Weak-Type Boundedness of the Fourier Transform on Rearrangement Invariant Function Spaces

We study several questions about the weak-type boundedness of the Fourier transform ℱ on rearrangement invariant spaces. In particular, we characterize the action of ℱ as a bounded operator from the minimal Lorentz space Λ(X) into the Marcinkiewicz maximal space M(X), both associated with a rearrangement invariant space X. Finally, we also prove some results establishing that the weak-type boundedness of ℱ, in certain weighted Lorentz spaces, is equivalent to the corresponding strong-type estimates.

NEW MARTINGALE INEQUALITIES IN REARRANGEMENT-INVARIANT FUNCTION SPACES

We establish various martingale inequalities in a rearrangement-invariant (RI) Banach function space. If $X$ is an RI space that is not too small, we associate with it RI spaces $\mathcal{H}_p(X)$ $(1\leq p\lt\infty)$ and $K(X)$, and discuss martingale inequalities in these spaces. One of our results is as follows. Let $1\leq p\lt\infty$, let $f=(f_n)$ be an $L_p$-bounded martingale, and let $|f|^p=g+h$ be the Doob decomposition of the submartingale $|f|^p=(|f_n|^p)$ into a martingale $g=(g_n)$ and a predictable non-decreasing process $h=(h_n)$ with $h_0=0$. Then, in the case where $1\ltp\lt\infty$, we obtain the inequalities$$ \|h_{\infty}^{1/p}\|_X\leq2\|f_{\infty}\|_{\mathcal{H}_p(X)}\quad\text{and}\quad \Big\|\sup_n|g_n|^{1/p}\Big\|_X\leq4\|f_{\infty}\|_{\mathcal{H}_p(X)}, $$and, in the case where $p=1$, we obtain the inequalities$$ \|h_{\infty}\|_X\leq\sup_{n\in\mathbb{Z}_{+}}\|f_n\|_{K(X)}\quad\text{and}\quad \sup_{n\in\mathbb{Z}_{+}}\|g_n\|_X\leq2\sup_{n\in\mathbb{Z}_{+}}\|f_n\|_{K(X)}. $$For some specific choices of $X$, we can give explicit expressions for $\mathcal{H}_p(X)$ and $K(X)$. For example, $\mathcal{H}_1(L_1)=L\log L$, $\mathcal{H}_p(L_{p,\infty})=L_{p,1}$, and so on. Furthermore, if the Boyd indices of $X$ satisfy $0\lt\alpha_{X}\leq\beta_{X}\lt1/p$ (respectively, $0\lt\alpha_{X}$), then $\mathcal{H}_p(X)=X$ (respectively, $K(X)=X$). In any case, $\mathcal{H}_p(X)$ is embedded in $K(X)$, and $K(X)$ is embedded in $X$.AMS 2000 Mathematics subject classification: Primary 60G42; 60G46. Secondary 46E30

About interpolation of subspaces of rearrangement invariant spaces generated by Rademacher system

The Rademacher series in rearrangement invariant function spaces “close” to the spaceL∞are considered. In terms of interpolation theory of operators, a correspondence between such spaces and spaces of coefficients generated by them is stated. It is proved that this correspondence is one-to-one. Some examples and applications are presented.

Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces

We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section 2 the equivalence between Shimogaki’s theorem and some martingale inequalities will be established, and in Section 3 the equivalence between Boyd’s theorem andmartingale inequalities with change of probability measure will be established.

Isometries of Hilbert space valued function spaces

Let X be a (real or complex) rearrangement-invariant function space on Ω (where Ω = [0, 1] or Ω ⊆ N) whose norm is not proportional to the L2-norm. Let H be a separable Hilbert space. We characterize surjective isometries of X (H). We prove that if T is such an isometry then there exist Borel maps a: Ω → + K and σ: Ω → Ω and a strongly measurable operator map S of Ω into B (H) so that for almost all ω, S(ω) is a surjective isometry of H, and for any f ∈ X(H), T f(ω) = a(ω)S(ω)(f(σ(ω))) a.e. As a consequence we obtain a new proof of the characterization of surjective isometries in complex rearrangement-invariant function spaces.