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Spherical summation : a problem of E.M. Stein

Antonio Cordoba, B. Lopez-Melero (1981)

Annales de l'institut Fourier

Writing ( T R λ f ) ^ ( ξ ) = ( 1 - | ξ | 2 / R 2 ) + λ f ^ ( ξ ) . E. Stein conjectured j | T R j λ f i | 2 1 / 2 p C j | f j | 2 1 / 2 p for λ > 0 , 4 3 p 4 and C = C λ , p . We prove this conjecture. We prove also f ( x ) = lim j T 2 j λ f ( x ) a.e. We only assume 4 3 + 2 λ < p < 4 1 - 2 λ .

Square functions associated to Schrödinger operators

I. Abu-Falahah, P. R. Stinga, J. L. Torrea (2011)

Studia Mathematica

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, L p and BMO of classical ℒ-square functions.

Square functions of Calderón type and applications.

Steve Hofmann, John L. Lewis (2001)

Revista Matemática Iberoamericana

We establish L2 and Lp bounds for a class of square functions which arises in the study of singular integrals and boundary value problems in non-smooth domains. As an application, we present a simplified treatment of a class of parabolic smoothing operators which includes the caloric single layer potential on the boundary of certain minimally smooth, non-cylindrical domains.

Systems of dyadic cubes in a doubling metric space

Tuomas Hytönen, Anna Kairema (2012)

Colloquium Mathematicae

A number of recent results in Euclidean harmonic analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen...

The Cauchy problem for viscous shallow water equations.

Weike Wang, Chao-Jiang Xu (2005)

Revista Matemática Iberoamericana

In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index s > 2 to obtain local solutions for any initial data, and global solutions for small initial data.

The fractional integral between weighted Orlicz and B M O φ spaces on spaces of homogeneous type

Gladis Pradolini, Oscar Salinas (2003)

Commentationes Mathematicae Universitatis Carolinae

In this work we give sufficient and necessary conditions for the boundedness of the fractional integral operator acting between weighted Orlicz spaces and suitable B M O φ spaces, in the general setting of spaces of homogeneous type. This result generalizes those contained in [P1] and [P2] about the boundedness of the same operator acting between weighted L p and Lipschitz integral spaces on n . We also give some properties of the classes of pairs of weights appearing in connection with this boundedness.

The higher order Riesz transform for Gaussian measure need not be of weak type (1,1)

Liliana Forzani, Roberto Scotto (1998)

Studia Mathematica

The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator L : = d 2 / d x 2 - 2 x d / d x , x ∈ ℝ, need not be of weak type (1,1). A function in L 1 ( d γ ) , where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.

The John-Nirenberg inequality for functions of bounded mean oscillation with bounded negative part

Min Hu, Dinghuai Wang (2022)

Czechoslovak Mathematical Journal

A version of the John-Nirenberg inequality suitable for the functions b BMO with b - L is established. Then, equivalent definitions of this space via the norm of weighted Lebesgue space are given. As an application, some characterizations of this function space are given by the weighted boundedness of the commutator with the Hardy-Littlewood maximal operator.

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