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1. CJM Online first

Chen, Xianghong; Seeger, Andreas
 Convolution powers of Salem measures with applications We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha$ of the form ${d}/{n}$, $n=2,3,\dots$ there exist $\alpha$-Salem measures for which the $L^2$ Fourier restriction theorem holds in the range $p\le \frac{2d}{2d-\alpha}$. The results rely on ideas of KÃ¶rner. We extend some of his constructions to obtain upper regular $\alpha$-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\in \mathbb{N}$. Keywords:convolution powers, Fourier restriction, Salem sets, Salem measures, random sparse sets, Fourier multipliers of Bochner-Riesz typeCategories:42A85, 42B99, 42B15, 42A61

2. CJM Online first

Nikolidakis, Eleftherios Nikolaos
 Extremal sequences for the Bellman function of the dyadic maximal operator and applications to the Hardy operator We prove that the extremal sequences for the Bellman function of the dyadic maximal operator behave approximately as eigenfunctions of this operator for a specific eigenvalue. We use this result to prove the analogous one with respect to the Hardy operator. Keywords:Bellman function, dyadic, Hardy operator, maximalCategory:42B25

3. CJM Online first

Günther, Christian; Schmidt, Kai-Uwe
 $L^q$ norms of Fekete and related polynomials A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and ErdÅs, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials $f_p(z)=\sum_{j=1}^{p-1}(j\,|\,p)\,z^j,$ where $p$ is an odd prime and $(\,\cdot\,|\,p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials. Keywords:character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomialCategories:11B83, 42A05, 30C10

4. CJM 2016 (vol 68 pp. 1257)

Cascante, Carme; Fàbrega, Joan; Ortega, Joaquín M.
 Sharp Norm Estimates for the Bergman Operator from Weighted Mixed-norm Spaces to Weighted Hardy Spaces In this paper we give sharp norm estimates for the Bergman operator acting from weighted mixed-norm spaces to weighted Hardy spaces in the ball, endowed with natural norms. Keywords:weighted Hardy space, Bergman operator, sharp norm estimateCategories:47B38, 32A35, 42B25, 32A37

5. CJM 2016 (vol 68 pp. 816)

Guo, Xiaoli; Hu, Guoen
 On the Commutators of Singular Integral Operators with Rough Convolution Kernels Let $T_{\Omega}$ be the singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$, where $\Omega$ is homogeneous of degree zero, has mean value zero and belongs to $L^q(S^{n-1})$ for some $q\in (1,\,\infty]$. In this paper, the authors establish the compactness on weighted $L^p$ spaces, and the Morrey spaces, for the commutator generated by $\operatorname{CMO}(\mathbb{R}^n)$ function and $T_{\Omega}$. The associated maximal operator and the discrete maximal operator are also considered. Keywords:commutator, singular integral operator, compact operator, completely continuous operator, maximal operator, Morrey spaceCategories:42B20, 47B07

6. CJM 2016 (vol 68 pp. 1159)

Yattselev, Maxim L.
 Strong Asymptotics of Hermite-PadÃ© Approximants for Angelesco Systems In this work type II Hermite-PadÃ© approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices. Keywords:Hermite-PadÃ© approximation, multiple orthogonal polynomials, non-Hermitian orthogonality, strong asymptotics, matrix Riemann-Hilbert approachCategories:42C05, 41A20, 41A21

7. CJM 2015 (vol 67 pp. 1161)

Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun
 Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights or in the class of $QC(\mathbb{R}^n)$ weights, and $L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\ge 2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for $p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and $w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$, the authors prove that the associated Riesz transform $\nabla L_w^{-1/2}$ is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical Hardy space $H_w^p(\mathbb{R}^n)$. Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transformCategories:42B30, 42B35, 35J70

8. CJM 2014 (vol 66 pp. 1358)

 Sharp Localized Inequalities for Fourier Multipliers In the paper we study sharp localized $L^q\colon L^p$ estimates for Fourier multipliers resulting from modulation of the jumps of LÃ©vy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on $2\times 2$ matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator . Keywords:Fourier multiplier, martingale, laminateCategories:42B15, 60G44, 42B20

9. CJM 2013 (vol 66 pp. 284)

Eikrem, Kjersti Solberg
 Random Harmonic Functions in Growth Spaces and Bloch-type Spaces Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v$ that fulfill a doubling condition. In the two-dimensional case let $u (re^{i\theta},\xi) = \sum_{j=0}^\infty (a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta)$ where $\xi =\{\xi_{ji}\}$ is a sequence of random subnormal variables and $a_{ji}$ are real; in higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients $a_{ji}$ which imply that $u$ is in $h^\infty_v(\mathbf B)$ almost surely. Our estimate improves previous results by Bennett, Stegenga and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces, which generalizes results by Anderson, Clunie and Pommerenke and by Guo and Liu. Keywords:harmonic functions, random series, growth space, Bloch-type spaceCategories:30B20, 31B05, 30H30, 42B05

10. CJM 2013 (vol 66 pp. 1382)

Wu, Xinfeng
 Weighted Carleson Measure Spaces Associated with Different Homogeneities In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two CalderÃ³n-Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong-Stein. Keywords:composition of operators, weighted Carleson measure spaces, dualityCategories:42B20, 42B35

11. CJM 2013 (vol 65 pp. 1217)

Cruz, Victor; Mateu, Joan; Orobitg, Joan
 Beltrami Equation with Coefficient in Sobolev and Besov Spaces Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$. Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, CalderÃ³n-Zygmund operatorsCategories:30C62, 35J99, 42B20

12. CJM 2012 (vol 65 pp. 510)

Blasco de la Cruz, Oscar; Villarroya Alvarez, Paco
 Transference of vector-valued multipliers on weighted $L^p$-spaces We prove restriction and extension of multipliers between weighted Lebesgue spaces with two different weights, which belong to a class more general than periodic weights, and two different exponents of integrability which can be below one. We also develop some ad-hoc methods which apply to weights defined by the product of periodic weights with functions of power type. Our vector-valued approach allow us to extend results to transference of maximal multipliers and provide transference of Littlewood-Paley inequalities. Keywords:Fourier multipliers, restriction theorems, weighted spacesCategories:42B15, 42B35

13. CJM 2012 (vol 66 pp. 102)

Birth, Lidia; Glöckner, Helge
 Continuity of convolution of test functions on Lie groups For a Lie group $G$, we show that the map $C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$, $(\gamma,\eta)\mapsto \gamma*\eta$ taking a pair of test functions to their convolution is continuous if and only if $G$ is $\sigma$-compact. More generally, consider $r,s,t \in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$ and a continuous bilinear map $b\colon E_1\times E_2\to F$ to a complete locally convex space $F$. Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$, $(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map. The main result is a characterization of those $(G,r,s,t,b)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed, as well as convolution of compactly supported $L^1$-functions and convolution of compactly supported Radon measures. Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigma-compactness, convolution, continuity, seminorm, product estimatesCategories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25

14. CJM 2012 (vol 65 pp. 299)

Grafakos, Loukas; Miyachi, Akihiko; Tomita, Naohito
 On Multilinear Fourier Multipliers of Limited Smoothness In this paper, we prove certain $L^2$-estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we extend the result of CalderÃ³n and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed. Keywords:multilinear Fourier multipliers, HÃ¶rmander multiplier theorem, Hardy spacesCategories:42B15, 42B20

15. CJM 2012 (vol 65 pp. 600)

Kroó, A.; Lubinsky, D. S.
 Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain - in particular this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex, under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings. Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functionsCategories:42C05, 42C99, 42B05, 60B20

16. CJM 2011 (vol 64 pp. 1036)

Koh, Doowon; Shen, Chun-Yen
 Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields In this paper we study the extension problem, the averaging problem, and the generalized ErdÅs-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial. Keywords:extension problems, averaging operator, finite fields, ErdÅs-Falconer distance problems, homogeneous polynomialCategories:42B05, 11T24, 52C17

17. CJM 2011 (vol 64 pp. 1201)

Aistleitner, Christoph; Elsholtz, Christian
 The Central Limit Theorem for Subsequences in Probabilistic Number Theory Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying $$\tag{1} f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad \operatorname{Var_{[0,1]}} f \lt \infty.$$ If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$ the distribution of $$\tag{2} \frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}}$$ converges to a Gaussian distribution. In the case $$1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty$$ there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$, and the limit distribution of (2). In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying (1) the distribution of $$\frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}}$$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property. Keywords:central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theoremCategories:60F05, 42A55, 11D04, 05C55, 11K06

18. CJM 2011 (vol 64 pp. 892)

Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongyong
 Boundedness of CalderÃ³n-Zygmund Operators on Non-homogeneous Metric Measure Spaces Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a CalderÃ³n-Zygmund operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a CalderÃ³n-Zygmund operator bounded on $L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition. Keywords:upper doubling, geometrical doubling, dominating function, weak type $(1,1)$ estimate, CalderÃ³n-Zygmund operator, maximal operatorCategories:42B20, 42B25, 30L99

19. CJM 2011 (vol 64 pp. 257)

Chen, Yanping; Ding, Yong; Wang, Xinxia
 Compactness of Commutators for Singular Integrals on Morrey Spaces In this paper we characterize the compactness of the commutator $[b,T]$ for the singular integral operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More precisely, we prove that if $b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO} (\mathbb R^n)$-closure of $C_c^\infty(\mathbb R^n)$, then $[b,T]$ is a compact operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and $0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and $[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$ for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$. Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right. Keywords:singular integral, commutators, compactness, VMO, Morrey spaceCategories:42B20, 42B99

20. CJM 2011 (vol 63 pp. 798)

Daws, Matthew
 Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra built out of these non-commutative $L^p$ spaces, say $A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to $L^1(G)$, generalising the abelian situation. Keywords:multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolationCategories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52

21. CJM 2011 (vol 63 pp. 689)

Olphert, Sean; Power, Stephen C.
 Higher Rank Wavelets A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in $L^2(\mathbb R^d)$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct \emph{Latin square wavelets} as rank~$2$ variants of Haar wavelets. Also we construct nonseparable scaling functions for rank $2$ variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable. Keywords: wavelet, multi-scaling, higher rank, multiresolution, Latin squaresCategories:42C40, 42A65, 42A16, 43A65

22. CJM 2010 (vol 63 pp. 181)

Ismail, Mourad E. H.; Obermaier, Josef
 Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials We characterize the continuous $q$-ultraspherical polynomials in terms of the special form of the coefficients in the expansion $\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$ being the Askey--Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the $L^2$ norm of the polynomials. A similar characterization is given for the discrete $q$-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big $q$-Jacobi polynomials is given. Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficientsCategories:33D45, 42C05

23. CJM 2010 (vol 62 pp. 1419)

Yang, Dachun; Yang, Dongyong
 BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures Let $\mu$ be a nonnegative Radon measure on $\mathbb{R}^d$ that satisfies the growth condition that there exist constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and $r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, the authors prove that if $f$ belongs to the $\textrm {BMO}$-type space $\textrm{RBMO}(\mu)$ of Tolsa, then the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an initial cube) and the inhomogeneous maximal function $\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube) associated with a given approximation of the identity $S$ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from $\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$-type space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous maximal operator $\mathcal{M}_S$ is bounded from the local $\textrm {BMO}$-type space $\textrm{rbmo}(\mu)$ to the local $\textrm {BLO}$-type space $\textrm{rblo}(\mu)$. Keywords:Non-doubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu)Categories:42B25, 42B30, 47A30, 43A99

24. CJM 2010 (vol 62 pp. 1182)

Yue, Hong
 A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$ A borderline case function $f$ for $Q_{\alpha}({\mathbb R^n})$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta>0$. On its support $I_0=[0, 1]^n$, $f(x)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for $\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a John--Nirenberg inequality for $Q_{\alpha}({\mathbb R^n})$. When $\beta\neq 1$, $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside $I_0$. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from $I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the graph of $f$ has a non-integer fractal dimension $n+1-\beta$. Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimensionCategories:42B35, 42C10, 30D50, 28A80

25. CJM 2010 (vol 62 pp. 827)

Ouyang, Caiheng; Xu, Quanhua
 BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm. Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spacesCategories:46E40, 42B25, 46B20
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