that fg L p(·) ≤ C p,p 1 f L p 1 (·) g L p 2 (·) holds for every f ∈ L p 1 (·) and g ∈ L p 2 (·) .
To prove Theorems 3.1 and 3.2, we shall decompose the symbol function σ as usual, indeed, we shall follow the method in [37].
Let f , g ∈ S(R n ) and {φ k } k∈N 0 be functions in R n as Definition 2.2. We write whereσ 1 (·, ·, ·) stands for the Fourier transform of σ (·, ·, ·) with respect to the first variable and Now we need only to estimate I 1 , since the estimate for I 2 will follow from the one for I 1 by interchanging the roles of k and j, f and g, and ξ and η. Using a partition of unity with respect to the variable ζ we write where the symbols are given by for j ≤ k. For σ j,k, we use the following estimates.

Lemma 3.4 (Lemma 3.2 in [37]) Let
where F 2n σ j,k, (x, ·, ·) denotes the Fourier transform in R 2n of σ j,k, (x, ·, ·) with respect to the last two variables. If a > 0 and N, M ∈ N 0 are even with M > a + n, then there is a constant C depending only on M, N , a and n such that for all x ∈ R n , j, k, ∈ N 0 with j ≤ k, σ ∈ BS 0 1,1 .
2 ) + C log (s) + n, then there exists a constant C depending only on N , M, n and p, q, s such that for all ∈ N 0 , f , g ∈ S(R n ) and σ ∈ BS 0 1,1 .
Proof Let {φ k } k∈N 0 and {ˆ k } k∈N 0 be functions in R n as Definition 2.2 and put f k :=ˆ k (D)f and g j :=ˆ j (D)g for j, k ∈ N 0 , j ≤ k. Then and we write From (13), for a > 0 with M > a + n, we have Q j,k, (x, -y, -z) 1 + 2 k |y| + 2 j |z| a f k (xy)g j (xz) (1 + 2 k |y| + 2 j |z|) a dy dz for all x ∈ R n . In the last inequality we used Lemma 3.4 and the definition of the Peetre maximal operator. Thus, we obtain, for all x ∈ R n , To prove (14), after adding in j ≤ k, multiplying by 2 ks(·) , we obtain Then taking the q(·) -norm in k we obtain Since M -n > n min(p -1 ,p -2 ,q -) +C log (s) by the assumption in item (a), we choose a > n min(p -1 ,p -2 ,q -) + C log (s) such that Mn > a. Using the generalized Hölder inequality and Lemma 2.4, we have This is the inequality (14).
Proof Let f , g ∈ S(R n ). We shall use the characterizations of Besov and Triebel-Lizorkin spaces with variable exponents by approximation as described in Lemma 2.14 and Theorem 2.15, which require the condition s ∈ C log loc ∩ L ∞ . For each fixed ∈ N 0 , we put It is enough to estimate h ω F s(·) p(·),q (·) and h ω B s(·) p(·),q(·) (see (3) and (4)) for an appropriate sequence of functions ω . To do so, we define the sequence ω := {ω k, } k∈N 0 as follows: Then we have We claim that This inclusion is induced by the fact that supp T σ j,k, (f , g) ⊂ ζ ∈ R n : |ζ | ≤ 2 ν+ +3 for all j, ν, ∈ N 0 , j ≤ ν, which is easy to check; see [37].
For s -> 0, by the generalized Hölder inequality, we have where in the last inequality we used Lemma 3.5. Similarly, we obtain where the first inequality follows from B s(·) p(·),q(·) → L p(·) if s -> 0 by Lemma 2.12 and Remark 2.13, and the last inequality follows from Lemma 3.5.
Notice that the ω 0, = 0 if ∈ N, ω 0,0 = T σ 0,0,0 (f , g) and that (17) implies that Thus we have for all ∈ N 0 . Using (18), we have for all ∈ N 0 . We now estimate {2 ks(·) |hω k, |} k∈N 0 L p(·) ( q(·) ) by breaking the sum in k into k ≤ -1 and k ≥ . Since ω k, = 0 if k ≤ -1, for the first part we obtain where the second inequality follows from Lemma 2.10 and the last inequality follows from (17). Now, we turn to an estimate of the second part (that is, when k ≥ ). Since T σ j,k-+ν, (f , g), where the third inequality follows from Lemma 3.5.
After these preparation, we now complete the proofs of Theorems 3.1 and 3.2.