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  • Author or Editor: V. Kolyada x
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Abstract  

It is known that theL p-norms of the sums of power series can be estimated from below and above by means of their coefficients, provided these coefficients are nonnegative. In the paper we prove analogous estimates for theL p-norms of the sums of Dirichlet series. Our main result gives exact lower and upper estimates for the BMO-norm of the sums of power series and Dirichlet series, respectively, by means of their coefficients.

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Abstract  

Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. LetfL(I n) (I≡[0,1]. Set

\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ \end{document}
, where the supremum is taken over all cubes containing the pointx. Forη=t α (0<α≤1) this definition was given by A.Calderón. In the paper we prove estimates of the maximal functions
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\mathcal{N}}_\eta f$$ \end{document}
, along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ \end{document}
, then
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\parallel {\mathcal{N}}_\sigma {f} {\parallel_{q,p}} \leqslant c \parallel {\mathcal{N}}_\eta {f} {\parallel_p} .$$ \end{document}
. Furthermore, we obtain estimates of
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\mathcal{N}}_\eta f$$ \end{document}
in terms of theL p-modulus of continuity off. We find sharp conditions for
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\mathcal{N}}_\eta f$$ \end{document}
to belong toL p(I n) and the Orlicz classϕ(L), too.

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Abstract

Moduli of p-continuity provide a measure of fractional smoothness of functions via p-variation. We prove a sharp estimate of the modulus of p-continuity in terms of the modulus of q-continuity (1<p<q<∞).

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Abstract  

We investigate the extension to Banach-space-valued functions of the classical inequalities due to Paley for the Fourier coefficients with respect to a general orthonormal system Φ. This leads us to introduce the notions of Paley Φ-type and Φ-cotype for a Banach space and some related concepts. We study the relations between these notions of type and cotype and those previously defined. We also analyze how the interpolation spaces inherit these characteristics from the original spaces, and use them to obtain sharp coefficient estimates for functions taking values in Lorentz spaces.

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