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Best constants for some operators associated with the Fourier and Hilbert transforms

B. Hollenbeck, N. J. Kalton, I. E. Verbitsky (2003)

Studia Mathematica

We determine the norm in L p ( ℝ ₊ ) , 1 < p < ∞, of the operator I - ℱ s ℱ c , where ℱ c and ℱ s are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the L p -norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real a,b. Best...

BMO harmonic approximation in the plane and spectral synthesis for Hardy-Sobolev spaces.

Joan Mateu, Joan Verdera Melenchón (1988)

Revista Matemática Iberoamericana

The spectral synthesis theorem for Sobolev spaces of Hedberg and Wolff [7] has been applied in combination with duality, to problems of Lq approximation by analytic and harmonic functions. In fact, such applications were one of the main motivations to consider spectral synthesis problems in the Sobolev space setting. In this paper we go the opposite way in the context of the BMO-H1 duality: we prove a BMO approximation theorem by harmonic functions and then we apply the ideas in its proof to produce...

Boundary integral representations of second derivatives in shape optimization

Karsten Eppler (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard differential calculus in order to get second derivatives by a straight forward "repetition of differentiation". Moreover, by using boundary value characerizations for more regular data, a complete boundary integral...

Brownian motion and generalized analytic and inner functions

Alain Bernard, Eddy A. Campbell, A. M. Davie (1979)

Annales de l'institut Fourier

Let f be a mapping from an open set in R p into R q , with p &gt; q . To say that f preserves Brownian motion, up to a random change of clock, means that f is harmonic and that its tangent linear mapping in proportional to a co-isometry. In the case p = 2 , q = 2 , such conditions signify that f corresponds to an analytic function of one complex variable. We study, essentially that case p = 3 , q = 2 , in which we prove in particular that such a mapping cannot be “inner” if it is not trivial. A similar result for p = 4 , q = 2 would solve...

Capacité analytique et le problème de Painlevé

Hervé Pajot (2003/2004)

Séminaire Bourbaki

Le problème de Painlevé consiste à trouver une caractérisation géométrique des sous-ensembles du plan complexe qui sont effaçables pour les fonctions holomorphes bornées. Ce problème d’analyse complexe a connu ces dernières années des avancées étonnantes, essentiellement grâce au dévelopement de techniques fines d’analyse réelle et de théorie de la mesure géométrique. Dans cet exposé, nous allons présenter et discuter une solution proposée par X. Tolsa en termes de courbure de Menger au problème...

Currently displaying 61 – 80 of 493