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Search: MSC category 42B25 ( Maximal functions, Littlewood-Paley theory )

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1. 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

2. CJM Online first

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

3. 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

4. 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

5. 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

6. CJM 2009 (vol 61 pp. 807)

Hong, Sunggeum; Kim, Joonil; Yang, Chan Woo
 Maximal Operators Associated with Vector Polynomials of Lacunary Coefficients We prove the $L^p(\mathbb{R}^d)$ ($1 Categories:42B20, 42B25 7. CJM 2008 (vol 60 pp. 1283) Ho, Kwok-Pun  Remarks on Littlewood--Paley Analysis Littlewood--Paley analysis is generalized in this article. We show that the compactness of the Fourier support imposed on the analyzing function can be removed. We also prove that the Littlewood--Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood--Paley analysis and obtain the corresponding renormalization'' for the convergence of this multiparameter Littlewood--Paley analysis. Keywords:Littlewood--Paley analysis, distributionsCategory:42B25 8. CJM 2007 (vol 59 pp. 276) Bernardis, A. L.; Martín-Reyes, F. J.; Salvador, P. Ortega  Weighted Inequalities for Hardy--Steklov Operators We characterize the pairs of weights$(v,w)$for which the operator$Tf(x)=g(x)\int_{s(x)}^{h(x)}f$with$s$and$h$increasing and continuous functions is of strong type$(p,q)$or weak type$(p,q)$with respect to the pair$(v,w)$in the case$0 Keywords:Hardy--Steklov operator, weights, inequalitiesCategories:26D15, 46E30, 42B25

9. CJM 2006 (vol 58 pp. 1121)

Bownik, Marcin; Speegle, Darrin
 The Feichtinger Conjecture for Wavelet Frames, Gabor Frames and Frames of Translates The Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a sup-adjoint Gabor frame can be written as the finite union of Riesz basic sequences. Finally, we show how existing techniques can be applied to determine whether frames of translates can be written as the finite union of Riesz basic sequences. We end by giving an example of a frame of translates such that any Riesz basic subsequence must consist of highly irregular translates. Keywords:frame, Riesz basic sequence, wavelet, Gabor system, frame of translates, paving conjectureCategories:42B25, 42B35, 42C40

10. CJM 2003 (vol 55 pp. 504)

Chen, Jiecheng; Fan, Dashan; Ying, Yiming
 Certain Operators with Rough Singular Kernels We study the singular integral operator $$T_{\Omega,\alpha}f(x) = \pv \int_{R^n} b(|y|) \Omega(y') |y|^{-n-\alpha} f(x-y)\,dy,$$ defined on all test functions $f$,where $b$ is a bounded function, $\alpha\geq 0$, $\Omega(y')$ is an integrable function on the unit sphere $S^{n-1}$ satisfying certain cancellation conditions. We prove that, for $1 Categories:42B20, 42B25, 42B15 11. CJM 1998 (vol 50 pp. 29) Ding, Yong; Lu, Shanzhen  Weighted norm inequalities for fractional integral operators with rough kernel Given function$\Omega$on${\Bbb R^n}$, we define the fractional maximal operator and the fractional integral operator by$$M_{\Omega,\alpha}\,f(x)=\sup_{r>0}\frac 1{r^{n-\alpha}} \int_{|\,y|1)$, homogeneous of degree zero. Categories:42B20, 42B25

12. CJM 1997 (vol 49 pp. 1010)

Lorente, Maria
 A characterization of two weight norm inequalities for one-sided operators of fractional type In this paper we give a characterization of the pairs of weights $(\w,v)$ such that $T$ maps $L^p(v)$ into $L^q(\w)$, where $T$ is a general one-sided operator that includes as a particular case the Weyl fractional integral. As an application we solve the following problem: given a weight $v$, when is there a nontrivial weight $\w$ such that $T$ maps $L^p(v)$ into $L^q(\w )$? Keywords:Weyl fractional integral, weightsCategories:26A33, 42B25
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