Methods of the Theory of Singular Integrals: Hilbert Transform and Calderón-Zygmund Theory

Dyn’kin, E. M.

doi:10.1007/978-3-662-02732-5_3

Cited by 30 publications

(18 citation statements)

References 16 publications

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“…I, p. 50) that if Γ is a Carleson curve, then S is bounded on L p (Γ, ω) if and only if ω ∈ A p (Γ). In the form presented here, Theorem 1.1 was perhaps first stated in [13], [14], [15] (also see [22]); a proof of the fact that S is bounded on L 2 (Γ) if it is bounded on L p (Γ, ω) is contained in [4]. From now on we always suppose that 1 < p < ∞, that Γ is a Carleson Jordan curve, and that ω ∈ A p (Γ).…”

Section: < P < ∞) If and Only If γ Is A Carleson Curve And ω Is A Mucmentioning

confidence: 96%

Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights

Böttcher

Karlovich

1999

Trans. Amer. Math. Soc.

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Abstract. We describe the spectra and essential spectra of Toeplitz operators with piecewise continuous symbols on the Hardy space H p (Γ, ω) in case 1 < p < ∞, Γ is a Carleson Jordan curve and ω is a Muckenhoupt weight in Ap(Γ). Classical results tell us that the essential spectrum of the operator is obtained from the essential range of the symbol by filling in line segments or circular arcs between the endpoints of the jumps if both the curve Γ and the weight are sufficiently nice. Only recently it was discovered by Spitkovsky that these line segments or circular arcs metamorphose into horns if the curve Γ is nice and ω is an arbitrary Muckenhoupt weight, while the authors observed that certain special so-called logarithmic leaves emerge in the case of arbitrary Carleson curves with nice weights. In this paper we show that for general Carleson curves and general Muckenhoupt weights the sets in question are logarithmic leaves with a halo, and we present final results concerning the shape of the halo.

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Section: < P < ∞) If and Only If γ Is A Carleson Curve And ω Is A Mucmentioning

confidence: 96%

Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights

Böttcher

Karlovich

1999

Trans. Amer. Math. Soc.

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“…In what follows, let |Ω| be the Lebesgue measure of a measurable set Ω ⊂ R. As is well known (see, e.g., [14], [32]), the Calderón-Zygmund operator (2.20) is of weak-type (1, 1), that is, there exists a finite constant C 1,1 > 0 such that…”

Section: Convolution Operators From the Viewpoint Of Pseudodifferentimentioning

confidence: 99%

“…In Section 2 we estimate the norms of convolution operators W 0 (a) on weighted Lebesgue spaces in terms of the quantities D γ a L ∞ (R) (γ = 0, 1, 2, 3) where (Da)(x) := xa (x) for x ∈ R. To this end we apply the pointwise estimates from [1] and the theory of pseudodifferential and Calderón-Zygmund operators on weighted Lebesgue spaces with Muckenhoupt weights (see [11], [14], [19], [32], [5]). …”

Section: Introductionmentioning

confidence: 99%

Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Karlovich

Hernández

2009

Integr. equ. oper. theory

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Fredholm conditions and an index formula are obtained for WienerHopf operators W (a) with slowly oscillating matrix symbols a on weighted Lebesgue spaces L p N (R+, w) where 1 < p < ∞, w is a Muckenhoupt weight on R and N ∈ N. The entries of matrix symbols belong to a Banach subalgebra of Fourier multipliers on L p (R, w) that are continuous on R and have, in general, different slowly oscillating asymptotics at ±∞. To define the Banach algebra SOp,w of corresponding slowly oscillating functions, we apply the theory of pseudodifferential and Calderón-Zygmund operators. Established sufficient conditions become a Fredholm criterion in the case of Muckenhoupt weights with equal indices of powerlikeness, and also for Muckenhoupt weights with different indices of powerlikeness under some additional condition on p, w and a.

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“…Оказывается, что функции µ 1 и µ имеют для почти всех ζ ∈ γ конечные угловые предельные значения µ 1 (ζ) и µ(ζ) со стороны G. Этот факт (его можно найти, например, в [59, теорема 2.22]) является результатом длитель-ного изучения поведения преобразования Коши мер (более подробно см., например, [59] и [89]). Таким образом, для почти всех точек ζ ∈ γ имеет место равенство…”

Section: главаunclassified

Approximation by polyanalytic polynomials

Fedorovskiy¹

2016

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Methods of the Theory of Singular Integrals: Hilbert Transform and Calderón-Zygmund Theory

Cited by 30 publications

References 16 publications

Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights

Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights

Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Approximation by polyanalytic polynomials

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