Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and , 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.
Abstract
We obtain the necessary and sufficient conditions for the boundedness of the weighted singular integral operator with power weights in grand Lebesgue spaces. Because of applications to singular integral equations, the underlying set on which the functions are defined is a Carleson curve in the complex plane. Note that weighted boundedness of an operator in grand Lebesgue space is not the same as the boundedness in weighted grand Lebesgue space.
We study the boundedness of the maximal operator in the weighted spacesLp(⋅)(ρ)over a bounded open setΩin the Euclidean spaceℝnor a Carleson curveΓin a complex plane. The weight function may belong to a certain version of a general Muckenhoupt-type condition, which is narrower than the expected Muckenhoupt condition for variable exponent, but coincides with the usual Muckenhoupt classApin the case of constantp. In the case of Carleson curves there is also considered another class of weights of radial type of the formρ(t)=∏k=1mwk(|t-tk|),tk∈Γ, wherewkhas the property thatr1p(tk)wk(r)∈Φ10, whereΦ10is a certain Zygmund-Bari-Stechkin-type class. It is assumed that the exponentp(t)satisfies the Dini–Lipschitz condition. For such radial type weights the final statement on the boundedness is given in terms of the index numbers of the functionswk(similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).
The maximal operator in weighted variable spacesLp(⋅)