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A stability result on Muckenhoupt's weights.

Juha Kinnunen (1998)

Publicacions Matemàtiques

We prove that Muckenhoupt's A1-weights satisfy a reverse Hölder inequality with an explicit and asymptotically sharp estimate for the exponent. As a by-product we get a new characterization of A1-weights.

A stable method for the inversion of the Fourier transform in R N

Leonede De Michele, Delfina Roux (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A general method is given for recovering a function f : R N C , N 1 , knowing only an approximation of its Fourier transform.

A strengthening of a theorem of Marcinkiewicz

Konstantin E. Tikhomirov (2011)

Banach Center Publications

We consider a problem of intervals raised by I. Ya. Novikov in [Israel Math. Conf. Proc. 5 (1992), 290], which refines the well-known theorem of J. Marcinkiewicz concerning structure of closed sets [A. Zygmund, Trigonometric Series, Vol. I, Ch. IV, Theorem 2.1]. A positive solution to the problem for some specific cases is obtained. As a result, we strengthen the theorem of Marcinkiewicz for generalized Cantor sets.

A strong convergence theorem for H¹(𝕋ⁿ)

Feng Dai (2006)

Studia Mathematica

Let ⁿ denote the usual n-torus and let S ̃ u δ ( f ) , u > 0, denote the Bochner-Riesz means of order δ > 0 of the Fourier expansion of f ∈ L¹(ⁿ). The main result of this paper states that for f ∈ H¹(ⁿ) and the critical index α: = (n-1)/2, l i m R 1 / l o g R 0 R ( | | S ̃ u α ( f ) - f | | H ¹ ( ) ) / ( u + 1 ) d u = 0 .

A subelliptic Bourgain–Brezis inequality

Yi Wang, Po-Lam Yung (2014)

Journal of the European Mathematical Society

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space N L ˙ 1 , Q by L functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for on the Heisenberg group n .

A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform

Sandra Pot (2007)

Studia Mathematica

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space . We show that if W and its inverse W - 1 both satisfy a matrix reverse Hölder property introduced by Christ and Goldberg, then the weighted Hilbert transform H : L ² W ( , ) L ² W ( , ) and also all weighted dyadic martingale transforms T σ : L ² W ( , ) L ² W ( , ) are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.

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