Dyadic Square Research Articles

On a Bellman function associated with the Chang–Wilson–Wolff theorem: a case study

The tail of distribution (i.e., the measure of the set { f ≥ x } \{f\ge x\} ) is estimated for those functions f f whose dyadic square function is bounded by a given constant. In particular, an estimate following from the Chang–Wilson–Wolf theorem is slightly improved. The study of the Bellman function corresponding to the problem reveals a curious structure of this function: it has jumps of the first derivative at a dense subset of the interval [ 0 , 1 ] [0,1] (where it is calculated exactly), but it is of C ∞ C^\infty -class for x > 3 x>\sqrt 3 (where it is calculated up to a multiplicative constant). An unusual feature of the paper consists of the usage of computer calculations in the proof. Nevertheless, all the proofs are quite rigorous, since only the integer arithmetic was assigned to a computer.

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.

Sharp Matrix Weighted Strong Type Inequalities for the Dyadic Square Function

In this paper we refine the recent sparse domination of the integrated p = 2 matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic square functions. We then use this to prove sharp two matrix weighted strong type inequalities for matrix weighted dyadic square functions when 1 < p ≤ 2.

The sharp constant in the weak (1,1) inequality for the square function: a new proof

In this note we give a new proof of the sharp constant C = e^{-1/2} + \int_0^1 e^{-x^2/2}\,dx in the weak (1, 1) inequality for the dyadic square function. The proof makes use of two Bellman functions \mathbb{L} and \mathbb{M} related to the problem, and relies on certain relationships between \mathbb{L} and \mathbb{M} , as well as the boundary values of these functions, which we find explicitly. Moreover, these Bellman functions exhibit an interesting behavior: the boundary solution for \mathbb{M} yields the optimal obstacle condition for \mathbb{L} , and vice versa.

A note on a two-weight estimate for the dyadic square function

We show that the two-weight estimate for the dyadic square function proved by Lacey–Li is sharp.

Duality for dyadic triangles

This paper is a direct continuation of the paper “Duality for dyadic intervals” by the same authors, and can be considered as its second part. Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent groupoid under the binary operation of arithmetic mean. The first paper dealt with the structure of finitely generated subgroupoids of the dyadic line, which were shown to be isomorphic to dyadic intervals. Then a duality between the class of dyadic intervals and the class of certain subgroupoids of the dyadic unit square was described. The present paper extends the results of the first paper, provides some characterizations of dyadic triangles, and describes a duality for the class of dyadic triangles. As in the case of intervals, the duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the dyadic unit cube, considered as (commutative, idempotent and entropic) groupoids with additional constant operations.

Characterizations of the exponential square class on spaces ofhomogeneous type and applications

Let $(X,d,\mu)$ be a space of homogeneous type.We establish three equivalentcharacterizations of the exponential square class associated to the classical dyadic square function on $(X,d,\mu)$: exponential square integrability, improved good-$\lambda$ inequalities, and the sharp $L^p$ lower bound for the classical dyadic square function. This result generalizes the famous Chang-Wislon-Wolff theorem on $\mathbb{R}^n$.We then apply this result to obtain sharp $L^p$ estimates for the square function associated to the Laplace-Beltrami operator on Riemannian manifolds with nonnegative Ricci curvature.

Dyadic product and crow lion algorithm based coefficient generation for privacy protection on cloud

Cloud systems are powerful computing resources used inevitably for data subscription and publication. Even though the cloud platform can handle the huge volume of data, privacy becomes a critical issue during data publishing. Hence, an effective technique for the privacy preservation of the data is required in the cloud computing environment. Accordingly, this paper proposes a technique for privacy protection using the dyadic product and an optimization algorithm. The privacy of the original database is protected by the construction of privacy preserved database using a dyadic square matrix obtained taking the dyadic product of two vectors, namely sensitive-utility (SU) coefficient and cumulative data key product. The selection of SU coefficient vector is based on the proposed (Crow search based Lion) C-Lion algorithm, which is designed by combining crow search algorithm with lion algorithm. The fitness of the proposed C-Lion algorithm is designed based on privacy and utility for the feasible selection of SU coefficient vector. The performance of the proposed privacy protection technique based on the C-Lion algorithm is evaluated using two factors, privacy, and utility. The experimental analysis shows that the proposed technique could attain the maximum utility of 0.909 with privacy 0.864 for the breast cancer dataset.

Weighted square function inequalities

For an integrable function $f$ on $[0,1)^d$, let $S(f)$ and $Mf$ denote the corresponding dyadic square function and the dyadic maximal function of $f$, respectively. The paper contains the proofs of the following statements. (i) If $w$ is a dyadic $A_1$ weight on $[0,1)^d$, then $$ ||S(f)||_{L^1(w)}\leq \sqrt{5}[w]_{A_1}^{1/2}||Mf||_{L^1(w)}. $$ The exponent $1/2$ is shown to be the best possible. (ii) For any $p>1$, there are no constants $c_p$, $\alpha_p$ depending only on $p$ such that for all dyadic $A_p$ weights $w$ on $[0,1)^d$, $$ ||S(f)||_{L^1(w)}\leq c_p[w]_{A_p}^{\alpha_p}||Mf||_{L^1(w)}. $$

Two-weight $L^{p}$-inequalities for dyadic shifts and the dyadic square function

We consider two weight $L^{p}\to L^{q}$-inequalities for dyadic shifts and the dyadic square function with general exponents $1<p,q<\infty$. It is shown that if a so-called quadratic $\mathscr{A}_{p,q}$-condition related to the measures holds, then a family of dyadic shifts satisfies the two weight estimate in an $\mathcal{R}$-bounded sense if and only if it satisfies the direct- and the dual quadratic testing condition. In the case $p=q=2$ this reduces to the result by T. Hyt\onen, C. P\'erez, S. Treil and A. Volberg. The dyadic square function satisfies the two weight estimate if and only if it satisfies the quadratic testing condition and the quadratic $\mathscr{A}_{p,q}$-condition holds. Again in the case $p=q=2$ we recover the result by F. Nazarov, S. Treil and A. Volberg. An example shows that in general the quadratic $\mathscr{A}_{p,q}$-condition is stronger than the Muckenhoupt type $A_{p,q}$-condition.

Weighted Inequalities for the Dyadic Square Function

We study Fefferman–Stein inequalities for the dyadic square function associated with an integrable, Hilbert-space-valued function on the interval [0, 1). The proof rests on a Bellman function method: the estimates are deduced from the existence of certain special functions enjoying appropriate majorization and concavity.

Exponential decay estimates for singular integral operators

The following subexponential estimate for commutators is proved $$\begin{aligned} |\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\le c\,e^{-\sqrt{\alpha \, t\Vert b\Vert _{BMO}}}\, |Q|, \qquad t>0. \end{aligned}$$ where $$c$$ and $$\alpha $$ are absolute constants, $$T$$ is a Calderón–Zygmund operator, $$M$$ is the Hardy Littlewood maximal function and $$f$$ is any function supported on the cube $$Q\subset \mathbb{R }^n$$ . We also obtain that $$\begin{aligned} |\{x\in Q: |f(x)-m_f(Q)|>tM_{\lambda _n;Q}^\#(f)(x) \}|\le c\, e^{-\alpha \,t}|Q|,\qquad t>0, \end{aligned}$$ where $$m_f(Q)$$ is the median value of $$f$$ on the cube $$Q$$ and $$M_{\lambda _n;Q}^\#$$ is Strömberg’s local sharp maximal function with $$\lambda _n=2^{-n-2}$$ . As a consequence we derive Karagulyan’s estimate: $$\begin{aligned} |\{x\in Q: |Tf(x)|> tMf(x)\}|\le c\, e^{-c\, t}\,|Q|\qquad t>0, \end{aligned}$$ from [21] improving Buckley’s theorem [3]. A completely different approach is used based on a combination of “Lerner’s formula” with some special weighted estimates of Coifman–Fefferman type obtained via Rubio de Francia’s algorithm. The method is flexible enough to derive similar estimates for other operators such as multilinear Calderón–Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calderón–Zygmund operators. In each case, $$M$$ will be replaced by a suitable maximal operator.

Sharp inequalities for the dyadic square function in the BMO setting

We introduce a method for the simultaneous study of a BMO function φ and its dyadic square function S(φ) that can yield sharp norm inequalities between the two. One of the applications is the sharp bound for the p-th moment of S(φ), 0<p<∞, which in turn implies the square-exponential integrability of the square function. We also present sharp refinements of these inequalities in the more restrictive case when φ is assumed to be bounded.

Sharp weighted estimates for classical operators

We give a general method based on dyadic Calderón–Zygmund theory to prove sharp one- and two-weight norm inequalities for some of the classical operators of harmonic analysis: the Hilbert and Riesz transforms, the Beurling–Ahlfors operator, the maximal singular integrals associated to these operators, the dyadic square function and the vector-valued maximal operator. In the one-weight case we prove the sharp dependence on the A p constant by finding the best value for the exponent α ( p ) such that ‖ T f ‖ L p ( w ) ⩽ C n , T [ w ] A p α ( p ) ‖ f ‖ L p ( w ) . For the Hilbert transform, the Riesz transforms and the Beurling–Ahlfors operator the sharp value of α ( p ) was found by Petermichl and Volberg (2007, 2008, 2002) [47–49]; their proofs used approximations by the dyadic Haar shift operators, Bellman function techniques, and two-weight norm inequalities. Our proofs again depend on dyadic approximation, but avoid Bellman functions and two-weight norm inequalities. We instead use a recent result due to A. Lerner (2010) [34] to estimate the oscillation of dyadic operators. By applying this we get a straightforward proof of the sharp dependence on the A p constant for any operator that can be approximated by Haar shift operators. In particular, we provide a unified approach for the Hilbert and Riesz transforms, the Beurling–Ahlfors operator (and their corresponding maximal singular integrals), dyadic paraproducts and Haar multipliers. Furthermore, we completely solve the open problem of sharp dependence for the dyadic square functions and vector-valued Hardy–Littlewood maximal function. In the two-weight case we use the very same techniques to prove sharp results in the scale of A p bump conditions. For the singular integrals considered above, we show they map L p ( v ) into L p ( u ) , 1 < p < ∞ , if the pair ( u , v ) satisfies sup Q ‖ u 1 / p ‖ A , Q ‖ v − 1 / p ‖ B , Q < ∞ , where A ¯ ∈ B p ′ and B ¯ ∈ B p are Orlicz functions. This condition is sharp. Furthermore, this condition characterizes (in the scale of these A p bump conditions) the corresponding two-weight norm inequality for the Hardy–Littlewood maximal operator M and its dual: i.e., M : L p ( v ) → L p ( u ) and M : L p ′ ( u 1 − p ′ ) → L p ( v 1 − p ′ ) . Muckenhoupt and Wheeden conjectured that these two inequalities for M are sufficient for the Hilbert transform to be bounded from L p ( v ) into L p ( u ) . Thus, in the scale of A p bump conditions, we prove their conjecture. We prove similar, sharp two-weight results for the dyadic square function and the vector-valued maximal operator.

Haar multipliers meet Bellman functions

Using Bellman function techniques, we obtain the optimal dependence of the operator norms in L^2(\mathbb{R}) of the Haar multipliers T_w^t on the corresponding RH^d_2 or A^d_2 characteristic of the weight w , for t=1,\pm 1/2 . These results can be viewed as particular cases of estimates on homogeneous spaces L^2(vd\sigma) , for \sigma a doubling positive measure and v\in A^d_2(d\sigma) , of the weighted dyadic square function S_{\sigma}^d . We show that the operator norms of such square functions in L^2(v d\sigma) are bounded by a linear function of the A^d_2(d\sigma ) characteristic of the weight v , where the constant depends only on the doubling constant of the measure \sigma . We also show an inverse estimate for S_{\sigma}^d . Both results are known when d\sigma=dx . We deduce both estimates from an estimate for the Haar multiplier (T_v^{\sigma})^{1/2} on L^2(d\sigma) when v\in A_2^d(d\sigma) , which mirrors the estimate for T_w^{1/2} in L^2(\mathbb{R}) when w\in A^d_2 . The estimate for the Haar multiplier adapted to the \sigma measure, (T_v^{\sigma})^{1/2} , is proved using Bellman functions. These estimates are sharp in the sense that the rates cannot be improved and be expected to hold for all \sigma , since the particular case d\sigma=dx , v=w , correspond to the estimates for the Haar multipliers T^{1/2}_w proven to be sharp.

Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón–Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.

Generalised finite radon transform for N× N images

This paper extends the domain of the finite radon transform (FRT) to apply to square arrays of arbitrary size. The FRT is a discrete formalism of the Radon transform that assumes the image is periodic over the finite array Z N 2 and requires only arithmetic operations for both forward and exact inverse transformation. The FRT is useful in image processing applications such as tomographic reconstruction [I. Svalbe, D. van der Spek, Reconstruction of tomographic images using analog projections and the digital Radon transform, Linear Algebra and Its Applications 339 (15) (2001) 125–145.], image representation [M. Do, M. Vetterli, Finite ridgelet transform for image representation, IEEE Transactions on Image Processing 12(1).] image convolution [D. Lun, T. Chan, T. Hsung, D. Feng, Y. Chan, Efficient blind image restoration using discrete periodic Radon transform, IEEE Transactions on Image Processing 13(2) (2004) 188–200.], image watermarking and encryption [A. Kingston, I. Svalbe, Projective transforms on periodic discrete image arrays, to appear in P. Hawkes (Ed), Advances in Imaging and Electron Physics (2006).], and robust data transmission [A. Kingston, I. Svalbe, Geometric shape effects in redundant keys used to encrypt data transformed by finite discrete Radon projections, In: Proc. 8th Int. Conf. on Digital Image Computing: Techniques and Applications (2005).]. The original definition by Matúš and Flusser in 1993 [F. Matúš, J. Flusser, Image representation via a finite Radon transform, IEEE Transactions on Pattern Analysis and Machine Intelligence 15(10) (1993) 996–1006.] was restricted to apply only to square arrays of prime size, p× p. Hsung, Lun and Siu developed an FRT that also applied to dyadic square arrays, 2 n ×2 n , called the discrete periodic radon transform (DPRT) [T. Hsung, D. Lun, W. Siu, The discrete periodic Radon transform, IEEE Transactions on Signal Processing 44(10) (1996) 2651–2657.]. Kingston further extended this to define an FRT that applies to prime-adic arrays, p n × p n . This paper defines a generalised FRT that applies to square arrays of arbitrary size, N× N for N ∈ N The Fourier slice theorem and convolution property (two important properties of the classical Radon transform) are established for this FRT. The original image can be reconstructed exactly from the FRT projections using Fourier inversion and back-projection. New methods are identified to correct for the over-representation of pixels due to the compositeness of N. A remarkable result is established that enables 2D sampling patterns to be corrected by an angle invariant 1D filter prior to back-projection.

Paraproducts and the exponential-square class

We prove that if a locally integrable f has a pointwise bounded dyadic square function, where the square function is defined with respect to a so-called “accretive” weight b, then f is locally exponentially square integrable. We generalize the result to d dimensions by means of “canonical” Haar functions for L 2( R d) .

An estimate for weighted Hilbert transform via square functions

We show that the norm of the Hilbert transform as an operator on the weighted space L 2 ( w ) L^2(w) is bounded by a constant multiple of the 3 / 2 3/2 power of the A 2 A_2 constant of w w , in other words by c sup I ( ⟨ ω ⟩ I ⟨ ω − 1 ⟩ I ) 3 / 2 c\, \sup _I (\langle \omega \rangle _I \langle \omega ^{-1} \rangle _I)^{3/2} . We also give a short proof for sharp upper and lower bounds for the dyadic square function.

$L^p$ weighted inequalities for the dyadic square function

$L^p$ weighted inequalities for the dyadic square function