We prove that, in arbitrary finite dimensions, the maximal operator for the Laguerre semigroup is of weak type (1,1). This extends Muckenhoupt's one-dimensional result.

We discuss the concept of Sobolev space associated to the Laguerre operator $L_{\alpha }=-yd²/dy²-d/dy+y/4+\alpha ²/4y$, y ∈ (0,∞). We show that the natural definition does not agree with the concept of potential space defined via the potentials ${\left(L_{\alpha }\right)}^{-s}$. An appropriate Laguerre-Sobolev space is defined in order to achieve that coincidence. An application is given to the almost everywhere convergence of solutions of the Schrödinger equation. Other Laguerre operators are also considered.

Given a probability measure μ with non-polar compact support K, we define the n-th Widom factor W²ₙ(μ) as the ratio of the Hilbert norm of the monic n-th orthogonal polynomial and the n-th power of the logarithmic capacity of K. If μ is regular in the Stahl-Totik sense then the sequence ${\left(W²ₙ\left(\mu \right)\right)}_{n=0}^{\infty }$ has subexponential growth. For measures from the Szegő class on [-1,1] this sequence converges to some proper value. We calculate the corresponding limit for the measure that generates the Jacobi polynomials, analyze...