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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 Hardy space related to the square root of the Poisson kernel

Jonatan Vasilis (2010)

Studia Mathematica

A real-valued Hardy space H ¹ ( ) L ¹ ( ) related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in H ¹ ( ) if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to H ¹ ( ) , and no Orlicz space...

A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem

Caroline Sweezy (2007)

Annales Polonici Mathematici

Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to L u = d i v f in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the L q ( Ω , d μ ) norm of |∇u| is dominated by the L p ( Ω , d v ) norms of d i v f and | f | . If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

A Littlewood-Paley-Stein estimate on graphs and groups

Nick Dungey (2008)

Studia Mathematica

We establish the boundedness in L q spaces, 1 < q ≤ 2, of a “vertical” Littlewood-Paley-Stein operator associated with a reversible random walk on a graph. This result extends to certain non-reversible random walks, including centered random walks on any finitely generated discrete group.

A new characterization of RBMO ( μ ) by John-Strömberg sharp maximal functions

Guoen Hu, Dachun Yang, Dongyong Yang (2009)

Czechoslovak Mathematical Journal

Let μ be a nonnegative Radon measure on d which only satisfies μ ( B ( x , r ) ) C 0 r n for all x d , r > 0 , with some fixed constants C 0 > 0 and n ( 0 , d ] . In this paper, a new characterization for the space RBMO ( μ ) of Tolsa in terms of the John-Strömberg sharp maximal function is established.

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