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Singular integrals and rectifiability.

Pertti Mattila (2002)

Publicacions Matemàtiques

We shall discuss singular integrals on lower dimensional subsets of Rn. A survey of this topic was given in [M4]. The first part of this paper gives a quick review of some results discussed in [M4] and a survey of some newer results and open problems. In the second part we prove some results on the Riesz kernels in Rn. As far as I know, they have not been explicitly stated and proved, but they are very closely related to some earlier results and methods.[Proceedings of the 6th International Conference...

Singular integrals with highly oscillating kernels on product spaces

Elena Prestini (2000)

Colloquium Mathematicae

We prove the L 2 ( 2 ) boundedness of the oscillatory singular integrals P 0 f ( x , y ) = ∫ D x e i ( M 2 ( x ) y ' + M 1 ( x ) x ' ) ο v e r x ' y ' f ( x - x ' , y - y ' ) d x ' d y ' for arbitrary real-valued L ∞ functions M 1 ( x ) , M 2 ( x ) and for rather general domains D x ⊆ 2 whose dependence upon x satisfies no regularity assumptions.

Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves

Tao Qian (1997)

Studia Mathematica

The paper presents a theory of Fourier transforms of bounded holomorphic functions defined in sectors. The theory is then used to study singular integral operators on star-shaped Lipschitz curves, which extends the result of Coifman-McIntosh-Meyer on the L 2 -boundedness of the Cauchy integral operator on Lipschitz curves. The operator theory has a counterpart in Fourier multiplier theory, as well as a counterpart in functional calculus of the differential operator 1/i d/dz on the curves.

Sobolev-Morrey Type Inequality for Riesz Potentials, Associated with the Laplace-Bessel Differential Operator

Guliyev, Vagif, Hasanov, Javanshir (2006)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 42B20, 42B25, 42B35We consider the generalized shift operator, generated by the Laplace- Bessel differential operator [...] The Bn -maximal functions and the Bn - Riesz potentials, generated by the Laplace-Bessel differential operator ∆Bn are investigated. We study the Bn - Riesz potentials in the Bn - Morrey spaces and Bn - BMO spaces. An inequality of Sobolev - Morrey type is established for the Bn - Riesz potentials.* This paper has been partially supported...

Some integral and maximal operators related to starlike sets

Sagun Chanillo, David Watson, Richard Wheeden (1993)

Studia Mathematica

We prove two-weight norm estimates for fractional integrals and fractional maximal functions associated with starlike sets in Euclidean space. This is seen to include general positive homogeneous fractional integrals and fractional integrals on product spaces. We consider both weak type and strong type results, and we show that the conditions imposed on the weight functions are fairly sharp.

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