For $ 1\le k <n$, we prove that for functions $F,G$ on $ {\Bbb R}^{n}$, any $k$-dimensional affine subspace $H \subset {\Bbb R}^{n}$, and $p,q,r \ge 2$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, one has the estimate
$$ {||(F*G)|_H||}_{L^{r}(H)} \leq {||F||}_{Λ^H_{2, p}({\Bbb R}^{n})} \cdot {||G||}_{Λ^H_{2, q}({\Bbb R}^{n})},$$ where the mixed norms on the right are defined by
$$ {||F||}_{Λ^H_{2,p}({\Bbb R}^{n})}={(\int_{H^*} {(\int {|\hat{F}|}^2 dH_ξ^{\perp})}^{\frac{p}{2}} dξ)}^{\frac{1}{p}},$$ with $dH_ξ^{\perp}$ the $(n-k)$-dimensional Lebesgue measure on the affine subspace $H_ξ^{\perp}:=ξ+ H^\perp$. Dually, one obtains restriction theorems for the Fourier transform for affine subspaces. Applied to $F(x^{1},...,x^{m})=\prod_{j=1}^m f_j(x^{j})$ on $\R^{md}$, the diagonal $H_0={(x,...,x): x \in {\Bbb R}^d}$ and suitable kernels $G$, this implies new results for multilinear convolution operators, including $L^p$-improving bounds for measures, an $m$-linear variant of Stein's spherical maximal theorem, estimates for $m$-linear oscillatory integral operators, certain Sobolev trace inequalities, and bilinear estimates for solutions to the wave equation.
The notions of higher-order weighted multilinear Poincaré and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibnitz-type rules in Campanato-Morrey spaces are established.
As the classical $(p,q)$-Poincaré inequality is known to fail for $0 < p < 1$, we introduce the notion of weighted multilinear Poincaré inequality as a natural alternative when $m$-fold products and $1/m < p$ are considered. We prove such weighted multilinear Poincaré inequalities in the subelliptic context associated to vector fields of Hörmader type. We do so by establishing multilinear representation formulas and weighted estimates for multilinear potential operators in spaces of homogeneous type.
In this note we prove a multilinear version of the reverse Hölder inequality in the theory of Muckenhoupt $A_p$ weights. We give two applications of this inequality to the study of multilinear weighted norm inequalities. First, we prove a structure theorem for multilinear $A_{\vec{p}}$ weights; second, we give a new sufficient condition for multilinear, two-weight norm inequalities for the maximal operator.
We prove Bloom type two-weight inequalities for commutators of multilinear singular integral operators including Calderón-Zygmund operators and their dyadic counterparts. Such estimates are further extended to a general higher order multilinear setting. The proof involves a pointwise sparse domination of multilinear commutators.
A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalities of Holder, Young, and Loomis-Whitney. This is a companion to a recent paper by the same authors, containing a different proof and discussing certain variants.
In this paper, we study the weighted inequality for multilinear fractional maximal operators and fractional integrals. We give sharp weighted estimates for both operators.
We prove weighted strong inequalities for the multilinear potential operator ${\cal T}_ϕ$ and its commutator, where the kernel $ϕ$ satisfies certain growth condition. For these operators we also obtain Fefferman-Stein type inequalities and Coifman type estimates. On the other hand we prove weighted weak type inequalities for the multilinear maximal operator $\mathcal{M}_{φ,B}$ associated to a essentially nondecreasing function $φ$ and to a submultiplicative Young function $B$. This result allows us to obtain a weighted weak type inequality for the operator ${\cal T}_ϕ$.
This note is devoted to establishing two-weight estimates for commutators of singular integrals. We combine multilinearity with product spaces. A new type of two-weight extrapolation result is used to yield the quasi-Banach range of estimates.
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…
Certain rearrangement inequalities of a type considered by Hardy, Riesz, and Brascamp-Lieb-Luttinger are studied. Subsets of the real line that extremize these inequalities are characterized. Our results apply only to special cases, and complement earlier results by Burchard treating other cases. This draft incorporates minor corrections and clarifications.
It is a well known general principle that the Fourier transform of a random measure is small, except at the zero frequency, in various senses for appropriate notions of randomness. In this note we develop analogues of this principle for two classes of random multilinear operators. Two ergodic theoretic applications, involving correlations over randomly generated sparse subsequences, are obtained as corollaries of the main results.
In this note, we provide various two-weight norm estimates of the multi-linear fractional maximal function and weighted maximal function between different Orlicz spaces. More precisely, we obtain Sawyer-type characterizations and norm estimates for these operators.
The aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by F. Berra, M. Carena and G. Pradolini \cite{BCP}. We prove that, under certain conditions on the weights, there exists a constant $C$ such that $$\Bigg\| \frac{\mathcal G_α(\vec f \,)}{v}\Bigg\|_{L^{q, \infty}(νv^q)} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(u_i)}},$$ where $\mathcal G_α(\vec f \,)$ is the multilinear maximal function $\mathcal M_α(\vec f\,)$ that was introduced by K. Moen in \cite{M} or the multilineal fractional integral $\mathcal I_α(\vec f \,)$. As an applica…
In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,...,w_m)$ and $ν= w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main result of the paper sentences that under different conditions on the weights we can obtain $$\Bigg\| \frac{T(\vec f\,)(x)}{v}\Bigg\|_{L^{\frac{1}{m}, \infty}(νv^\frac{1}{m})} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(w_i)}}, $$ where $T$ is a multilinear Calderón-Zygmund operator. To obtain this result we first prove it for the $m$-fold product of the Hardy-Littlewood maximal operator…
This paper establishes a necessary and sufficient condition for $L^p$-boundedness of a class of multilinear functionals which includes both the Brascamp-Lieb inequalities and generalized Radon transforms associated to algebraic incidence relations. The testing condition involves bounding the average of an inverse power of certain Jacobian-type quantities along fibers of associated projections and covers many widely-studied special cases, including convolution with measures on nondegenerate hypersurfaces or on nondegenerate curves. The heart of the proof is based on Guth's visibility lemma in o…
In this paper we establish sharp estimates (up to logarithmic losses) for the multilinear oscillatory integral operator studied by Phong, Stein, and Sturm and Carbery and Wright on any product $\prod_{j=1}^d L^{p_j}(\mathbb R)$ with each $p_j \geq 2$, expanding the known results for this operator well outside the previous range $\sum_{j=1}^d p_j^{-1} = d-1$. Our theorem assumes second-order nondegeneracy condition of Varchenko type, and as a corollary reproduces Varchenko's theorem and implies Fourier decay estimates for measures of smooth density on degenerate hypersurfaces in $\mathbb R^d$.
Let $\vecω=( ω_{1},...,ω_{m})$ be a multiple weight and $\{Ψ_{j}\}^{m}_{j=1}$ be a sequence of Young functions. Let $\mathcal{M}_{\mathcal{R}}^{\vecΨ}$ be the multilinear strong maximal function with Orlicz norms which is defined by $$\mathcal{M}_{\mathcal{R}}^{\vecΨ}(\vec{f})(x)=\sup_{R\in \mathcal{R},R\ni x}\prod^{m}_{j=1}\|f_{j}\|_{Ψ_{j},R}$$ where the supremum is taken over all rectangles with sides parallel to the coordinate axes. If $Ψ_j(t)=t$, then $\mathcal{M}_{\mathcal{R}}^{\vec{t}}$ coincides with the multilinear strong mximal function $\mathcal{M}_{\mathcal{R}}$ defined and studied …
We prove mixed inequalities for commutators of Calderón-Zygmund operators (CZO) with multilinear symbols. Concretely, let $m\in\mathbb{N}$ and $\mathbf{b}=(b_1,b_2,\dots, b_m)$ be a vectorial symbol such that each component $b_i\in \mathrm{Osc}_{\mathrm{exp}\, L^{r_i}}$, with $r_i\geq 1$. If $u\in A_1$ and $v\in A_\infty(u)$ we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T_\mathbf{b}(fv)(x)|}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\|\mathbf{b}\|\frac{|f(x)|}{t}\right)u(x)v(x)\,dx\] holds for every $t>0$, where $Φ(t)=t(1+\log^+t)^r$, with $1/r=\sum_{i=1}^…
In this paper, the multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},α}$ associated with rectangles and corresponding multiple weights $A_{(\vec{p},q),\mathcal{R}}$ are introduced. Under the dyadic reverse doubling condition, a necessary and sufficient condition for two-weight inequalities is given. As consequences, we first obtain a necessary and sufficient condition for one-weight inequalities. Then, we give a new proof for the weighted estimates of multilinear fractional maximal operator $\mathcal{M}_α$ associated with cubes and multilinear fractional integral operato…
We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on $\ell^r$-valued extensions of linear operators. We show that for certain $1 \leq p, q_1, \dots, q_m, r \leq \infty$, there is a constant $C\geq 0$ such that for every bounded multilinear operator $T\colon L^{q_1}(μ_1) \times \cdots \times L^{q_m}(μ_m) \to L^p(ν)$ and functions $\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(μ_1), \dots, \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(μ_m)$, the following inequality holds \begin{equation}\label{MZ ineq abstract} (1) \quad \quad \left\Vert \left(\sum_{k_1, \dots…
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.
Basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered. Lebesgue space norm inequalities are established for multilinear integral operators of Calderon-Zygmund type which incorporate oscillatory factors exp(iP), where P is a real-valued polynomial with large coefficients. A related problem concerning upper bounds for measures of sublevel sets is solved.
Let $σ=(σ_{1},σ_{2},\dots,σ_{n})\in \mathbb{S}^{n-1}$ and $dσ$ denote the normalised Lebesgue measure on $\mathbb{S}^{n-1},~n\geq 2$. For functions $f_1, f_2,\dots,f_n$ defined on $\R$ consider the multilinear operator given by $$T(f_{1},f_{2},\dots,f_{n})(x)=\int_{\mathbb{S}^{n-1}}\prod^{n}_{j=1}f_{j}(x-σ_j)dσ, ~x\in \R.$$ In this paper we obtain necessary and sufficient conditions on exponents $p_1,p_2,\dots,p_n$ and $r$ for which the operator $T$ is bounded from $\prod_{j=1}^n L^{p_j}(\R)\rightarrow L^r(\R),$ where $1\leq p_j,r\leq \infty, j=1,2,\dots,n.$ This generalizes the results obtain…
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 $\mathbb{Z}^n$.
In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes $A_{\vec{p}}$, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators. Furthermore, we go beyond the classes $A_{\vec{p}}$ and extrapolate within the classes $A_{\vec{p},\vec{r}}$ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several a…
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 consider the Brascamp--Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. We obtain two new proofs of Lieb's fundamental theorem concerning exhaustion by gaussians. Our techniques are partly based upon monotonicity formulas for po…
Using the multilinear theory of oscillatory integral operators, we establish a square function inequality with implications to moment inequalities for discrete exponential sums which frequencies lie on curved hyper-surfaces. In particular a new distributional estimate for toral eigenfunctions is established.
In this paper, we obtain the continuity for some multilinear operators related to certain non-convolution operators on the Triebel--Lizorkin space. The operators include Littlewood--Paley operator and Marcinkiewicz operator.
In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding ``finite'' case obtained by Duoandikoetxea, which wa…
We prove multiple vector-valued and mixed-norm estimates for multilinear operators in $\rr R^d$, more precisely for multilinear operators $T_k$ associated to a symbol singular along a $k$-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension $d \geq 2$, the input functions are not necessarily in $L^p(\rr R^d)$ and can instead be elements of mixed-norm spaces $L^{p_1}_{x_1} \ldots L^{p_d}_{x_d}$. Such a result has interesting consequences especially when $L^\infty$ spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type i…
We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weights $w_1,\,w_2$ for the bilinear Hilbert transform to satisfy weighted norm inequalities of the form \[ BH : L^{p_1}(w_1^{p_1}) \times L^{p_2}(w_2^{p_2}) \longrightarrow L^p(w^p), \] where $w=w_1w_2$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}<\frac…
We prove a sparse bound for the $m$-sublinear form associated to vector-valued maximal functions of Fefferman-Stein type. As a consequence, we show that the sparse bounds of multisublinear operators are preserved via $\ell^r$-valued extension. This observation is in turn used to deduce vector-valued, multilinear weighted norm inequalities for multisublinear operators obeying sparse bounds, which are out of reach for the extrapolation theory recently developed by Cruz-Uribe and Martell. As an example, vector-valued multilinear weighted inequalities for bilinear Hilbert transforms are deduced fr…
This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators, $$\mathfrak{M}_{α,Ω}(\vec{f})(x)=\sup\limits_{0<r<{\rm dist}(x,Ω^c)}\frac{r^α}{|B(x,r)|^m}\prod\limits_{i=1}^m\int_{B(x,r)}|f_i(y)|dy,\quad \hbox{for \ }0\leqα<mn,$$ where $Ω$ is a subdomain in $\mathbb{R}^n$, $Ω^c=\mathbb{R}^n\backslashΩ$ and $B(x,r)$ is the ball in $\mathbb{R}^n$ centered at $x$ with radius $r$. Several new pointwise estimates for the derivative of the local multilinear maximal function $\mathfrak{M}_{0,Ω}$ and the fractional maxi…
In this paper we prove sharp $L^\infty$-$L^\infty$-$L^\infty$ decay for certain trilinear oscillatory integral forms of convolution type on $\mathbb R^2$. These estimates imply earlier $L^2$-$L^2$-$L^2$ results obtained by the second author as well as corresponding sharp, stable sublevel set estimates of the form studied by Christ and Christ, Li, Tao, and Thiele. New connections to the multilinear results of Phong, Stein, and Sturm are also considered.
The Hölder-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized Hölder-Brascamp-Lieb type inequality to hold and establish sufficient conditions for extremizers to exist when the underlyi…
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…
In this paper, we prove analogues of O'Neil's inequalities for the convolution in the weighted Lebesgue spaces. We also establish the weighted two-sided norm inequalities for the potential operator.
We obtain convolution inequalities in Lebesgue and Lorentz spaces with power weights when the functions involved are assumed to be radially symmetric. We also present applications of these results to inequalities for Riesz potentials of radial functions in weighted Lorentz spaces and embedding theorems for radial Besov spaces with power weights.
We consider various versions of fractional Leibniz rules (also known as Kato-Ponce inequalities) with polynomial weights $\langle x\rangle^a = (1+|x|^2)^{a/2}$ for $a\ge 0$. We show that the weighted Kato-Ponce estimate with the inhomogeneous Bessel potential $J^s = (1- \De)^{{s}/{2}}$ holds for the full range of bilinear Lebesgue exponents, for all polynomial weights, and for the sharp range of the degree $s$. This result, in particular, demonstrates that neither the classical Muckenhoupt weight condition nor the more general multilinear weight condition is required for the weighted Kato-Ponc…
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}(…
We give a $L^2\times L^2 \rightarrow L^2$ convolution estimate for singular measures supported on transversal hypersurfaces in $\mathbb{R}^n$, which improves earlier results of Bejenaru, Herr & Tataru as well as Bejenaru & Herr. The arising quantities are relevant in the study of the validity of bilinear estimates for dispersive partial differential equations. We also prove a class of global, nonlinear Brascamp-Lieb inequalities with explicit constants in the same spirit.
In this paper, the index groups for which the weighted Young's inequalities hold in both continuous case and discrete case are characterized. As applications, the index groups for the product inequalities on modulation spaces are characterized, we also obtain the weakest conditions for the boundedness of bilinear Fourier multipliers on modulation spaces in some sense. For the fractional integral operator, the sharp conditions for the boundedness of power weighted Lp-Lq estimates in both continuous case and discrete case are obtained. By a quite different approach from others, our theorems opti…
We study $L^p(μ) \to L^q(ν)$ mapping properties of the convolution operator $ T_λf(x)=λ*(fμ)(x)$ and of the corresponding maximal operator $ {\mathcal T}_λf(x)=\sup_{t>0} |λ_t*(fμ)(x)|$, where $λ$ is a tempered distribution, and $μ$ and $ν$ are compactly supported measures satisfying the polynomial growth bounds $μ(B(x,r)) \leq Cr^{s_μ}$ and $ν(B(x,r)) \leq Cr^{s_ν}$. As a result, we prove variants of the classical $L^p$-improving (Littman; Strichartz) and maximal (Stein) inequalities in a setting where the Plancherel formula is not available. Connections with the David-Semmes conjecture are a…
