Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63)

Stein, Elias M.

doi:10.1515/9781400881871

Cited by 614 publications

(737 citation statements)

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“…Note that the optimal order of the constant C p above is different from that in the commutative case, which is (p − 1) −1 as p → 1. Part ii) is the noncommutative analogue of Stein's maximal ergodic inequality (see [St2]). Note that in the case where τ is normalized (i.e.…”

Section: Noncommutative Maximal Ergodic Theorems 387mentioning

confidence: 99%

“…For the proof of part ii) of Theorem 0.1, we adapt Stein's arguments in [St2] to the noncommutative setting.…”

Section: Noncommutative Maximal Ergodic Theorems 387mentioning

confidence: 99%

“…It is Stein's original approach that suits well to the noncommutative setting. Thus we will follow the same pattern as in [St2,Chapter III]. Throughout the remainder of this section T will be fixed as in Theorem 5.1.…”

Section: Maximal Inequalities For Symmetric Contractionsmentioning

confidence: 99%

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Noncommutative maximal ergodic theorems

Junge¹,

Xu²

2006

J. Amer. Math. Soc.

165

245

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This paper is devoted to the study of various maximal ergodic theorems in noncommutative L p L_p -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on L p L_p and the analogue of Stein’s maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

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Section: Noncommutative Maximal Ergodic Theorems 387mentioning

confidence: 99%

“…For the proof of part ii) of Theorem 0.1, we adapt Stein's arguments in [St2] to the noncommutative setting.…”

Section: Noncommutative Maximal Ergodic Theorems 387mentioning

confidence: 99%

See 1 more Smart Citation

Noncommutative maximal ergodic theorems

Junge¹,

Xu²

2006

J. Amer. Math. Soc.

165

245

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“…Let {T t } be a symmetric diffusion semigroup of operators acting on measurable functions on ‫ޒ(‬ n , dµ), with a second order differential operator −L, (symmetric in L 2 (dµ)) as its infinitesimal generator. In this context, the following operators can be considered; see [13], Riesz potentials:…”

Section: Theorem 21 Let V Be a Positive Measurable Function The Fomentioning

confidence: 99%

Hermite Function Expansions Versus Hermite Polynomial Expansions

Abu-Falahah¹,

Torrea²

2006

Glasgow Math. J.

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Abstract. We consider expansions with respect to the multi-dimensional Hermite functions and to the multi-dimensional Hermite polynomials. They are respectively eigenfunctions of the Harmonic oscillator L = − + |x| 2 and of the OrnsteinUhlenbeck operator L = − + 2x · ∇. The corresponding heat semigroups and Riesz transforms are considered and results on both aspects (polynomials and functions) are obtained.2000 Mathematics Subject Classification. 42C10, 42B20, 42B25.

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“…A class of operators £ for which the above program is particularly natural are the generators of symmetric diffusion semigroups [7]: £ > 0, and the operators e~l t for / > 0 preserve pointwise positivity and are contractions on L p 9 1 < p < oo. The classic example of such an operator £ is the Laplace-Beltrami operator on a compact Riemannian manifold.…”

mentioning

confidence: 99%

Book Review: Hardy spaces on homogeneous groups

Goodman¹

1983

Bull. Amer. Math. Soc.

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Integral formulas are difficult to discuss in just one chapter. For one thing, there are many different formulas; for another, their construction and use has remained a rather technical enterprise. Krantz limits himself to constructing and estimating the "Leray-Stokes" formula of Henkin in the case n -2, a choice which strikes a good compromise between generality and technicality.As a text, this book should be excellent for a second course on complex analysis. It covers many of the basic results and connects them up with harmonic analysis and P.D.E.; and the final three chapters provide an introduction to more specialized material.

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Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63)

Cited by 614 publications

References 0 publications

Noncommutative maximal ergodic theorems

Noncommutative maximal ergodic theorems

Hermite Function Expansions Versus Hermite Polynomial Expansions

Book Review: Hardy spaces on homogeneous groups

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