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An explicit right inverse of the divergence operator which is continuous in weighted norms

Ricardo G. Durán, Maria Amelia Muschietti (2001)

Studia Mathematica

The existence of a continuous right inverse of the divergence operator in W ₀ 1 , p ( Ω ) ⁿ , 1 < p < ∞, is a well known result which is basic in the analysis of the Stokes equations. The object of this paper is to show that the continuity also holds for some weighted norms. Our results are valid for Ω ⊂ ℝⁿ a bounded domain which is star-shaped with respect to a ball B ⊂ Ω. The continuity results are obtained by using an explicit solution of the divergence equation and the classical theory of singular integrals...

An improved maximal inequality for 2D fractional order Schrödinger operators

Changxing Miao, Jianwei Yang, Jiqiang Zheng (2015)

Studia Mathematica

The local maximal operator for the Schrödinger operators of order α > 1 is shown to be bounded from H s ( ℝ ² ) to L² for any s > 3/8. This improves the previous result of Sjölin on the regularity of solutions to fractional order Schrödinger equations. Our method is inspired by Bourgain’s argument in the case of α = 2. The extension from α = 2 to general α > 1 faces three essential obstacles: the lack of Lee’s reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable...

An oscillatory singular integral operator with polynomial phase

Josfina Alvarez, Jorge Hounie (1999)

Studia Mathematica

We prove the continuity of an oscillatory singular integral operator T with polynomial phase P(x,y) on an atomic space H P 1 related to the phase P. Moreover, we show that the cancellation condition to be imposed on T holds under more general conditions. To that purpose, we obtain a van der Corput type lemma with integrability at infinity.

Analytic capacity, Calderón-Zygmund operators, and rectifiability

Guy David (1999)

Publicacions Matemàtiques

For K ⊂ C compact, we say that K has vanishing analytic capacity (or γ(K) = 0) when all bounded analytic functions on CK are constant. We would like to characterize γ(K) = 0 geometrically. Easily, γ(K) &gt; 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) &lt; 1. Thus only the case when dim(K) = 1 is interesting. So far there is no characterization of γ(K) = 0 in general, but the special case when the Hausdorff measure H1(K) is finite was recently settled. In this...

Banach algebra of the Fourier multipliers on weighted Banach function spaces

Alexei Karlovich (2015)

Concrete Operators

Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.

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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