Dyadic Cubes Research Articles

ON $L^p$–$L^q$ TRACE INEQUALITIES

We give necessary and sufficient conditions in order that inequalities of the type \[ \| T_K f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \quad f \in L^p(d\sigma), \] hold for a class of integral operators $T_K f(x) = \int_{R^n} K(x,y) f(y)\,d \sigma(y)$ with nonnegative kernels, and measures $d \mu$ and $d\sigma$ on $\mathbb{R}^n$ , in the case where $p>q>0$ and $p>1$ . An important model is provided by the dyadic integral operator with kernel $K_{\mathcal D}(x, y)= \sum_{Q\in{\mathcal D}} K(Q)\chi_Q(x) \chi_Q(y)$ , where $\mathcal D=\{Q\}$ is the family of all dyadic cubes in $\mathbb{R}^n$ , and $K(Q)$ are arbitrary nonnegative constants associated with $Q \in{\mathcal D}$ . The corresponding continuous versions are deduced from their dyadic counterparts. In particular, we show that, for the convolution operator $T_k f = k \star f$ with positive radially decreasing kernel $k(|x-y|)$ , the trace inequality \[\| T_k f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d x)}, \quad f \in L^p(dx),\] holds if and only if ${\mathcal W}_{k}[\mu] \in L^s (d\mu)$ , where $s = q(p-1)/(p-q)$ . Here ${\mathcal W}_{k}[\mu]$ is a nonlinear Wolff potential defined by ${\mathcal W}_{k}[\mu](x)=\int_0^{+\infty} k(r) \bar{k}(r)^{1/(p-1)} \mu(B(x,r))^{1/(p-1)} r^{n-1} \, dr$ , and $\bar{k}(r)=(1/r^n)\int_0^r k(t) t^{n-1} \, dt$ . Analogous inequalities for $1\le q were characterized earlier by the authors using a different method which is not applicable when $q .

Classification with minimax fast rates for classes of Bayes rules with sparse representation

We consider the classification problem on the cube [0,1]d when the Bayes rule is known to belong to some new functions classes. These classes are made of prediction rules satisfying some conditions regarding their coefficients when developed over the (overcomplete) basis of indicator functions of dyadic cubes of [0,1]d. The main concern of the paper is on the thorough analysis of the approximation term, which is in general bypassed in the classification literature. An adaptive classifier is designed to achieve the minimax rate of convergence (up to a logarithmic factor) over these functions classes. Lower bounds on the convergence rate over these classes are established when the underlying marginal of the design is comparable to the Lebesgue measure. Connections with some existing models for classification (RKHS and “boundary fragements”) are established.

Weak weighted inequalities for a dyadic one-sided maximal function in $ \mathbb {R}^{n}$

In this note we introduce a dyadic one-sided maximal function defined as formula math. where Q + is a certain cube associated with the dyadic cube Q and f ∈ L 1 loc (R n ). We characterize the pair of weights (w,v) for which the maximal operator M +,d applies L p (v) into weak-L p (w) for 1 ≤ p < oo.

Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels

We study trace inequalities of the type ∥T k f∥ L q (dμ) ≤ C∥f∥ L p (dσ) , f ∈ L p (dσ), in the upper triangle case 1 ≤ q < p for integral operators T k with positive kernels, where dσ and dp are positive Borel measures on R n . Our main tool is a generalization of Th. Wolff's inequality which gives two-sided estimates of the energy E k,σ [μ] = R n(T k [μ]) p' dσ through the L 1 (dμ)-norm of an appropriate nonlinear potential W k,σ [μ] associated with the kernel k and measures dp, dσ. We initially work with a dyadic integral operator with kernel K D (x,y) = Σ Q ∈ D K(Q)X Q (x)X Q (y), where D = {Q} is the family of all dyadic cubes in R n , and K: D → R + . The corresponding continuous versions of Wolff's inequality and trace inequalities are derived from their dyadic counterparts.

COMBINATORIAL APPROXIMATION BY DEVANEY-CHAOTIC OR PERIODIC VOLUME PRESERVING HOMEOMORPHISMS

We modify a combinatorial idea of Peter Lax, to uniformly approximate any volume preserving homeomorphism of the closed unit cube In, n≥2, by another one which either: (i) rigidly permutes an infinite family of dyadic cubes of total volume one, or (ii) is chaotic in the sense of Devaney. Our methods apply as well to arbitrary compact manifolds endowed with a nonatomic, locally positive Borel measure.

Imbedding and multiplier theorems for discrete Littlewood-Paley spaces

We prove imbedding and multiplier theorems for discrete Littlewood—Paley spaces introduced by M. Prazier and B. Jawerth in their theory of wavelet-type decompositions of Ίriebel—Lizorkin spaces. The corresponding inequalities for discrete spaces defined in terms of characteristi c functions of dyadic cubes, with respect to an arbitrary positive locally finite measure on the Euclidean space, are useful in the theory of tent spaces, weighted inequalities, duality theorems, interpolation by analytic and harmonic functions, etc. Our main tools are vector-valued maximal inequalities, a dyadic version of the Carleson measure theorem, and Pisier's factorization lemma. We also consider more general inequalities, with an arbitrary family of measurable functions in place of characteristic functions of dyadic cubes, which occur in the factorization theory of operators.

A discrete transform and decompositions of distribution spaces

We study a representation formula of the form ƒ = ∑ Q 〈ƒ, ϑ Q 〉ψ Q for a distribution ƒ on R n . This formula is obtained by discretizing and localizing a standard Littlewood-Paley decomposition. The map taking ƒ to the sequence {〈ƒ, ϑ Q 〉} Q , with Q running over the dyadic cubes in R n , is called the ϑ-transform. The functions ϑ Q and ψ Q have a particularly simple form. Moreover, most of the familiar distribution spaces ( L p -spaces, 1 < p < +∞, H p spaces, 0 < p ⩽ 1, Sobolev and potential spaces, BMO, Besov and Triebel-Lizorkin spaces) are characterized by the magnitude of the ϑ-transform. This enables us to carry out a discrete Littlewood-Paley theory on the sequence spaces corresponding to these distribution spaces. The sequence space norms depend only on magnitudes; cancellation is accounted for in the ϑ Q 's and ψ Q 's. Consequently, analysis on the sequence space level is often easy. With this we can simplify, extend, and unify a variety of results in harmonic analysis. We obtain conditions for the boundedness of linear operators on these distribution spaces by considering corresponding conditions for matrices on the associated sequence spaces. Applications include a general version of the Hörmander (Fourier) multiplier theorem and results for kernel operators of Calderón-Zygmund type. We discuss certain other, more general, decomposition methods, including the “smooth atomic decomposition,” and the “generalized ϑ-transform.” The smooth atomic decomposition yields a simple method for dealing with restriction and extension phenomena for hyperplanes in R n . We also consider pointwise multipliers. For the characteristic function of a domain, we obtain boundedness results for a general class of domains which properly includes Lipschitz domains. Several interpolation methods are easily analyzed via the sequence spaces. For real interpolation, we obtain, among other things, an extension to the case p = 0. This in turn gives a new approach to the traditional atomic decomposition of Hardy spaces.

BMO from dyadic BMO

Then clearly BMO c BMO, with \\φ\\d £\\φ\\, but BMO and BMO, are not the same space; the function log\x9 \X{ttlj>0] is in BMO, but not in BMO. In analysis BMO is more important than BMO, because BMO is translation invariant, but BMO, is not. On the other hand, BMO, is very much the easier space to work with because dyadic cubes are nested (if two open daydic cubes intersect then one of them is contained in the other). For example, for BMO the original proofs [1], [6], [8], [11] of the four theorems stated below were rather technical, while for BMO, the analogous results are comparatively trivial. In this paper we derive the four theorems from their dyadic counterparts. Here is the idea. Let Taφ(x) = φ(x — a). Then