We consider the multilinear pseudo-differential operators with symbols in a generalized $S_{0,0}$-type class and prove the boundedness of the operators from $(L^2,\ell^{q_1}) \times \dots \times (L^2,\ell^{q_N})$ to $(L^2,\ell^{r})$, where $(L^2, \ell^{q})$ denotes the $L^2$-based amalgam space. This extends the previous result by the same authors, which treated the bilinear pseudo-differential operators and gave the $L^2 \times L^2 $ to $(L^2, \ell^{1})$ boundedness.
In this paper, we first introduce several new classes of weighted amalgam spaces. Then we discuss both strong type and weak type estimates for certain multilinear $θ$-type Calderón--Zygmund operators $T_θ$ recently introduced in the literature on products of these spaces with multiple weights. Furthermore, the strong type and weak end-point estimates for both multilinear commutators and iterated commutators of $T_θ$ and pointwise multiplication with BMO functions are established as well.
The present article focuses on the bounds of weighted multilinear $p$-adic Hardy operators on the product of Herz spaces and Morrey-Herz spaces. The corresponding norm on both the cases are also obtain.
The multilinear pseudo-differential operators with symbols in the multilinear Hörmander class $S_{0,0}$ are considered. A complete identification of the cases where those operators define bounded operators between local Hardy spaces is given. Some results for the boundedness between Wiener amalgam spaces are also given. These are extensions and improvements of the results known in the bilinear case.
This paper will be devoted to study the two-weight norm inequalities of the multilinear fractional maximal operator $\mathcal{M}_α$ and the multilinear fractional integral operator $\mathcal{I}_α$. The entropy conditions in the multilinear setting will be introduced and the entropy bounds for $\mathcal{M}_α$ and $\mathcal{I}_α$ will be given.
We introduce multilinear analogues of dyadic paraproduct operators and Haar Multipliers, and study boundedness properties of these operators and their commutators. We also characterize dyadic BMO functions via the boundedness of certain paraproducts and also via the boundedness of the commutators of multilinear Haar Multipliers and paraproduct operators.
In this paper, we study multilinear Fourier multiplier operators on Hardy spaces. In particular, we prove that the multilinear Fourier multiplier operator of Hörmander type is bounded from $H^{p_1} \times \cdots \times H^{p_m}$ to $H^p$ for $0<p_1,\dots,p_m\le 1$ with $1/p_1 + \cdots 1/p_m = 1/p$, under suitable cancellation conditions. As a result, we extend the trilinear estimates of the authors(arXiv:2107.00225) to general multilinear ones and improve the boundedness result of the authors, Lee, Heo, Hong, Park, and Yang(Math. Ann. 381 : 499-555, 2021) in limiting situations.
We extend and improve the known results about the boundedness of the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class $BS^{m}_{0,0}(\mathbb{R}^n)$. We consider wider classes of symbols and improve estimates for the corresponding operators. A key idea is to consider the operators in Wiener amalgam spaces.
In this paper, we established the boundedness of m-linear Marcinkiewicz integral on Campanato type spaces. We showed that if the $m$-linear Marcinkiewicz integral is finite for one point, then it is finite almost everywhere. Moreover, the following norm inequality holds, $$\|μ(\vec{f})\|_{\mathcal{E}^{α,p}} \leq C\prod_{j=1}^m\|f_j\|_{\mathcal{E}^{α_j,p_j}},$$ where $\mathcal{E}^{α,p}$ is the classical Campanato spaces.
We consider the boundedness of the multilinear pseudo-differential operators with symbols in the multilinear Hörmander class $S_{0,0}$. The aim of this paper is to discuss smoothness conditions for symbols to assure the boundedness between local Hardy spaces.
We deal with the boundedness properties of higher order commutators related to some generalizations of the multilinear fractional integral operator of order $m$, $I_α^m$, from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces, extending some previous estimates for the linear case. Our study includes two different types of commutators and sufficient conditions on the weights in order to guarantee the continuity properties described above. We also exhibit the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defini…
Let $(X,d,μ)$ be a non-homogeneous metric measure space satisfying both the geometrically doubling and the upper doubling measure conditions. In this paper, the boundedness of multilinear fractional integral operator in this setting is proved. Via a sharp maximal operator, the boundedness of commutators generated by multilinear fractional integral operator with $RBMO(μ)$ function on non-homogeneous metric measure spaces in Lebesgue spaces is obtained.
We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in weighted Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander $S^{m}_{ρ, δ}$ classes. These results are new in the case $ρ< 1$, that is, outwith the scope of multilinear Calderón-Zygmund theory.
In this paper, we will study the boundedness properties of multilinear Calderon--Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.
In this article, we investigate the boundedness properties of the multilinear dyadic paraproduct operators in the weighted setting. We also obtain weighted estimates for the multilinear Haar multipliers and their commutators with dyadic BMO functions.
Let $T$ be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of $\mathbb R^n$. We assume that the associated kernel of $T$ satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. We then show the boundedness for $T$ and the boundedness of the commutator of $T$ with BMO functions on products of weighted Lebesgue spaces of $\mathbb R^n$. As an application, we obtain the weighted norm inequalities of multilinear Fourier multipliers and of the…
In this note we give a characterization of $\ell^{p}\times ...\times \ell^{p}\to\ell^q$ boundedness of maximal operators associated to multilinear convolution averages over spheres in $\mathbb{Z}^n$.
We establish the boundedness of the multilinear Calderón-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multil…
We study the boundedness on the Wiener amalgam spaces $W^{p,q}_s$ of Fourier multipliers with symbols of the type $e^{iμ(ξ)}$, for some real-valued functions $μ(ξ)$ whose prototype is $|ξ|^β$ with $β\in (0,2]$. Under some suitable assumptions on $μ$, we give the characterization of $W^{p,q}_s\rightarrow W^{p,q}$ boundedness of $e^{iμ(D)}$, for arbitrary pairs of $0< p,q\leq \infty$. Our results are an essential improvement of the previous known results, for both sides of sufficiency and necessity, even for the special case $μ(ξ)=|ξ|^β$ with $1<β<2$.
Characterizations for multilinear fractional maximal and integral operators and their commutators on generalized weighted Morrey spaces and applications
In this work we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited decay. The a.e. convergence is deduced from the $L^2\times\cdots\times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.
In dimensions $n\ge 2$ we obtain $L^{p_1}(\mathbb R^n) \times\dots\times L^{p_m}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results. Moreover, we obtain weak type and Lorentz space estimates as well as counterexamples in the endpoint cases.
In this paper, the main aim is to consider the boundedness of commutators of multilinear Calderón-Zygmund operators with Lipschitz functions in the context of the variable exponent Lebesgue spaces. Furthermore, the variable versions of the Lipschitz spaces are also discussed, and with the help of a key tool of a pointwise estimate involving the sharp maximal operator of the multilinear fractional commutator and certain associated maximal operators.
In this paper, we establish the boundedness of the commutators of multilinear Hausdorff operators on the product of some weighted Morrey-Herz type spaces with variable exponent with their symbols belong to both Lipschitz space and central BMO space. By these, we generalize and strengthen some previous known results.
The aim of the present paper is to give necessary and sufficient conditions for the boundedness of a general class of multilinear Hausdorff operators that acts on the product of some weighted function spaces with variable exponent such as the weighted Lebesgue, Herz, central Morrey and Morrey-Herz type spaces with variable exponent. Our results improve and generalize some previous known results.
Multilinear interpolation is a powerful tool used in obtaining strong type boundedness for a variety of operators assuming only a finite set of restricted weak-type estimates. A typical situation occurs when one knows that a multilinear operator satisfies a weak $L^q$ estimate for a single index $q$ (which may be less than one) and that all the adjoints of the multilinear operator are of similar nature, and thus they also satisfy the same weak $L^q$ estimate. Under this assumption, in this expository note we give a general multilinear interpolation theorem which allows one to obtain strong typ…
In this paper, we consider the boundedness properties of multilinear $θ$-type Calderón--Zygmund operators $T_θ$ recently introduced in the literature. First, we prove strong type and weak type estimates for multilinear $θ$-type Calderón--Zygmund operators on products of weighted Morrey spaces with multiple weights. Then we discuss strong type estimates for both multilinear commutators and iterated commutators of $T_θ$ on products of these spaces with multiple weights. Furthermore, the weak end-point estimates for commutators of $T_θ$ and pointwise multiplication with functions in bounded mean …
We prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calderón-Zygmund operators as several functions can now depend on the same one-dimensional variable. The study of this class is motivated by examples related to the two-dimensional bilinear Hilbert transform and to bilinear ergodic averages. This paper is a sequel to the prior paper arXiv:1108.0917 by the first author.
In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying the results, we improve the multilinear Hormander's multiplier theorem of Tomita, that of Grafakos-Si, and the boundedness results for bilinear pseudo-differential operators, given by Koezuka-Tomita.
This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction and geometric averaging operators. It is shown that, under very general circumstances, the boundedness of such functionals is equivalent to a geometric inequality for measures which has recently appeared in work by D. Oberlin (Math Proc. Cambridge. Philos. Soc., 129, 2000) and Bak, Oberlin, and Seeger (J. Aust. Math. Soc., 85, 2008).
In this paper, we investigated the boundedness of multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},α}$ associated with rectangles or related to more general basis with multiple weights $A_{(\vec{p},q),\mathcal{R}}$. In the rectangles setting, we first gave an end-point estimate of $\mathcal{M}_{\mathcal{R},α}$, which not only extended the famous linear result of Jessen, Marcinkiewicz and Zygmund, but also extended the multilinear result of Grafakos, Liu, Pérez and Torres ($α=0$) to the case $0<α<mn.$ Then, in one weight case, we gave several equivalent characterizations…
We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving $L^p$-averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the $L^p$-boundedness proved in \cite{MTT} and entails as a corollary a rich multilinear weighted theory. In particular, we obtain $L^{q_1}(v_1) \times L^{q_2}(v_2)$-boundedness of the bilinear Hilbert transform when the weights $v_j$ belong to the class $A_{\frac{q+1}{2}}\cap RH_2$. Our proof relies on a stopping time cons…
In this paper, the weighted Lp boundedness of multilinear commutators and iterated commutators of multilinear singular integral operators with generalized kernels is established, where the weight is multiple weight. Our results are generalizations of the corresponding results for multilinear singular integral operators with standard kernels and Dini kernels.
Let $m(ξ,η)$ be a measurable locally bounded function defined in $\mathbb R^2$. Let $1\leq p_1,q_1,p_2,q_2<\infty $ such that $p_i=1$ implies $q_i=\infty $. Let also $0<p_3,q_3<\infty $ and $1/p=1/p_1+1/p_2-1/p_3$. We prove the following transference result: the operator $$ {\mathcal C}_m(f,g)(x)=\int_{\bbbr} \int_{\bbbr} \hat f(ξ) \hat g(η) m(ξ,η) e^{2πi x(ξ+η)}dξdη$$ initially defined for integrable functions with compact Fourier support, extends to a bounded bilinear operator from $L^{p_1,q_1}(\bbbr)\times L^{p_2,q_2}(\bbbr)$ into $L^{p_3,q_3}(\bbbr)$ if and only if the family of operators …
In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function $\Mm$ introduced in \cite{LOPTT} and for multilinear Calderón-Zygmund operators. In particular we obtain a sharp mixed "$A_p-A_{\infty}$" bound for $\Mm$, some partial results related to a Buckley-type estimate for $\Mm$, and a sufficient condition for the boundedness of $\Mm$ between weighted $L^p$ spaces with different weights taking into account the precise bounds. Next we get a bound for multilinear Calderón-Zygmund operators in terms of dyadic positive multilinear operators in the spirit…
We give a weak factorization proof of the Hardy space $H^{p}(\mathbb{R}^{n})$ in the multilinear setting, for $\frac{n}{n+1} < p <1$. As a consequence, we obtain a characterization of the boundedness of the commutator $[b,T]$ from $L^{r_{1}}(\mathbb R^n) \text{~x ... x~} L^{r_{m}} (\mathbb R^n) \text{ to } L^{q^\prime} (\mathbb R^n)$, where $b \in \text{Lip}_α(\mathbb R^n)$, and $\fracα{n} = \sum_{i=1}^{m} \frac{1}{r_{i}} +\frac{1}{q} - 1$.
Let $T$ be a multilinear Calderón-Zygmund operator of type $ω$ with $ω(t)$ being nondecreasing and satisfying a kind of Dini's type condition. Let $T_{Π\vec{b}}$ be the iterated commutators of $T$ with $BMO$ functions. The weighted strong and weak $L(\log{L})$-type endpoint estimates for $T_{Π\vec{b}}$ with multiple weights are established. Some boundedness properties on weighted variable exponent Lebesgue spaces are also obtained. As applications, multiple weighted estimates for iterated commutators of paraproducts and bilinear pseudo-differential operators with mild regularity are given.
Let $κ\ge 2, λ> 1$ and define the multilinear Littlewood-Paley-Stein operators by $$g_{λ,μ}^*(\vec{f})(x) = \bigg(\iint_{\mathbb{R}^{n+1}_{+}} \vartheta_t(x, y) \bigg|\int_{\mathbb{R}^{n κ}} s_t(y,\vec{z}) \prod_{i=1}^κ f_i(z_i) \ dμ(z_i)\bigg|^2 \frac{dμ(y) dt}{t^{m+1}}\bigg)^{\frac12}, $$ where $\vartheta_t(x, y)=\big(\frac{t}{t + |x - y|}\big)^{m λ}$. In this paper, our main aim is to investigate the boundedness of $g_{λ,μ}^*$ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that $g_{λ,μ}^*$ is bounded from $L^{p_1}(…
This paper is devoted to give several characterizations on a more general level for the boundedness of $τ$-Wigner distributions acting from weighted modulation spaces to weighted modulation and Wiener amalgam spaces. As applications, sharp exponents are obtained for the boundedness of $τ$-Wigner distributions on modulation spaces with power weights. We also recapture the main theorems of Wigner distribution obtained in \cite{CorderoNicola2018IMRNI,Cordero2020a}. As consequences, the characterizations of the boundedness on weighted modulation spaces of several types of pseudodifferential operat…
We deal with the boundedness of the multilinear fractional integral operator $I_{γ,m}$ from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We characterize the classes of weights for which the problem described above holds and show the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region. We further exhibit examples…
We study $m$-linear homogeneous rough singular integral operators $\mathcal{L}_Ω$ associated with integrable functions $Ω$ on $\mathbb{S}^{mn-1}$ with mean value zero. We prove boundedness for $\mathcal{L}_Ω$ from $L^{p_1}\times \cdots \times L^{p_m}$ to $L^p$ when $1<p_1,\dots, p_m<\infty$ and $1/p=1/p_1+\cdots +1/p_m$ in the largest possible open set of exponents when $Ω\in L^q(\mathbb S^{mn-1})$ and $q\ge 2$. This set can be described by a convex polyhedron in $\mathbb R^m$.
The $L^p$ boundedness theory of convolution operators is \linebreak based on an initial $L^2\to L^2$ estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view of the unavailability of Plancherel's identity in this setting, and up to now it has not been clear what a natural initial estimate might be. In this work we achieve exactly this goal, i.e., obtain an initial $L^2\times\cdots\times L^2\to L^{2/m}$ estimate for general building blocks of $m$-linear multiplier operators. We apply this result to deduce analogo…
