We establish sharp (H1,L1,q) and local (L logrL,L1,q) mapping properties for rough one-dimensional multipliers. In particular, we show that the multipliers in the Marcinkiewicz multiplier theorem map H1 to L1,∞ and L log1/2L to L1,∞, and that these estimates are sharp.

Suppose Δ̃ is the Laplace-Beltrami operator on the sphere $S^{d-1},{\Delta }_{\rho }^{k}f\left(x\right)={\Delta }_{\rho }{\Delta }_{\rho }^{k-1}f\left(x\right)$ and ${\Delta }_{\rho }f\left(x\right)=f\left(\rho x\right)-f\left(x\right)$ where ρ ∈ SO(d). Then ${\omega }^m{(f,t)}_{L_p\left(S^{d-1}\right)}\equiv sup\parallel {\Delta }_{\rho }^{m}f{\parallel }_{L_p\left(S^{d-1}\right)}:\rho \in SO\left(d\right),ma{x}_{x\in S^{d-1}}\rho x·x\ge cost$ and $K̃ₘ{(f,t^m)}_p\equiv inf\parallel f-g{\parallel }_{L_p\left(S^{d-1}\right)}+t^m\parallel {(-\Delta ̃)}^{m/2}g{\parallel }_{L_p\left(S^{d-1}\right)}:g\in \left({(-\Delta ̃)}^{m/2}\right)$ are equivalent for 1 < p < ∞. We note that for even m the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for $L_p\left(S^{d-1}\right)$ given in this paper plays a significant role in the proof.

Generalizing the classical BMO spaces defined on the unit circle with vector or scalar values, we define the spaces $BM{O}_{{\psi }_q}\left(\right)$ and $BM{O}_{{\psi }_q}(,B)$, where ${\psi }_q\left(x\right)=e^{x^q}-1$ for x ≥ 0 and q ∈ [1,∞[, and where B is a Banach space. Note that $BM{O}_{{\psi }_1}\left(\right)=BMO\left(\right)$ and $BM{O}_{{\psi }_1}(,B)=BMO(,B)$ by the John-Nirenberg theorem. Firstly, we study a generalization of the classical Paley inequality and improve the Blasco-Pełczyński theorem in the vector case. Secondly, we compute the idempotent multipliers of $BM{O}_{{\psi }_q}\left(\right)$. Pisier conjectured that the supports of idempotent multipliers of $L^{{\psi }_q}\left(\right)$ form a Boolean...

Let $b_1,b_2\in \mathrm{BMO}\left({ℝ}^n\right)$ and $T_{\sigma }$ be a bilinear Fourier multiplier operator with associated multiplier $\sigma$ satisfying the Sobolev regularity that ${sup}_{\kappa \in \mathbb{Z}}{\parallel {\sigma }_{\kappa }\parallel }_{W^{s_1,s_2}\left({ℝ}^{2n}\right)}<\infty$ for some $s_1,s_2\in (n/2,n]$. In this paper, the behavior on $L^{p_1}\left({ℝ}^n\right)\times L^{p_2}\left({ℝ}^n\right)$$(p_1,...$

We prove that if $\Lambda \in M_p\left({ℝ}^N\right)$ and has compact support then Λ is a weak summability kernel for 1 < p < ∞, where $M_p\left({ℝ}^N\right)$ is the space of multipliers of $L^p\left({ℝ}^N\right)$.

We study Fourier multipliers resulting from martingale transforms of general Lévy processes.

Cet article est consacré à l’étude des espaces $A_{\omega }=ℱL^2({\mathbf{R}}^n;\omega \left(x\right)dx)$ qui sont des algèbres de Banach. On démontre que les multiplicateurs ponctuels de $A_{\omega }$ sont les fonctions qui appartiennent localement et uniformément à $A_{\omega }$ si et seulement si $A_{\omega }$ contient des fonctions à support compact.

We investigate generalized Bochner-Riesz means at the critical index on spaces generated by smooth blocks and give some approximation theorems.

Connections between Hankel transforms of different order for $L^p$-functions are examined. Well known are the results of Guy [Guy] and Schindler [Sch]. Further relations result from projection formulae for Bessel functions of different order. Consequences for Hankel multipliers are exhibited and implications for radial Fourier multipliers on Euclidean spaces of different dimensions indicated.

The Littlewood-Paley theory is extended to weighted spaces of distributions on [-1,1] with Jacobi weights $w\left(t\right)={(1-t)}^{\alpha }{(1+t)}^{\beta }$. Almost exponentially localized polynomial elements (needlets) ${\phi }_{\xi }$, ${\psi }_{\xi }$ are constructed and, in complete analogy with the classical case on ℝⁿ, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $⟨f,{\phi }_{\xi }⟩$ in respective sequence spaces.

Józef Marcinkiewicz’s (1910-1940) name is not known by many people, except maybe a small group of mathematicians, although his influence on the analysis and probability theory of the twentieth century was enormous. This survey of his life and work is in honour of the ${100}^{th}$ anniversary of his birth and ${70}^{th}$ anniversary of his death. The discussion is divided into two periods of Marcinkiewicz’s life. First, 1910-1933, that is, from his birth to his graduation from the University of Stefan Batory in Vilnius,...