We give a characterization of the Hölder-Zygmund spaces ${𝒞}^{\sigma }\left(G\right)$ ($0<\sigma <\infty$) on a stratified Lie group $G$ in terms of Littlewood-Paley type decompositions, in analogy to the well-known characterization of the Euclidean case. Such decompositions are defined via the spectral measure of a sub-Laplacian on $G$, in place of the Fourier transform in the classical setting. Our approach mainly relies on almost orthogonality estimates and can be used to study other function spaces such as Besov and Triebel-Lizorkin spaces...

Let $E\subset \mathbf{R}$ be a closed null set. We prove an equivalence between the Littlewood-Paley decomposition in $L^p$ with respect to the complementary intervals of $E$ and Fourier multipliers of Hörmander-Mihlin and Marcinkiewicz type with singularities on $E$. Similar properties are studied in ${\mathbf{R}}^2$ for a union of rays from the origin. Then there are connections with the maximal function operator with respect to all rectangles parallel to these rays. In particular, this maximal operator is proved to be bounded on $L^p$, $1\&lt;p\&lt;\infty$,...

For certain non compact Riemannian manifolds with ends which may or may not satisfy the doubling condition on the volume of geodesic balls, we obtain Littlewood-Paley type estimates on (weighted) $L^p$ spaces, using the usual square function defined by a dyadic partition.

Let 𝔾 be a homogeneousgroup on ℝⁿ whose multiplication and inverse operations are polynomial maps. In 1999, T. Tao proved that the singular integral operator with Llog⁺L function kernel on ≫ is both of type (p,p) and of weak type (1,1). In this paper, the same results are proved for the Littlewood-Paley g-functions on 𝔾

We study the weak type (1,1) and the $L^p$-boundedness, 1 < p ≤ 2, of the so-called vertical (i.e. involving space derivatives) Littlewood-Paley-Stein functions and ℋ respectively associated with the Poisson semigroup and the heat semigroup on a complete Riemannian manifold M. Without any assumption on M, we observe that and ℋ are bounded in $L^p$, 1 < p ≤ 2. We also consider modified Littlewood-Paley-Stein functions that take into account the positivity of the bottom of the spectrum. Assuming that...

Let $M_S$ denote the strong maximal operator. Let $M_x$ and $M_y$ denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that ${\int }_Q{M}_y{M}_{x}h<\infty$ but ${\int }_Q{M}_x{M}_{y}h=\infty$. It is shown that if f is a function supported on Q such that ${\int }_Q{M}_y{M}_{x}f<\infty$ but ${\int }_Q{M}_x{M}_{y}f=\infty$, then there exists a set A of finite measure in ℝ² such that ${\int }_A{M}_{S}f=\infty$.

The paper deals with local means and wavelet bases in weighted and unweighted function spaces of type $B_{pq}^s$ and $F_{pq}^s$ on ℝⁿ and on ⁿ.

The paper deals with local means and wavelet bases in function spaces of Besov and Triebel-Lizorkin type with local Muckenhoupt weights.

We consider two standard group representations: one acting on functions by translations and dilations, the other by translations and modulations, and we study local Toeplitz operators based on them. Local Toeplitz operators are the averages of projection-valued functions $g\mapsto P_{g,\varphi }$, where for a fixed function ϕ, $P_{g,\varphi }$ denotes the one-dimensional orthogonal projection on the function $U_g\varphi$, U is a group representation and g is an element of the group. They are defined as integrals ${ʃ}_W{P}_{g,\varphi }dg$, where W is an open, relatively...