Perturbation of Orthogonal Fourier Expansions

Guadalupe, José; Pérez, Mario

doi:10.1006/jath.1997.3129

Cited by 8 publications

(4 citation statements)

References 16 publications

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“…The (δ, R)-BMO condition which appeared in [3,4] is also called a small BMO condition and has been used in various work as an appropriate substitute for the Sarason VMO (vanishing mean oscillation [27]) condition (see, e.g., [3,4,11,14,20,28,30] Another obstacle in obtaining global integrability results is the roughness of the boundary of the ground domain. Even for smooth coefficients, global integrability of gradients of solutions to (1.1) over domains with bad boundary may not be true.…”

Section: Introductionmentioning

confidence: 99%

Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains

Mengesha

Phuc

2011

Journal of Differential Equations

108

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Global weighted L p estimates are obtained for the gradient of solutions to nonlinear elliptic Dirichlet boundary value problems over a bounded nonsmooth domain. Morrey and Hölder regularity of solutions are also established, as a consequence. These results generalize various existing estimates for nonlinear equations. The nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation. The boundary of the domain, on the other hand, may exhibit roughness but assumed to be sufficiently flat in the sense of Reifenberg. Our approach uses maximal function estimates and Vitali covering lemma, and also known regularity results of solutions to nonlinear homogeneous equations.

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Section: Introductionmentioning

confidence: 99%

Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains

Mengesha

Phuc

2011

Journal of Differential Equations

108

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“…where lim n→∞ κ n−m (|V | 2 dμ) κ n (dμ) = 1 from (2) and (8). The behaviour of ϕ n (dμ; ζ j ) and K n−1 (dμ; ζ j , ζ j ) when n → ∞, is given by (4) and (22), respectively.…”

Section: Proof Of Theorems 1 Andmentioning

confidence: 97%

“…In the real case, the works by Guadalupe-Pérez, Badkov or Golinskii (see [8,2,6]) show that it is possible to obtain some properties of Fourier series from perturbations of the orthogonality measure. Because of this, we consider that an interesting future challenge is to study properties of Fourier series in terms of properties of Verblunsky coefficients.…”

Section: Introductionmentioning

confidence: 98%

Asymptotic behaviour of Verblunsky coefficients

Alfaro

Montaner

2006

Journal of Mathematical Analysis and Applications

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Let V (z) = m j =1 (z − ζ j ), ζ h = ζ k , h = k and |ζ j | = 1, j = 1, . . . , m, and consider the polynomials orthogonal with respect to |V | 2 dμ, ϕ n (|V | 2 dμ; z), where μ is a finite positive Borel measure on the unit circle with infinite points in its support, such that the reciprocal of its Szegő function has an analytic extension beyond |z| < 1. In this paper we deduce the asymptotic behaviour of their Verblunsky coefficients. By means of this result, an asymptotic representation for these polynomials inside the unit circle is also obtained. (M.P. Alfaro), mbello@dmc.unirioja.es (M.B. Hernández), montaner@unizar.es (J.M. Montaner).

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“…For general nonlinearities A(x, ξ) of at most linear growth, i.e., p = 2, the above (γ, R 0 )-BMO condition was introduced in [4], whereas such a condition for general p > 1 appears first in [26]. We remark that the (γ, R 0 )-BMO condition allows the nonlinearity A(x, ξ) to have certain discontinuity in x, and it can be used as an appropriate substitute for the Sarason VMO condition (vanishing mean oscillation [28], see also [2,4,9,13,24,29,33]).…”

Section: Introductionmentioning

confidence: 99%