We discuss the concept of Sobolev space associated to the Laguerre operator $L_{α} = - y d²/dy² - d/dy + y/4 + α²/4y$, y ∈ (0,∞). We show that the natural definition does not agree with the concept of potential space defined via the potentials $(L_{α})^{-s}$. 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.
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average $Aₙf(x) = 2^{-n} ∫_{x}^{x+2ⁿ} f$. The square function is defined as $Sf(x) = (∑_{n=-∞}^{∞} |Aₙf(x) - A_{n-1}f(x)|²)^{1/2}$. The local version of this operator, namely the operator $S₁f(x) = (∑_{n=-∞}^{0} |Aₙf(x) - A_{n-1}f(x)|²)^{1/2}$, is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps $L^{p}$ into itself (p > 1) and $L^{∞}$ into BMO. We prove that the operator S not only maps $L^{∞}$ into BMO but it also maps BMO into BMO. We also prove that the $L^{p}$ boundedness still holds if one replaces Lebesgue measure by a measure of the form w(x)dx if, and only if, the weight w belongs to the $A⁺_{p}$ class introduced by E. Sawyer [8]. Finally we prove that the one-sided Hardy-Littlewood maximal function maps BMO into itself.
3
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
In the half-space $ℝ^{d} × ℝ₊$, consider the Hermite-Schrödinger equation i∂u/∂t = -Δu + |x|²u, with given boundary values on $ℝ^{d}$. We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary by means of this link.
4
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We obtain weighted $L^{p}$ boundedness, with weights of the type $y^{δ}$, δ > -1, for the maximal operator of the heat semigroup associated to the Laguerre functions, ${ℒ_{k}^{α}}_{k}$, when the parameter α is greater than -1. It is proved that when -1 < α < 0, the maximal operator is of strong type (p,p) if p > 1 and 2(1+δ)/(2+α) < p < 2(1+δ)/(-α), and if α ≥ 0 it is of strong type for 1 < p ≤ ∞ and 2(1+δ)/(2+α) < p. The behavior at the end points of the intervals where there is strong type is studied in detail and sharp results about the existence or not of strong, weak or restricted types are given.
5
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, $L^{p}$ and BMO of classical ℒ-square functions.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.