Let $P(\lambda ,D)={\sum }_{\left|\alpha \right|\le m}{a}_{\alpha }\left(\lambda \right)D^{\alpha }$ be a differential operator with constant coefficients $a_{\alpha }$ depending analytically on a parameter $\lambda$. Assume that the family $\{$ P($\lambda$,D)$\}$ is of constant strength. We investigate the equation $P(\lambda ,D){𝔣}_{\lambda }\equiv g_{\lambda }$ where ${𝔤}_{\lambda }$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution ${𝔣}_{\lambda }$ is required to depend analytically on $\lambda$, too. As a special case we obtain a regular fundamental solution of P($\lambda$,D) which depends analytically on $\lambda$. This result answers a question of L. Hörmander. ...

Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^p\left(ℝ\right)$-boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m}f\left(x\right)=p.v.\int k(x-y)m(x+y)f\left(y\right)dy$ where $k\left(x\right)={\sum }_{j\in \mathbb{Z}}{2}^j{\phi }_j\left(2^{j}x\right)$ for a family of functions ${{\phi }_j}_{j\in \mathbb{Z}}$ satisfying conditions (1.1)-(1.3) given below.

We investigate the Fourier transforms of functions in the Sobolev spaces $W_1^{r_1,...,r_n}$. It is proved that for any function $f\in W_1^{r_1,...,r_n}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n/r,1}$, where $r=n{({\sum }_{j=1}^{n}1/r_j)}^{-1}\le n$. Furthermore, we derive from this result that for any mixed derivative $D^{s}f(f\in C_0^{\infty },s=(s_1,...,s_n))$ the weighted norm $\parallel {\left(D^{s}f\right)}^{\wedge }{\parallel }_{L^1\left(w\right)}(w\left(\xi \right)={\left|\xi \right|}^{-n})$ can be estimated by the sum of $L^1$-norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$ which are the local versions of $H^p\left({ℝ}^n\right)$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$ and discuss the $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$-boundedness of Calderón-Zygmund operators. Similar results can also be obtained...

The authors obtain some multiplier theorems on $H^p$ spaces analogous to the classical $L^p$ multiplier theorems of de Leeuw. The main result is that a multiplier operator ${\left(Tf\right)}^{(}x)=\lambda \left(x\right)f̂\left(x\right)$ $\left(\lambda \in C\left({ℝ}^n\right)\right)$ is bounded on $H^p\left({ℝ}^n\right)$ if and only if the restriction ${\lambda \left(\epsilon m\right)}_{m\in \Lambda }$ is an $H^p\left(T^n\right)$ bounded multiplier uniformly for ε>0, where Λ is the integer lattice in ${ℝ}^n$.

It is proved that if ${\phi }_n$ is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form ${\sum }_{k=1}^{\infty }{c}_k{\phi }_k\left(x\right)$, where $c_k\in l_q$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f\in L_{\mu }^p[0,1]=f:{ʃ}_0^1{\left|f\left(x\right)\right|}^p\mu \left(x\right)dx<\infty$ there are numbers ${\varepsilon }_k$, k=1,2,…, ${\varepsilon }_k$ = 1 or 0, such that $li{m}_{n\to \infty }{ʃ}_0^1{|{\sum }_{k=1}^n{\varepsilon }_k{c}_k{\phi }_k\left(x\right)-f\left(x\right)|}^p\mu \left(x\right)dx=0.$ 2. For every p ∈ [1,2) and $f\in L_{\mu }^p[0,1]$ there are a function $g\in L^1[0,1]$ with g(x) = f(x) on E and numbers ${\delta }_k$, k=1,2,…, ${\delta }_k=1$ or 0,...

In this paper, the authors establish the phi-transform and wavelet characterizations for some Herz and Herz-type Hardy spaces by means of a local version of the discrete tent spaces at the origin.

L’auteur reprend l’étude classique de la représentation spectrale d’un opérateur auto-adjoint $A$ dans un espace de Hilbert $\mathscr{H}$. Il y ajoute des précisions nouvelles qui conduisent à la définition du projecteur infinitésimal $P^{\left)\lambda \right)}$ sur l’espace des vecteurs propres généralisés ${\mathscr{H}}^{\left(\lambda \right)}$. Il obtient, par conséquent, des énoncés plus précis de bien des théorèmes classiques. Il introduit ensuite la notion de “$A$-expansibilité” d’un sous-ensemble $S\subset \mathscr{H}$. Cette notion est appliquée à l’étude des espaces fonctionnels...

It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space $H_{p,q}$ to $L_{p,q}$ (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type $(L_1,L_1)$. As a consequence we show that the dyadic integral of a ∞ function $f\in L_1$ is dyadically differentiable and its derivative is f a.e.