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Almost everywhere summability of Laguerre series

Krzysztof Stempak (1991)

Studia Mathematica

We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions n a ( x ) = ( n ! / Γ ( n + a + 1 ) ) 1 / 2 e - x / 2 L n a ( x ) , n = 0,1,2,..., in L 2 ( + , x a d x ) , a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function f L p ( x a d x ) , 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.

Almost everywhere summability of Laguerre series. II

K. Stempak (1992)

Studia Mathematica

Using methods from [9] we prove the almost everywhere convergence of the Cesàro means of Laguerre series associated with the system of Laguerre functions L n a ( x ) = ( n ! / Γ ( n + a + 1 ) ) 1 / 2 e - x / 2 x a / 2 L n a ( x ) , n = 0,1,2,..., a ≥ 0. The novel ingredient we add to our previous technique is the A p weights theory. We also take the opportunity to comment and slightly improve on our results from [9].

Bases d'ondelettes sur les courbes corde-arc, noyau de Cauchy et spaces de Hardy associés.

Pascal Auscher, Philippe Tchamitchian (1989)

Revista Matemática Iberoamericana

Se construyen dos bases incondicionales de L2(R) adaptadas al estudio de la integral de Cauchy sobre una curva cuerda-arco, y se extiende la construcción a L2(Rd). Esto permite obtener una prueba simple del "Teorema T(b)" de G. David, J.L. Journé u S. Semmes. Se define un espacio de Hardy ponderado Hb1(Rd) caracterizado por las bases anteriores. Finalmente se aplican estos métodos al estudio del potencial de doble capa sobre una superficie lipschitziana.

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