A Kolmogorov-Szego-Krein type condition for weighted Sobolev spaces (original) (raw)

2005, Indiana University Mathematics Journal

§0. Introduction and main results Let p ∈ [1, +∞), k ∈ N, and let µ = (µ 0 , µ 1 ,. .. , µ k) be a (k + 1)-tuple of positive finite Borel measures on the unit circle T = {z : |z| = 1} in the complex plane. Consider the continuous mapping Π : C k (T) → k j=0 C(T), given by Πf = f, f ,. .. , f (k) , where f (z) = df dz (all spaces of functions that we consider are complex-valued). Note that df dz (e iθ) = −ie −iθ d dθ f (e iθ). Definition. The abstract Sobolev space W k,p (µ) = W k,p (µ 0 ,. .. , µ k) is the closure of ΠC k (T) in the space k j=0 L p (T, µ j). We refer to [17] for the classical theory of Sobolev spaces in domains of R n. We refer to [13, 18, 5, 12] for the theory of weighted Sobolev spaces in domains of R n ; in [11, 14], one can find applications of this topic to partial differential equations. We consider in W k,p (µ) the usual norm f k,p,µ = k j=0 f j p p,µ j 1/p , f = (f 0 ,. .. , f k). Each function f in C k (T) has its image Πf in W k,p (µ), and these images are dense in W k,p (µ). In many cases, an element g = (g 0 ,. .. , g k) in W k,p (µ) is completely determined by its first component g 0 , so that W k,p (µ) can be identified with a certain space of measurable functions g 0 , and the components g 1 ,. .. , g k can be thought of as a kind of generalized derivatives of g 0. In general, however, elements of W k,p (µ) cannot be identified with scalar functions on T. This setting of abstract Sobolev spaces is the most natural for us. (See [2], [22]-[26] in order to know when W k,p (µ) is a space of functions.) This space plays a central role in the theory of orthogonal polynomials with respect to Sobolev inner products (see [2], [15], [16] and [23]; in [2] and [16], the authors consider measures supported in compact sets in the complex plane). In fact, if the multiplication operator (M f)(z) = zf (z) is bounded in W k,2 (µ), then every 1 The research of the first author was partially supported by two grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain. 2 The research of the second author was partially supported by the Ramón y Cajal Programme by the Ministry of Science and Technology of Spain.