Hermite Conjugate Functions and Rearrangement Invariant Spaces

1973 ◽  
Vol 16 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Weak Type ◽  
Conjugate Function ◽  
Poisson Integral ◽  
Conjugate Functions ◽  
Rearrangement Invariant Spaces ◽  
Image Position ◽  
Invariant Spaces

The Hermite conjugate Poisson integral of a given f ∊ L1(μ), dμ(y)= exp(—y2) dy, was defined by Muckenhoupt [5, p. 247] aswhereIf the Hermite conjugate function operator T is defined by (Tf) a.e., then one of the main results of [5] is that T is of weak-type (1, 1) and strongtype (p,p) for all p>l.

scholarly journals The K-functional of certain pairs of rearrangement invariant spaces

1983 ◽  
Vol 27 (2) ◽  
pp. 249-257 ◽  
Author(s):  
Jonathan Arazy
Keyword(s):  
Fundamental Function ◽  
Rearrangement Invariant Spaces ◽  
Image Position ◽  
Invariant Spaces

Let X, Y be rearrangement invariant spaces and let M = M(Y, X) be the space of all multipliers of Y into X. It is shown that if X = YM and some technical conditions are satisfied, then the K-functional K(t, f, X, Y) is equivalent to the expressionwhere ψ is the inverse of the fundamental function ϕM of M, defined by

A Dimension-Free Weak-Type Estimate for Operators on UMD-Valued Functions

2000 ◽  
Vol 43 (3) ◽  
pp. 355-361
Author(s):  
Brian P. Kelly
Keyword(s):  
Weak Type ◽  
Stochastic Integral ◽  
Conjugate Function ◽  
Summation Method ◽  
Lexicographical Order ◽  
Type Estimate ◽  
Umd Spaces ◽  
Stochastic Integral Representation

AbstractLet T denote the unit circle in the complex plane, and let X be a Banach space that satisfies Burkholder’s UMD condition. Fix a natural number, N ∈ . Let P denote the reverse lexicographical order on ZN. For each f ∈ L1(TN, X), there exists a strongly measurable function such that formally, for all . In this paper, we present a summation method for this conjugate function directly analogous to the martingale methods developed by Asmar and Montgomery-Smith for scalar-valued functions. Using a stochastic integral representation and an application of Garling’s characterization of UMD spaces, we prove that the associated maximal operator satisfies a weak-type (1, 1) inequality with a constant independent of the dimension N.

scholarly journals New classes of rearrangement-invariant spaces appearing in extreme cases of weak interpolation

2006 ◽  
Vol 4 (3) ◽  
pp. 275-304 ◽  
Author(s):  
Evgeniy Pustylnik ◽  
Teresa Signes
Keyword(s):  
Weak Type ◽  
Interpolation Theorem ◽  
Optimal Interpolation ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Type Interpolation ◽  
Rearrangement Invariant Spaces ◽  
Invariant Spaces ◽  
Quasiconcave Function

We study weak type interpolation for ultrasymmetric spacesL?,Ei.e., having the norm??(t)f*(t)?E˜, where?(t)is any quasiconcave function andE˜is arbitrary rearrangement-invariant space with respect to the measuredt/t. When spacesL?,Eare not “too close” to the endpoint spaces of interpolation (in the sense of Boyd), the optimal interpolation theorem was stated in [13]. The case of “too close” spaces was studied in [15] with results which are optimal, but only among ultrasymmetric spaces. In this paper we find better interpolation results, involving new types of rearrangement-invariant spaces,A?,b,EandB?,b,E, which are described and investigated in detail.

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