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( H p , L p ) -type inequalities for the two-dimensional dyadic derivative

Ferenc Weisz (1996)

Studia Mathematica

It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space H p , q to L p , q (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type ( L 1 , L 1 ) . As a consequence we show that the dyadic integral of a ∞ function f L 1 is dyadically differentiable and its derivative is f a.e.

A characterization of Fourier transforms

Philippe Jaming (2010)

Colloquium Mathematicae

The aim of this paper is to show that, in various situations, the only continuous linear (or not) map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups ℤ/nℤ, the integers ℤ, the torus 𝕋 and the real line. We also ask a related question for the twisted convolution.

A class of Fourier multipliers on H¹(ℝ²)

Michał Wojciechowski (2000)

Studia Mathematica

An integral criterion for being an H 1 ( 2 ) Fourier multiplier is proved. It is applied in particular to suitable regular functions which depend on the product of variables.

A class of pairs of weights related to the boundedness of the Fractional Integral Operator between L p and Lipschitz spaces

Gladis Pradolini (2001)

Commentationes Mathematicae Universitatis Carolinae

In [P] we characterize the pairs of weights for which the fractional integral operator I γ of order γ from a weighted Lebesgue space into a suitable weighted B M O and Lipschitz integral space is bounded. In this paper we consider other weighted Lipschitz integral spaces that contain those defined in [P], and we obtain results on pairs of weights related to the boundedness of I γ acting from weighted Lebesgue spaces into these spaces. Also, we study the properties of those classes of weights and compare...

A complete characterization of R-sets in the theory of differentiation of integrals

G. A. Karagulyan (2007)

Studia Mathematica

Let s be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis s differentiates the integral of f if s ∉ S, and D ̅ s f ( x ) = l i m s u p d i a m ( R ) 0 , x R s | R | - 1 R f = almost everywhere if s ∈ S. If the condition D ̅ s f ( x ) = holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a G δ (resp. a G δ σ ).

A counter-example in singular integral theory

Lawrence B. Difiore, Victor L. Shapiro (2012)

Studia Mathematica

An improvement of a lemma of Calderón and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is O(|log r|), let f C ¹ ( N 0 ) and suppose f vanishes outside of a compact subset of N , N ≥ 2. Also, let k(x) be a Calderón-Zygmund kernel of spherical harmonic type. Suppose f(x) = O(|log r|) as r → 0 in the L p -sense. Set F ( x ) = N k ( x - y ) f ( y ) d y x N 0 . Then F(x) = O(log²r) as r → 0 in the L p -sense, 1 < p < ∞....

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