Let be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ of arbitrarily small diameter containing x. The collection is said to be a density basis provided that, given a measurable set A ⊂ ℝⁿ, for a.e. x ∈ ℝⁿ we have $li{m}_{k\to \infty }1/\left|R_k\right|{\int }_{R_k}{\chi }_A={\chi }_A\left(x\right)$ for any sequence $R_k$ of sets in containing x whose diameters tend to 0. The geometric maximal operator $M_{}$ associated to is defined on L¹(ℝⁿ) by $M_{}f\left(x\right)=su{p}_{x\in R\in }1/\left|R\right|{\int }_R\left|f\right|$. The halo function ϕ of is defined on (1,∞) by $\varphi \left(u\right)=sup1/\left|A\right||x\in ℝⁿ:M_{}{\chi }_A\left(x\right)>1/u|:0<|A|<\infty$ and on [0,1] by ϕ(u) = u. It is shown that the halo...

In this paper, we obtain some strong and weak type continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator on Hardy spaces, Hardy type spaces and weak Hardy type spaces.

We determine the convolution operators $T_{\mu }:=\mu *$ on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction of hyperbolic type on the location of zeros of the Fourier transform μ̂(z).

We prove three theorems on linear operators $T_{\tau ,p}:H_p\left(ℬ\right)\to H_p$ induced by rearrangement of a subsequence of a Haar system. We find a sufficient and necessary condition for $T_{\tau ,p}$ to be continuous for 0 < p < ∞.

In this work we define and study wavelets and continuous wavelet transform on semisimple Lie groups G of real rank l. We prove for this transform Plancherel and inversion formulas. Next using the Abel transform A on G and its dual A*, we give relations between the continuous wavelet transform on G and the classical continuous wavelet transform on Rl, and we deduce the formulas which give the inverse operators of the operators A and A*.

On montre que si $T$ est une contraction à spectre dénombrable et telle que, pour tout $ϵ\&gt;0$$$\parallel T^{-1}...$$

For the Schrödinger equation, $(i{\partial }_t+\nabla )u=0$ on a torus, an arbitrary non-empty open set $\Omega$ provides control and observability of the solution: $∥u{\left|{}_{t=0}\right.∥}_{L^2\left({𝕋}^2\right)}\le K_T{∥u∥}_{L^2([0,T]\times \Omega )}$. We show that the same result remains true for $(i{\partial }_t+\nabla -V)u=0$ where $V\in L^2\left({𝕋}^2\right)$, and ${𝕋}^2$ is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for $V\in C\left({𝕋}^2\right)$ and conjectured for $V\in L^{\infty }\left({𝕋}^2\right)$. The higher dimensional generalization remains open for $V\in L^{\infty }\left({𝕋}^n\right)$.