The purpose of this paper is to provide a method of reduction of some problems concerning families $A_t={\left(A\left(t\right)\right)}_{t\in }$ of linear operators with domains ${{(}_t)}_{t\in }$ to a problem in which all the operators have the same domain . To do it we propose to construct a family ${\left({\Psi }_t\right)}_{t\in }$ of automorphisms of a given Banach space X having two properties: (i) the mapping $t\mapsto {\Psi }_t$ is sufficiently regular and (ii) ${\Psi }_t\left(\right){=}_t$ for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition with a parameter;...

In this paper, the authors introduce a kind of local Hardy spaces in Rn associated with the local Herz spaces. Then the authors investigate the regularity in these local Hardy spaces of some nonlinear quantities on superharmonic functions on R2. The main results of the authors extend the corresponding results of Evans and Müller in a recent paper.

We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the ${\ell }_{\infty }$-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily...

The relationship between the spectral properties of the transfer operator corresponding to a wavelet refinement equation and the $L^p$-Sobolev regularity of solution for the equation is established.

In this paper we consider the regularity problem for the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j{)}_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${ℝ}^n$. More precisely, we prove that these commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded from $L^p$ into the Besov space ${\dot{B}}_p^{s,p}$ for $1\<p\<+\infty$ and $0\<s\<1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space ${\dot{F}}_{\infty }^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space ${\dot{H}}^s$ if and only if $b$ is in the $BMO$-Sobolev...

For s>0, we consider bounded linear operators from $D\left({ℝ}^n\right)$ into $D^{\text{'}}\left({ℝ}^n\right)$ whose kernels K satisfy the conditions $|{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|}$ for x≠y, |γ|≤ [s]+1, $|{\nabla }_y{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|-1}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^2\left({ℝ}^n\right)$ into the homogeneous Sobolev space ${Ḣ}^s\left({ℝ}^n\right)$. This is an extension of the well-known T(1) Theorem due to David and Journé. Our arguments make use of the function T(1) and the BMO-Sobolev space. We give some applications to the Besov and Triebel-Lizorkin spaces as well as some other potential...

We prove inclusion relations between generalizing Waterman's and generalized Wiener's classes for functions of two variable.

We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator $I_{\alpha }$ maps weak weighted Orlicz$-\phi$ spaces into appropriate weighted versions of the spaces $BM{O}_{\psi }$, where $\psi \left(t\right)=t^{\alpha /n}{\phi }^{-1}(1/t)$. This generalizes known results about boundedness of $I_{\alpha }$ from weak $L^p$ into Lipschitz spaces for $p>n/\alpha$ and from weak $L^{n/\alpha }$ into $BMO$. It turns out that the class of weights corresponding to $I_{\alpha }$ acting on weak$-L_{\phi }$ for $\phi$ of lower type equal or greater than $n/\alpha$, is the same as the one solving the problem for weak...

Dans le chapitre I on indique la croissance de $\omega$ et de la fonction convexe $C$ pour que de $log\left|F\right(z\left)\right|\le \pi \left|y\right|-\omega \left(\right|y\left|\right)$$(y=\mathrm{Im}z)...$

We prove F. Riesz’ inequality assuming the boundedness of the norm of the first arithmetic mean of the functions ${\left|\phi ₙ\right|}^p$ with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.