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Orlicz boundedness for certain classical operators

E. Harboure, O. Salinas, B. Viviani (2002)

Colloquium Mathematicae

Let ϕ and ψ be functions defined on [0,∞) taking the value zero at zero and with non-negative continuous derivative. Under very mild extra assumptions we find necessary and sufficient conditions for the fractional maximal operator M Ω α , associated to an open bounded set Ω, to be bounded from the Orlicz space L ψ ( Ω ) into L ϕ ( Ω ) , 0 ≤ α < n. For functions ϕ of finite upper type these results can be extended to the Hilbert transform f̃ on the one-dimensional torus and to the fractional integral operator I Ω α , 0...

Orlicz bounds for operators of restricted weak type

Paul Alton Hagelstein (2005)

Colloquium Mathematicae

It is shown that if T is a sublinear translation invariant operator of restricted weak type (1,1) acting on L¹(𝕋), then T maps simple functions in L log L(𝕋) boundedly into L¹(𝕋).

Orlicz-Morrey spaces and the Hardy-Littlewood maximal function

Eiichi Nakai (2008)

Studia Mathematica

We prove basic properties of Orlicz-Morrey spaces and give a necessary and sufficient condition for boundedness of the Hardy-Littlewood maximal operator M from one Orlicz-Morrey space to another. For example, if f ∈ L(log L)(ℝⁿ), then Mf is in a (generalized) Morrey space (Example 5.1). As an application of boundedness of M, we prove the boundedness of generalized fractional integral operators, improving earlier results of the author.

Orthogonal polynomials and the Lanczos method

C. Brezinski, H. Sadok, M. Redivo Zaglia (1994)

Banach Center Publications

Lanczos method for solving a system of linear equations is well known. It is derived from a generalization of the method of moments and one of its main interests is that it provides the exact answer in at most n steps where n is the dimension of the system. Lanczos method can be implemented via several recursive algorithms known as Orthodir, Orthomin, Orthores, Biconjugate gradient,... In this paper, we show that all these procedures can be explained within the framework of formal orthogonal polynomials....

Oscillating multipliers on the Heisenberg group

E. K. Narayanan, S. Thangavelu (2001)

Colloquium Mathematicae

Let ℒ be the sublaplacian on the Heisenberg group Hⁿ. A recent result of Müller and Stein shows that the operator ℒ - 1 / 2 s i n √ ℒ is bounded on L p ( H ⁿ ) for all p satisfying |1/p - 1/2| < 1/(2n). In this paper we show that the same operator is bounded on L p in the bigger range |1/p - 1/2| < 1/(2n-1) if we consider only functions which are band limited in the central variable.

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