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] Qian , C. , Boundedness and asymptotic behaviour of solutions of a second-order nonlinear system , Bull, London Math, Soc. 24 ( 1992 ), 281 – 288 . MR 93d :34065 [8

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Abstract  

A well-known result of Pták extending the Uniform Boundedness Principle involves a partial substitution of linearity for continuity, and has become a staple argument in automatic continuity. The results of this paper use that argument to extend the Uniform Boundedness Principle by weakening the hypothesis of pointwise-boundedness, and by reducing the continuity requirements on the maps involved. Extensions of the Arzelà-Ascoli theorem are also proved, as well as interpretations of the result which may enable them to be extended both to systems theory and automatic continuity.

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A perturbed Cantor set (without the uniform boundedness condition away from zero of contraction ratios) whose upper Cantor dimension and lower Cantor dimension coincide has its Hausdorff dimension of the same value of Cantor dimensions. We will show this using an energy theory instead of Frostman's density lemma which was used for the case of the perturbed Cantor set with the uniform boundedness condition. At the end, we will give a nontrivial example of such a perturbed Cantor set.

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Summary  

The boundedness of anisotropic singular integral operators with the domains of definition and ranges in various anisotropic spaces of Banach-valued functions is analyzed from a unified point of view. A number of parameterized classes of sufficient conditions are obtained that are expressed in terms of the approximation D-functional. Our sufficient conditions are weaker then their known counterparts in the same settings. The inhomogeneity of the dependence on certain parameters is revealed. The results obtained are also applicable to nonsingular (in the ordinary sense) integral operators, for example, to potential-type operators. The main results are presented in the style of the Caldern-Zygmund theory. The approach is based on the study of decompositions of operators and some properties of the related function spaces.

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Abstract  

Lubinsky and Totik’s decomposition [11] of the Cesàro operators σ n (α,β) of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖w (a,b) σ n (α,β)Cw (a,b) f, whenever α,β ≧ −1/2 and a, b are within the square around (α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations.

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Abstract  

We introduce the generalized fractional integrals (generalized B-fractional integrals) generated by the ΔB Laplace-Bessel differential operator and give some results for them. We obtain O’Neil type inequalities for the B-convolutions and give pointwise rearrangement estimates of the generalized B-fractional integrals. Then we get the L p,γ-boundedness of the generalized B-convolution operator, the generalized B-Riesz potential and the generalized fractional B-maximal function. Finally, we prove a sharp pointwise estimate of the nonincreasing rearrangement of the generalized fractional B-maximal function.

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Abstract  

The paper is concerned with bounds for integrals of the type
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int\limits_{U_k (x)} {\left| {p_n^{(\alpha , \beta )} (t)} \right|^p w^{(a, b)} } (t) dt, p \geqq 0$$ \end{document}
, involving Jacobi polynomials p n (α,β) and Jacobi weights w (a,b) depending on α,β, a, b > −1, where the subsets U k(x) ⊂ [−1, 1] located around x and are given by
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$U_k (x) = \left[ {x - \tfrac{{\phi _k (x)}} {k}, x + \tfrac{{\phi _k (x)}} {k}} \right] \cap [ - 1, 1]$$ \end{document}
with
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\phi _k (x) = \sqrt {1 - x^2 } + \tfrac{1} {k}$$ \end{document}
. The functions to be integrated will also be of the type
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left| {\tfrac{{p_n^{(\alpha , \beta )} (t)}} {{x - t}}} \right|$$ \end{document}
on the domain [−1,1] t/ U k(x). This approach uses estimates of Jacobi polynomials modified Jacobi weights initiated by Totik and Lubinsky in [1]. Various bounds for integrals involving Jacobi weights will be derived. The results of the present paper form the basis of the proof of the uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms in [3].
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