For multiparameter bilinear paraproduct operators $B$ we prove the estimate $$
B: L^p X L^q --> L^r, 1<p,q\le{}\infty. $$ Here, $1/p+1/q=1/r$ and special attention is paid to the case of $0<r<1$. (Note that the families of multiparameter paraproducts are much richer than in the one parameter case.) These estimates are the essential step in the version of the multiparameter Coifman-Meyer theorem proved by C. Muscalu, J. Pipher, T. Tao, and C. Thiele. We offer a different proof of these inequalities.
Analogues of multi-paramter multiplier operators on R^d are defined on the torus T^d. It is shown that these operators satisfy the classical Coifman-Meyer theorem. In addition, L log L and L (log L)^n end-point estimates are proved.
We complete our boundedness theory of commutators of bilinear bi-parameter singular integrals by establishing the following result. If $T$ is a bilinear bi-parameter singular integral satisfying suitable $T1$ type assumptions, $\|b\|_{\operatorname{bmo}(\mathbb{R}^{n+m})} = 1$ and $1 < p, q \le \infty$ and $1/2 < r < \infty$ satisfy $1/p+1/q = 1/r$, then we have $$ \|[b, T]_1(f_1, f_2)\|_{L^r(\mathbb{R}^{n+m})} \lesssim \|f_1\|_{L^p(\mathbb{R}^{n+m})} \|f_2\|_{L^q(\mathbb{R}^{n+m})}. $$ Previously the range $r \le 1$ was proved only in the paraproduct free situation. The main novelty lies in t…
In the first part of the paper we prove a bi-parameter version of a well known multilinear theorem of Coifman and Meyer. As a consequence, we generalize the Kato-Ponce inequality in nonlinear PDE, obtaining a fractional Leibnitz rule for derivatives in the $x_1$ and $x_2$ directions simultaneously. Then, we show that the double bilinear Hilbert transform does not satisfy any $L^p$ estimates.
We give an explicit formula for one possible Bellman function associated with the $L^p$ boundedness of dyadic paraproducts regarded as bilinear operators or trilinear forms. Then we apply the same Bellman function in various other settings, to give self-contained alternative proofs of the estimates for several classical operators. These include the martingale paraproducts of Bañuelos and Bennett and the paraproducts with respect to the heat flows.
We introduce another notion of bounded logarithmic mean oscillation in the N-torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little BMO of Cotlar-Sadosky to the product BMO of Chang-Fefferman. We also obtain a sufficient condition for the boundedness of iterated commutators with Hilbert transforms betweeen the strong notions of these two spaces.
Muscalu, Pipher, Tao and Thiele \cite{MPTT} showed that the tensor product between two one dimensional paraproducts (also known as bi-parameter paraproduct) satisfies all the expected $L^p$ bounds. In the same paper they showed that the tensor product between two bilinear Hilbert transforms is unbounded in any range. They also raised the question about $L^p$ boundedness of the bilinear Hilbert transform tensor product with a paraproduct. We answer their question by obtaining a wide range of estimates for this hybrid bilinear operator. Our method relies on new vector valued estimates for a fami…
In this paper, we prove some uniform estimates between Lebesgue and Hardy spaces for operators closely related to the multilinear paraproducts on R^d. We are looking for uniformity with respect to parameters, which allow us to disturb the geometry and the metric on $R^d$.
We study boundedness properties of a class of multiparameter paraproducts on the dual space of the dyadic Hardy space H_d^1(T^N), the dyadic product BMO space BMO_d(T^N). For this, we introduce a notion of logarithmic mean oscillation on the polydisc. We also obtain a result on the boundedness of iterated commutators on BMO([0,1]^2).
We prove $L^p$ estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in arXiv:1011.6140.
We prove new results for multi-parameter singular integrals. For example, we prove that bi-parameter singular integrals in $\mathbb{R}^{n+m}$ satisfying natural $T1$ type conditions map $L^q(\mathbb{R}^n; L^p(\mathbb{R}^m;E))$ to $L^q(\mathbb{R}^n; L^p(\mathbb{R}^m;E))$ for all $p,q \in (1,\infty)$ and UMD function lattices $E$. This result is shown to hold even in the $\mathcal{R}$-boundedness sense for all suitable families of bi-parameter singular integrals. On the technique side we demonstrate how many dyadic multi-parameter operators can be bounded by using, and further developing, the th…
The main result of this paper is a bi-parameter T(b) theorem for the case that b is a tensor product of two pseudo-accretive functions. In the proof, we also discuss the L^2 boundedness of different types of the b-adapted bi-parameter paraproducts.
We show that the product BMO space can be characterized by iterated commutators of a large class of Calderón-Zygmund operators. This result follows from a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hytönen's representation theorem of Calderón-Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.
Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding for a scale of Dirichlet spaces from the b…
We prove multi-parameter Leibniz rules corresponding to flag paraproducts of arbitrary complexity in mixed-norm spaces, including endpoint estimates. The proof relies on multi-linear harmonic analysis techniques and a quantitative treatment of the commutators introduced by Bourgain and Li. The argument is robust and applicable to a generic class of multipliers, including (symmetric) Mikhlin multipliers of positive order and asymmetric variants of partial differential operators and Mikhlin multipliers of positive order.
Let $S$ be the dyadic bi-parameter square function $$Sf(x)^{2} = \sum_{R \in \mathcal{D}} |\langle f, h_{R} \rangle|^{2} \frac{1_{R}(x)}{|R|}.$$ We prove that if $T$ is a bi-parameter martingale transform and $f,g$ are suitable test functions, then there exists a sparse collection of rectangles $\mathcal{S}$ such that $$|\langle Tf, g \rangle| \lesssim \sum_{R \in \mathcal{S}} |R|(Sf)_{R}(Sg)_{R}.$$ We also extend this estimate to the case where $T$ is a bi-parameter cancellative dyadic shift and when $T$ is a paraproduct-free singular integral of Journé type. Weighted estimates follow from th…
We consider iterated commutators of multiplication by a symbol function and tensor products of Hilbert or Riesz transforms. We establish mixed BMO classes of symbols that characterize boundedness of these objects in $L^p$. Little BMO and product BMO, big Hankel operators and iterated commutators are the base cases of our results. We use operator theoretical methods and existing profound results on iterated commutators for the Hilbert transform case, while the general result in several variables is obtained through the construction of a Journé operator that models the behavior of the multiple H…
We study the boundedness properties of commutators formed by $b$ and $T$, where $T$ is a bilinear bi-parameter singular integral satisfying natural $T1$ type conditions and $b$ is a little BMO function. For paraproduct free bilinear bi-parameter singular integrals $T$ we prove that $[b, T]_1 \colon L^p(\mathbb{R}^{n+m}) \times L^q(\mathbb{R}^{n+m}) \to L^r(\mathbb{R}^{n+m})$ in the full range $1 < p, q \le \infty$, $1/2 < r < \infty$ satisfying $1/p+1/q = 1/r$. A special case is when $T$ is a bilinear bi-parameter multiplier. We also prove the corresponding Banach range result for all singular…
The famous $T1$ theorem for classical Calderón-Zygmund operators is a characterisation for their boundedness in $L^{2}$. In the bi-parameter case, on the other hand, the current $T1$ theorem is merely a collection of sufficient conditions. This difference in mind, we study a particular dyadic bi-parameter singular integral operator, namely the full mixed bi-parameter paraproduct $P$, which is precisely the operator responsible for the outstanding problems in the bi-parameter theory. We make several remarks about $P$, the common theme of which is to demonstrate the delicacy of the problem of fi…
We give an alternative proof of several sharp commutator estimates involving Riesz transforms, Riesz potentials, and fractional Laplacians. Our methods only involve harmonic extensions to the upper half-space, integration by parts, and trace space characterizations. The commutators we investigate are Jacobians, more generally Coifman-Rochberg-Weiss commutators, Chanillo's commutator with the Riesz potential, Coifman-Meyer or Kato-Ponce-Vega type commutators, and the Da Lio-Rivière three-term commutators. We also give a new limiting $L^1$-estimate for a double commutator of Coifman-Rochberg-Wei…
The main purpose of this paper is to study $L^r$ Hölder type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for $f,g,h\in \mathcal{S}(\mathbb{R}^{2})$, the bi-parameter trilinear flag Fourier multiplier operators we consider are defined by $$ T_{m_1,m_2}(f,g,h)(x):=\int_{\mathbb{R}^{6}}m_1(ξ,η,ζ)m_2(η,ζ)\hat f(ξ) \hat g(η)\hat h(ζ)e^{2πi(ξ+η+ζ)\cdot x}dξdηdζ, $$ when $m_1,m_2$ are two bi-parameter symbols. We will show that our problem can be reduced…
Let $ M_b$ be the operator of pointwise multiplication by $b$, that is $\operatorname M_b f=bf$. Set $[ A,B]={} AB- BA$. The Reisz potentials are the operators $$ R_αf(x)=\int f(x-y)\frac{dy}{\abs y ^α},\qquad 0<α<1. $$ They map $L^p\mapsto L^q$, for $1-α+\frac1q=\frac1p$, a fact we shall take for granted in this paper. A Theorem of Chanillo \cite{MR84j:42027} states that one has the equivalence $$ \norm [ M_b, R_α].p\to q.\simeq \norm b.\operatorname{BMO}. $$ with the later norm being that of the space of functions of bounded mean oscillation. We discuss a new proof of this result in a discre…
We generalize the universal power series of Seleznev to several variables and we allow the coefficients to depend on parameters. Then, the approximable functions may depend on the same parameters. The universal approximation holds on products $K = \displaystyle\prod_{i = 1}^d K_i$, where $K_i \subseteq \mathbb{C}$ are compact sets and $\mathbb{C} \setminus K_i$ are connected, $i = 1, \dots, d$ and $0 \notin K$. On such $K$ the partial sums approximate uniformly any polynomial. Finally, the partial sums may be replaced by more general expressions. The phenomenon is topologically and algebraical…
We establish generic existence of Universal Taylor Series on products $Ω= \prod Ω_i$ of planar simply connected domains $Ω_i$ where the universal approximation holds on products $K$ of planar compact sets with connected complements provided $K \cap Ω= \emptyset$. These classes are with respect to one or several centers of expansion and the universal approximation is at the level of functions or at the level of all derivatives. Also, the universal functions can be smooth up to the boundary, provided that $K \cap \overlineΩ = \emptyset$ and $\{\infty\} \cup [\mathbb{C} \setminus \overlineΩ_i]$ i…
In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities and give new proofs for known results with elementary arguments.
