Dyadic Rectangles Research Articles

Optimal orders of the best constants in the Littlewood-Paley inequalities

Let {Pt}t>0 be the classical Poisson semigroup on Rd and GP the associated Littlewood-Paley g-function operator:GP(f)=(∫0∞t|∂∂tPt(f)|2dt)12. The classical Littlewood-Paley g-function inequality asserts that for any 1<p<∞ there exist two positive constants Lt,pP and Lc,pP such that(Lt,pP)−1‖f‖p≤‖GP(f)‖p≤Lc,pP‖f‖p,f∈Lp(Rd). We determine the optimal orders of magnitude on p of these constants as p→1 and p→∞. We also consider similar problems for more general test functions in place of the Poisson kernel.The corresponding problem on the Littlewood-Paley dyadic square function inequality is investigated too. Let Δ be the partition of Rd into dyadic rectangles and SR the partial sum operator associated to R. The dyadic Littlewood-Paley square function of f isSΔ(f)=(∑R∈Δ|SR(f)|2)12. For 1<p<∞ there exist two positive constants Lc,p,dΔ and Lt,p,dΔ such that(Lt,p,dΔ)−1‖f‖p≤‖SΔ(f)‖p≤Lc,p,dΔ‖f‖p,f∈Lp(Rd). We show thatLt,p,dΔ≈d(Lt,p,1Δ)d and Lc,p,dΔ≈d(Lc,p,1Δ)d.All the previous results can be equally formulated for the d-torus Td. We prove a de Leeuw type transference principle in the vector-valued setting.

A Bilinear Rubio de Francia Inequality for Arbitrary Rectangles

AbstractLet $\mathscr{R}$ be a collection of disjoint dyadic rectangles $R$, let $\pi _R$ denote the non-smooth bilinear projection onto $R$and let $r&amp;gt;2$. We show that the bilinear Rubio de Francia operator associated with $\mathscr{R}$ given by $$\begin{equation*} f,g \mapsto \left(\sum_{R\in\mathscr{R}} |\pi_{R} (f,g)|^r \right)^{1/r} \end{equation*}$$ is $L^p \times L^q \rightarrow L^s$ bounded whenever $1/p + 1/q = 1/s$, $r^{\prime}&amp;lt;p,q&amp;lt;r$. This extends from squares to rectangles a previous result by the same authors in [7], and as a corollary extends in the same way a previous result from [2] for smooth projections, albeit in a reduced range.

The n-linear embedding theorem for dyadic rectangles

Let $\sg_i$, $i=1,\ldots,n$, denote reverse doubling weights on $\R^d$, let $\cdr(\R^d)$ denote the set of all dyadic rectangles on $\R^d$ (Cartesian products of usual dyadic intervals) and let $K:\,\cdr(\R^d)\to[0,\8)$ be a~map. In this paper we give the $n$-linear embedding theorem for dyadic rectangles. That is, we prove the $n$-linear embedding inequality for dyadic rectangles \[ \sum_{R\in\cdr(\R^d)} K(R)\prod_{i=1}^n\lt|\int_{R}f_i\,{\rm d}\sg_i\rt| \le C \prod_{i=1}^n \|f_i\|_{L^{p_i}(\sg_i)} \] can be characterized by simple testing condition \[ K(R)\prod_{i=1}^n\sg_i(R) \le C \prod_{i=1}^n\sg_i(R)^{\frac{1}{p_i}} \quad R\in\cdr(\R^d), \] in the range $1 1$. As a~corollary to this theorem, for reverse doubling weights, we verify a~necessary and sufficient condition for which the weighted norm inequality for the multilinear strong positive dyadic operator and for strong fractional integral operator to hold.

Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

Refining the Two-Dimensional Signed Small Ball Inequality

The two-dimensional signed small ball inequality states that for all possible choices of signs, $$\begin{aligned} \left\| \sum _{|R| = 2^{-n}}{ \varepsilon _R h_R} \right\| _{L^{\infty }} \gtrsim n, \end{aligned}$$ where the summation runs over all dyadic rectangles in the unit square and $$h_R$$ denotes the associated Haar function. This inequality first appeared in the work of Talagrand, and alternative proofs are due to Temlyakov and Bilyk & Feldheim (who showed that the supremum equals $$n+1$$ in all cases). We prove a stronger result: for all integers $$0\le k \le n+1$$ , all possible choices of signs, and all dyadic rectangles Q with $$|Q| \ge 2^{-n-1}$$ , $$\begin{aligned} \left| \left\{ x \in Q: \sum _{|R| = 2^{-n}}{ \varepsilon _R h_R} = n + 1 - 2k\right\} \right| = \frac{|Q|}{2^{n+1}}\left( {\begin{array}{c}n+1 k\end{array}}\right) . \end{aligned}$$

Equivalence of sparse and Carleson coefficients for general sets

We remark that sparse and Carleson coefficients are equivalent for every countable collection of Borel sets and hence, in particular, for dyadic rectangles, the case relevant to the theory of bi-parameter singular integrals. The key observation is that a dual refomulation by I. E. Verbitsky for Carleson coefficients over dyadic cubes holds also for Carleson coefficients over general sets. We give a simple proof for this reformulation.

On an equivalence for differentiation bases of dyadic rectangles

The paper considers differentiation properties of rare bases of dyadic rectangles corresponding to increasing sequences $\{\nu_k\}$ of integers. We prove that the condition \begin{equation*} \sup_k(\nu_{k+1}-\nu_k) \lt \infty \end{equation*} is necessary

The Two-Dimensional Small Ball Inequality and Binary Nets

In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this inequality and discrepancy theory, namely the construction of two-dimensional binary nets, i.e. finite sets which are perfectly distributed with respect to dyadic rectangles. This relation allows one to generate all possible point distributions of this type. In addition, we outline a potential approach to the higher-dimensional small ball inequality by a dimension reduction argument. In particular this gives yet another proof of the two-dimensional signed (i.e. coefficients $$\pm 1$$ ) small ball inequality by reducing it to a simple one-dimensional estimate. However, we show that an analogous estimate fails to hold for arbitrary coefficients.

LOCALIZATION AND PROJECTIONS ON BI-PARAMETER BMO

We prove that for any operator $T$ on bi--parameter BMO the identity factors through $T$ or $I - T$. Bourgain's localization method provides the conceptual framework of our proof. It consists in replacing the factorization problem on the non--separable bi--parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by exploiting the geometry and combinatorics of colored dyadic rectangles.

Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection

This article is devoted to nonlinear approximation and estimation via piecewise polynomials built on partitions into dyadic rectangles. The approximation rate is studied over possibly inhomogeneous and anisotropic smoothness classes that contain Besov classes. Highlighting the interest of such a result in statistics, adaptation in the minimax sense to both inhomogeneity and anisotropy of a related multivariate density estimator is proved. Besides, that estimation procedure can be implemented with a computational complexity simply linear in the sample size.

Counting dyadic equipartitions of the unit square

A dyadic interval is an interval of the form [ j/2 k ,( j+1)/2 k ], where j and k are integers, and a dyadic rectangle is a rectangle with sides parallel to the axes whose projections on the axes are dyadic intervals. Let u n count the number of ways of partitioning the unit square into 2 n dyadic rectangles, each of area 2 − n . One has u 0=1, u 1=2 and u n =2 u n−1 2− u n−2 4. This paper determines an asymptotic formula for a solution to this nonlinear recurrence for generic real initial conditions. For almost all real initial conditions there are real constants ω and β (depending on u 0, u 1) with ω>0 such that for all sufficiently large n one has the exact formula u n=ω 2 n g(βλ n), where λ=2 5 −4≈0.472 , and g(z)=∑ j=0 ∞ c jz j , in which c 0=(−1+ 5 )/2 , c 1=(2− 5 )/2 , all coefficients c j lie in the field Q( 5 ) , and the power series converges for | z|<0.16. These results apply to the initial conditions u 0=1, u 1=2 with ω≈1.845 and β≈0.480. The exact formula for u n then holds for all n⩾2. The proofs are based on an analysis of the holomorphic dynamics of iterating the rational function R( z)=2−1/ z 2.

Random dyadic tilings of the unit square

AbstractA “dyadic rectangle” is a set of the form R = [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n‐tilings, which consist of 2n nonoverlapping dyadic rectangles, each of area 2−n, whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n‐tilings, and study some limiting properties of random tilings. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 225–251, 2002

A Fast Algorithm for Adapted Time–Frequency Tilings

We first consider orthonormal bases of open face RNconsisting of discretized rescaled Walsh functions, whereNis a power of two. Given a vector, the best basis with respect to an additive cost function is found with an algorithm of orderO(NlogN). The algorithm operates in the time–frequency plane by constructing a tiling of minimal cost among all possible tilings with dyadic rectangles of area one. Then we discuss generalizations replacing the Walsh group, which controls the structure of the time–frequency plane, by other finite abelian groups. The main example here involves the Fast Fourier Transform.

A discrete transform and Triebel-Lizorkin spaces on the bidisc

We use a discrete transform to study the Triebel-Lizorkin spaces on bidisc F ˙ p α q , f ˙ p α q \dot F_p^{\alpha q},\,\dot f_p^{\alpha q} and establishes the boundedness of transform S ϕ : F ˙ p α q → f ˙ p α q {S_\phi }:\dot F_p^{\alpha q} \to \dot f_p^{\alpha q} and T ψ : f ˙ p α q → F ˙ p α q {T_\psi }:\dot f_p^{\alpha q} \to \dot F_p^{\alpha q} . We also define the almost diagonal operator and prove its boundedness. With the use of discrete transform and Journé lemma, we get the atomic decomposition of f ˙ p α q \dot f_p^{\alpha q} for 0 &gt; p ⩽ 1 , p ⩽ q &gt; ∞ 0 &gt; p \leqslant 1,\,p \leqslant q &gt; \infty . The atom supports on an open set, not a rectangle. Duality ( f ˙ 1 α q ) ∗ = f ˙ ∞ − α q ′ , 1 q + 1 q ′ = 1 , q &gt; 1 , α ∈ R {(\dot f_1^{\alpha q})^{\ast }} = \dot f_\infty ^{ - \alpha q’},\,\tfrac {1} {q} + \tfrac {1} {{q’}} = 1,\,q &gt; 1,\,\alpha \in R , is established, too. The case for F ˙ p α q \dot F_p^{\alpha q} is similar.