Weighted estimates of singular integrals and their applications

Dyn’kin, E. M.; Осиленкер, Б. П.

doi:10.1007/bf02105397

Cited by 51 publications

(26 citation statements)

References 184 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…IV]) that ρ ∈ A ∞ (R) if and only if ρ belongs to the Muckenhoupt A p (R) class for some p ≥ 1. It should be mentioned that some essential properties of the Muckenhoupt A p classes defined on rectangles has been studied in [11,23] (see also [8], [16,Ch. IV]).…”

Section: And Only Ifmentioning

confidence: 99%

Two-Weight Estimates for Strong Fractional Maximal Functions and Potentials With Multiple Kernels

Kokilashvili¹,

Meskhi²

2009

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

Section: And Only Ifmentioning

confidence: 99%

Two-Weight Estimates for Strong Fractional Maximal Functions and Potentials With Multiple Kernels

Kokilashvili¹,

Meskhi²

2009

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

“…It was then observed by several authors (see e.g. [15] or [20], Vol. 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).…”

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

confidence: 69%

“…Shelepov, A.P. Calderón, and others (see, e.g., [15]). For general Carleson curves a proof was first given by E.A.…”

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

confidence: 90%

“…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: 95%

See 1 more Smart Citation

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

Böttcher

Karlovich

1999

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

“…The role of the Muckenhoupt condition and the reverse Hölder inequality in the theory of singular integrals is described in any book concerning this subject, for example, in the monograph "Harmonic Analysis" by Stein [5], or in the survey [6] by Dyn kin and Osilenker.…”

mentioning

confidence: 99%

The sharp constant in the reverse Hölder inequality for Muckenhoupt weights

Vasyunin

2003

St. Petersburg Math. J.

View full text Add to dashboard Cite

Abstract. Coifman and Fefferman proved that the "reverse Hölder inequality" is fulfilled for any weight satisfying the Muckenhoupt condition. In order to illustrate the power of the Bellman function technique, Nazarov, Volberg, and Treil showed (among other things) how this technique leads to the reverse Hölder inequality for the weights satisfying the dyadic Muckenhoupt condition on the real line. In this paper the proof of the reverse Hölder inequality with sharp constants is presented for the weights satisfying the usual (rather than dyadic) Muckenhoupt condition on the line. The results are a consequence of the calculation of the true Bellman function for the corresponding extremal problem.In [1] Coifman and Fefferman proved that for any weight (i.e., for any nonnegative function) w on R n satisfying the Muckenhoupt condition(the supremum is taken over all cubes Q with edges parallel to the coordinate axes) the "reverse Hölder inequality" is fulfilled:for some q > 1, with a constant C independent of the cube Q. Here and later, ϕ Q stands for the average of the function ϕ over the set Q:In [2] Nazarov, Treil, and Volberg illustrated the power of the Bellman function technique. In particular, they deduced the reverse Hölder inequality for dyadic A ∞ -weights on the real line by constructing a Bellman function. In the present paper we refine this technique to prove the reverse Hölder inequality for arbitrary A p -weights (1 ≤ p ≤ ∞) on the real line with a sharp constant C depending on p and q (the exponent in the reverse Hölder inequality), and on the A p -"norm" δ of w. Furthermore, we find the sharp bound for the exponents q for which the reverse Hölder inequality is true. Instead of Muckenhoupt's original definition of A ∞ -weights (used, e.g., in [1]), we employ the equivalent definition introduced by Khrushchëv [3]. Namely, the symbol A δ ∞ (J) will denote the set of all nonnegative functions w ∈ L 1 (J) such that (1) sup I⊂J w I exp(− log w I ) ≤ δ < ∞.2000 Mathematics Subject Classification. Primary 42B20, 42B25.

show abstract

Weighted estimates of singular integrals and their applications

Cited by 51 publications

References 184 publications

Two-Weight Estimates for Strong Fractional Maximal Functions and Potentials With Multiple Kernels

Two-Weight Estimates for Strong Fractional Maximal Functions and Potentials With Multiple Kernels

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

The sharp constant in the reverse Hölder inequality for Muckenhoupt weights

Contact Info

Product

Resources

About