Estimates of bilinear pseudodifferential operators associated to bilinear Hörmander classes in Besov and Triebel–Lizorkin spaces with variable exponents
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Abstract
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.
Keywords
Variable exponent Triebel–Lizorkin space Besov space Bilinear pseudodifferential operatorMSC
47G30 46E35 42B25 42B351 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 \(\mathcal{S}_{\rho,\delta}^{0}\), \(0\leq\delta\leq\rho<1\), whose corresponding pseudodifferential operators are bounded on \({L}^{2} ( \mathbb{{R}}^{{n}} )\), the classes \({B} \mathcal{S}_{\rho,\delta}^{0}\) (its definition is in Sect. 2) contain symbols for which the corresponding bilinear pseudodifferential operators do not map any product \({L}^{{P}_{1}} ( \mathbb{{R}}^{{n}} ) \times {L}^{{P}_{2}} ( \mathbb{{R}}^{{n}} )\), into any \({L}^{{P}} ( \mathbb{{R}}^{{n}} )\) with \({1} / {{P}} = {1} / {{P}_{1}} + {1} / {{P}_{2}}\); see [6]. Moreover, \({B} \mathcal{S}_{1,1}^{0}\) contains symbols for which the corresponding bilinear operators are unbounded from any \({L}^{{P}_{1}} ( \mathbb{{R}}^{{n}} ) \times{L}^{{P}_{2}} ( \mathbb{{R}}^{{n}} )\) into any \({L}^{{P}} ( \mathbb{{R}}^{{n}} )\) with \({1} / {{P}} = {1} / {{P}_{1}} + {1} / {{P}_{2}}\). Nevertheless, the operators with symbols in \({B} \mathcal{S}_{1,1}^{0}\) are proved to be bounded on products of Sobolev spaces with positive smoothness in [8]. However, the classes \({B} \mathcal{S}_{\rho,\delta}^{0}\) with \(0\leq\delta<1\), 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 \({B} \mathcal{S}_{0,0}^{{m}}\) 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 \({B} \mathcal{S}_{\rho,\delta}^{{m}}\). 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.
Since variable exponent function spaces have widely used in many fields such as electrorheological fluid [43], differential equations [19, 23, 41] and image restoration [9, 22, 29, 34, 46], many classical constant exponent function spaces have been generalized to variable exponent setting, such as variable exponent Bessel potential spaces [4, 21], variable Hajłasz–Sobolev spaces [5], variable exponent Besov and Triebel–Lizorkin spaces [3, 13, 16, 27, 30, 31, 49], variable exponent Hardy spaces [38, 56], variable exponent Morrey spaces [2], variable exponent Herz spaces [1, 26, 44], variable exponent Herz-type Hardy spaces [18, 28, 48], variable exponent Herz–Morrey Hardy spaces [50], variable exponent Herz-type Besov and Triebel–Lizorkin spaces [14, 17, 45, 52], variable exponent Morrey-type Besov and Triebel–Lizorkin spaces [20], Herz–Morrey-type Besov and Triebel–Lizorkin spaces with variable exponents [15], Triebel–Lizorkin-type spaces with variable exponents [57], variable weak Hardy spaces [53], Besov-type spaces with variable smoothness and integrability [58], variable integral and smooth exponent Triebel–Lizorkin spaces associated with a non-negative self-adjoint operator [51], variable exponent Hardy spaces associated with operators [55], and variable Hardy spaces associated with operators [54, 59, 60]. For the boundedness of integral operators in variable function spaces, we recommend [32] and [33]. In [39], Noi gave Fourier multiplier theorems for Besov and Triebel–Lizorkin spaces with variable exponents. Motivated by the mentioned work, we shall present the boundedness of the bilinear pseudodifferential operator associated to bilinear Hörmander classes in Besov and Triebel–Lizorkin spaces with variable exponents. Indeed, by using the embedding properties of the Besov and Triebel–Lizorkin spaces with variable exponents, we shall establish corresponding Leibnitz-type inequalities for the Besov and Triebel–Lizorkin spaces with variable exponents.
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.
2 Preliminaries
If \({a}\leq{cb}\) and \(b\leq{ca}\) we will write \({a}\approx {b}\). C is always a positive constant but it may change from line to line.
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
- (i)The function g is called locally log-Hölder continuous, abbreviated \({g}\in{C}_{{\mathrm{loc}}}^{\log} \), if there exists \({C}_{\log} > 0\) such that$$\bigl\vert g(x) - g(y) \bigr\vert \le\frac{C_{\log}}{\log ( e + 1 / \vert {x} - {y} \vert )}, \quad x,y \in \mathbb{R}^{n}, \vert x - y \vert < \frac{1}{2}. $$
- (ii)The function g is called globally log-Hölder continuous, abbreviated \({g}\in{C}_{\log} \), if it is locally log-Hölder continuous and there exists \({g}_{\infty} \in\mathbb{{R}}\) such that$$\bigl\vert g(x) - g_{\infty} \bigr\vert \le\frac{C_{\log}}{\log ( e + \vert {x} \vert )}, \quad \forall x \in\mathbb{R}^{n}. $$
If \({p}\in\mathcal{P}^{\log} \), then convolution with a radially decreasing \({L}^{1}\)-function is bounded on \({L}^{{P}({\cdot})}\): \(\Vert \varphi *f \Vert _{{p}({\cdot})} \leq{c} \Vert \varphi \Vert _{1} \Vert {f} \Vert _{{p}({\cdot})} \).
Definition 2.2
Let ψ be a function in \(\mathcal{S} ( \mathbb{{R}}^{{n}} )\) satisfying \(\psi ( {x} ) =1\) for \(| {x}|\leq1\) and \(\psi ( {x} ) =0\) for \(| {x}|\geq2\). We let \(\hat{\varphi}_{0} ( {x} ) := \psi ( {x} )\), \(\hat{\varphi} ( 2{x} ) := \psi ( {x} ) - \psi ( 2{x} )\) and \(\varphi_{{j}} ( {x} ) := 2^{{jn}} \varphi( 2^{{j}} {x})\) for \({j}\in\mathbb{{N}}\) and for all \({x}\in \mathbb{{R}}^{{n}}\). Then \(\sum_{{k}\in\mathbb{{N}}_{0}} \hat{\varphi}_{{k}} =1\).
Thus we obtain the Littlewood–Paley decomposition \({f}= \sum_{{v}=0}^{\infty} \varphi_{{v}} *{f}\) for all \({f}\in \mathcal{S}' ( \mathbb{{R}}^{{n}} )\) (convergence in \(\mathcal{S}' ( \mathbb{{R}}^{{n}} )\)).
For an appropriate function h, \(h(D)\) will stand for the multiplier operator given \(\hat{{h}({D}){f}} ={h} \hat{{f}}\) for \({f}\in\mathcal{S}' ( \mathbb{{R}}^{{n}} )\).
Definition 2.3
- (i)Let \({p},{q}\in \mathcal{P}_{0}^{\log} ( \mathbb{{R}}^{{n}} )\) and let \({s} \in {C}_{{\mathrm{loc}}}^{\log} ( \mathbb{{R}}^{{n}} )\). Thenwhere$$F_{p( \cdot),q( \cdot)}^{s( \cdot)} \bigl(\mathbb{R}^{n} \bigr): = \bigl\{ f \in\mathcal {{S}}' \bigl(\mathbb{R}^{n} \bigr): \Vert f \Vert _{F_{p( \cdot),q( \cdot )}^{s( \cdot)}}^{\varphi} < \infty \bigr\} , $$$$\Vert f \Vert _{F_{p( \cdot),q( \cdot)}^{s( \cdot)}}^{\varphi}: = \bigl\Vert \bigl( 2^{js( \cdot)}\varphi_{j} * f \bigr)_{j} \bigr\Vert _{L^{p( \cdot)} ( \ell^{q( \cdot)} )}. $$
- (ii)Let \({p},{q}\in\mathcal{P}_{0}^{\log} ( \mathbb{{R}}^{{n}} )\) and let \({s}\in{C}_{{\mathrm{loc}}}^{\log} ( \mathbb{{R}}^{{n}} )\).where$$B_{p( \cdot),q( \cdot)}^{s( \cdot )}\bigl(\mathbb{R}^{n}\bigr): = \bigl\{ f \in\mathcal {{S}}'\bigl(\mathbb{R}^{n}\bigr): \Vert f \Vert _{B_{p( \cdot),q( \cdot )}^{s( \cdot)}}^{\varphi} < \infty \bigr\} , $$$$\Vert f \Vert _{B_{p( \cdot),q( \cdot)}^{s( \cdot)}}^{\varphi}: = \bigl\Vert \bigl( 2^{ks( \cdot)}\varphi_{k} * f \bigr)_{k} \bigr\Vert _{\ell^{q( \cdot)} ( L^{p( \cdot)} )}. $$
Lemma 2.4
(Theorem 14 in [31])
- (i)If\({a} > \frac{{n}+ {C}_{\log} ( {1} / {{q}} )}{{p}^{-}} + {C}_{\log} ( {s} )\), then, for all\({f}\in\mathcal{S}' ( \mathbb{{R}}^{{n}} )\), we have$$ \Vert f \Vert _{B_{p( \cdot),q( \cdot)}^{s( \cdot)}} \approx \bigl\Vert \bigl\{ 2^{ks( \cdot)} ( \Phi_{k} * f ) \bigr\} _{k = 0}^{\infty} \bigr\Vert _{\ell^{q( \cdot)} ( L^{p( \cdot)} )} \approx \bigl\Vert \bigl\{ 2^{ks( \cdot)} \Phi_{k}^{ * a}f \bigr\} _{k = 0}^{\infty} \bigr\Vert _{\ell^{q( \cdot)} ( L^{p( \cdot)} )}. $$(1)
- (ii)If\({a} > \frac{{n}}{\min( {p}^{-}, {q}^{-} )} + {C}_{\log} ( {s} )\), then, for all\({f}\in \mathcal{S}' ( \mathbb{{R}}^{{n}} )\), we have$$ \Vert f \Vert _{F_{p( \cdot),q( \cdot)}^{s( \cdot )}} \approx \bigl\Vert \bigl\{ 2^{ks( \cdot)} ( \Phi_{k} * f ) \bigr\} _{k = 0}^{\infty} \bigr\Vert _{L^{p( \cdot)} ( \ell^{q( \cdot)} )} \approx \bigl\Vert \bigl\{ 2^{ks( \cdot)} \Phi_{k}^{ * a}f \bigr\} _{k = 0}^{\infty} \bigr\Vert _{L^{p( \cdot)} ( \ell^{q( \cdot)} )}. $$(2)
Denote \(\eta_{{v},{m}} := 2^{{nv}} (1+ 2^{{v}} | {x} | )^{-{m}}\), for \({v} \in\mathbb{{N}}_{0}\), \({m}\in\mathbb{{R}}\) and \({x}\in\mathbb{{R}}^{{n}}\).
Lemma 2.5
(Lemma A.3 in [13])
Here the implicit constant depends only on m and n.
Lemma 2.6
(Lemma A.6 in [13])
Lemma 2.7
(Theorem 3.2 in [13])
Lemma 2.8
(Lemma 4.7 in [3])
Lemma 2.9
(Lemma 6.1 in [13])
Lemma 2.10
(Lemma 9 in [31])
Lemma 2.11
(Theorem 3.6 in [3])
Let\({p},{q} \in\mathcal{P}\). If either\(\frac{1}{ {p}} + \frac{1}{{q}} \leq1\)pointwise, orqis a constant, then\(\Vert {\cdot} \Vert _{\ell^{{q}({\cdot})} ( {L}^{{p}({\cdot})} )}\)is a norm.
Lemma 2.12
(Theorem 6.1 in [3])
- (i)
If\({q}_{0} \leq{q}_{1}\)then\({B}_{{p}({\cdot }), {q}_{0} ({\cdot})}^{{s}({\cdot})} \hookrightarrow {B}_{{p}({\cdot}), {q}_{1} ({\cdot})}^{{s}({\cdot})}\).
- (ii)
If\(( {s}_{0} - {s}_{1} )^{-} > 0\), then\({B}_{{p}({\cdot}), {q}_{0} ({\cdot})}^{{s}_{0} ({\cdot})} \hookrightarrow {B}_{{p}({\cdot}), {q}_{1} ({\cdot})}^{{s}_{1} ({\cdot})}\).
- (iii)
If\({p}^{+}, {q}^{+} < \infty\), then\({B}_{{p}({\cdot}),\min \{ {p}({\cdot}),q({\cdot}) \}}^{{s}({\cdot})} \hookrightarrow {F}_{{p}({\cdot}),q({\cdot})}^{{s}( {\cdot})} \hookrightarrow{B}_{{p}({\cdot}),\max \{ {p}({\cdot}),q({\cdot}) \}}^{{s}({\cdot})}\).
Remark 2.13
If \({p}\in\mathcal{P}^{\log} ( \mathbb{{R}}^{{n}} )\) with \(1< p^{-}\le p^{+}<\infty\), then Theorem 12.5.7 in [12] says that \({F}_{{p} ( {\cdot} ),2}^{0} ( \mathbb{{R}}^{{n}} ) = {L}^{{p} ( {\cdot} )}\).
We shall use characterizations of \({B}_{{p} ( {\cdot} ),{q} ( {\cdot} )}^{{s} ( {\cdot} )} ( \mathbb{{R}}^{{n}} ) \) and \({F}_{{p} ( {\cdot} ),{q} ( {\cdot} )}^{{s} ( {\cdot} )} ( \mathbb{{R}}^{{n}} )\) by approximation, which are a generalization of the classical Besov and Triebel–Lizorkin spaces. For the latter, see [47].
Lemma 2.14
(Theorem 8.1 in [3])
Theorem 2.15
Proof
The following generalized Hölder inequality will often be used in the sequel. It is Theorem 2.3 in [25].
Lemma 2.16
Lemma 2.17
Proof
3 Main results
Theorem 3.1
Theorem 3.2
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].
Lemma 3.3
(Lemma 3.1 in [37])
Lemma 3.4
(Lemma 3.2 in [37])
Lemma 3.5
- (a)If\({M} > \frac{{n}}{\min ( {p}_{1}^{-}, {p}_{2}^{-}, {q}^{-} )} + {C}_{\log} ( {s} ) +{n}\), then there exists a positive constantCdepending onN, M, n, p, q, ssuch thatfor all\(\ell\in\mathbb{{N}}_{0}\), \({f}, {g} \in\mathcal{S} ( \mathbb{{R}}^{{n}} )\)and\(\sigma\in{B} \mathcal{S}_{1,1}^{0}\).$$ \Biggl\Vert \Biggl\{ 2^{ks ( \cdot )}\sum_{j = 0}^{k} \bigl\vert {T}_{\sigma_{j,k,\ell}} ({f},{g}) \bigr\vert \Biggr\} _{k \in\mathbb{N}_{0}} \Biggr\Vert _{L^{p( \cdot)}(\ell^{q( \cdot)})} \le C \Vert \sigma \Vert _{N,M}2^{ - \ell N} \Vert f \Vert _{F_{p_{1}( \cdot),q( \cdot)}^{s( \cdot)}} \Vert g \Vert _{F_{p_{2}( \cdot ),1}^{0}} $$(14)
- (b)If\({M} > \frac{{n}+ {C}_{\log} ( {1} / {{q}} )}{\min ( {p}_{1}^{-}, {p}_{2}^{-} )} + {C}_{\log} ( {s} ) +{n}\), then there exists a constantCdepending only onN, M, nandp, q, ssuch thatfor all\(\ell\in\mathbb{{N}}_{0}\), \({f}, {g} \in\mathcal{S} ( \mathbb{{R}}^{{n}} )\)and\(\sigma\in{B} \mathcal{S}_{1,1}^{0}\).$$ \Biggl\Vert \Biggl\{ 2^{ks ( \cdot )}\sum_{j = 0}^{k} \bigl\vert {T}_{\sigma_{j,k,\ell}} ({f},{g}) \bigr\vert \Biggr\} _{k \in\mathbb{N}_{0}} \Biggr\Vert _{\ell^{q( \cdot)}(L^{p( \cdot)})} \le C \Vert \sigma \Vert _{N,M}2^{ - \ell N} \Vert f \Vert _{B_{p_{1}( \cdot),q( \cdot)}^{s( \cdot)}} \Vert g \Vert _{F_{p_{2}( \cdot ),1}^{0}} $$(15)
Proof
Lemma 3.6
- (a)If\({s} \in {C}_{{\mathrm{loc}}}^{\log} \cap {L}^{\infty}\)with\({s}^{-} > 0\), \({s}^{+} < \infty\), \({M} > \frac{{n}}{\min ( {p}_{1}^{-}, {p}_{2}^{-}, {q}^{-} )} + {C}_{\log} ( {s} ) +{n}\), then there exists a constantCdepending only onN, M, nand\({p}_{1}\), \({p}_{2}\), q, ssuch thatfor all\(\ell\in\mathbb{{N}}_{0}\), \({f}, {g}\in\mathcal{S} ( \mathbb{{R}}^{{n}} )\)and\(\sigma\in{B} \mathcal{S}_{1,1}^{0}\).$$\bigg\| \sum_{j,k \in\mathbb{N}_{0},j \le k} {T}_{\sigma_{j,k,\ell}} ({f},{g}) \bigg\| _{F_{p( \cdot),q( \cdot)}^{s( \cdot)}} \le C \Vert \sigma \Vert _{N,M}2^{ ( s^{ +} - N )\ell} \Vert f \Vert _{F_{p_{1}( \cdot),q( \cdot)}^{s( \cdot)}} \Vert g \Vert _{F_{p_{2}( \cdot),1}^{0}} $$
- (b)If\({s} \in {C}_{{\mathrm{loc}}}^{\log} \cap {L}^{\infty}\)with\({s}^{-} > 0\), \({s}^{+} < \infty\)and\({N} > \frac{{n}+ {C}_{\log} ( {1} / {{q}} )}{\min ( {p}_{1}^{-}, {p}_{2}^{-} )} + {C}_{\log} ( {s} ) +{n}\), then there exists a constantCdepending only onN, M, nand\({p}_{1}\), \({p}_{2}\), q, ssuch thatfor all\(\ell\in\mathbb{{N}}_{0}, {f}, {g}\in\mathcal{S} ( \mathbb{{R}}^{{n}} )\)and\(\sigma\in{B} \mathcal{S}_{1,1}^{0}\).$$\bigg\| \sum_{j,k \in\mathbb{N}_{0},j \le k} {T}_{\sigma_{j,k,\ell}} ({f},{g}) \bigg\| _{B_{p( \cdot),q( \cdot)}^{s( \cdot)}} \le C \Vert \sigma \Vert _{N,M}2^{ ( s^{ +} - N )\ell} \Vert f \Vert _{B_{p_{1}( \cdot),q( \cdot)}^{s( \cdot)}} \Vert g \Vert _{F_{p_{2}( \cdot),1}^{0}} $$
Proof
After these preparation, we now complete the proofs of Theorems 3.1 and 3.2.
Proofs of Theorems 3.1 and 3.2
Notes
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Funding
The work is supported by the National Natural Science Foundation of China (Grant No. 11761026 and 11761027) and Hainan Province Natural Science Foundation of China (2018CXTD338).
Competing interests
The authors declare that they have no competing interests.
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