The spaces of multi-Morrey type for positive Radon measures satisfying a growth condition on ${ℝ}^d$ are introduced. After defining the spaces, we investigate the multilinear maximal function, the multilinear fractional integral operator and the multilinear Calderón-Zygmund operators, respectively, from multi-Morrey spaces to Morrey spaces.

Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^p\left(\nu \right)$, $T₁,...,T_k$, $n̅=(n₁,...,n_k)\in {\mathbb{N}}^k$ and $\alpha ̅=(\alpha ₁,...,{\alpha }_k)$ with $0<{\alpha }_j\le 1$, we define the ergodic Cesàro-α̅ averages ${}_{n̅,\alpha ̅}f=1/\left({\prod }_{j=1}^k{A}_{n_j}^{{\alpha }_j}\right){\sum }_{i_k=0}^{n_k}\cdots {\sum }_{i₁=0}^{n₁}{\prod }_{j=1}^k{A}_{n_j-i_j}^{{\alpha }_j-1}{T}_k^{i_k}\cdots T{₁}^{i₁}f$. For these averages we prove the almost everywhere convergence on X and the convergence in the $L^p\left(\nu \right)$ norm, when $n₁,...,n_k\to \infty$ independently, for all $f\in L^p\left(d\nu \right)$ with p > 1/α⁎ where $\alpha ⁎=mi{n}_{1\le j\le k}{\alpha }_j$. In the limit case p = 1/α⁎, we prove that the averages ${}_{n̅,\alpha ̅}f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $\Lambda (1/\alpha ⁎,{\phi }_{m-1})$ with $\phi ₘ\left(t\right)=t{(1+log⁺t)}^m$. To obtain the result in the limit case we need to study...

We prove that classical Coifman-Meyer theorem holds on any polidisc Td or arbitrary dimension d ≥ 1.

We prove $L^p$-boundedness for a class of singular integral operators and maximal operators associated with a general $k$-parameter family of dilations on ${\mathbf{R}}^n$. This class includes homogeneous operators defined by kernels supported on homogeneous manifolds. For singular integrals, only certain “minimal” cancellation is required of the kernels, depending on the given set of dilations.

We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(\alpha ₁,...,{\alpha }_N)$ for some $0<\alpha ₁,...,{\alpha }_N<1$ and its marginal functions satisfy $f(·,x₂,...,x_N)\in Lip\beta ₁,...,f(x₁,...,x_{N-1},·)\in Lip{\beta }_N$ for some $0<\beta ₁,...,{\beta }_N<1$ uniformly in the indicated variables $x_l$, 1 ≤ l ≤ N, then $f{̃}^{(\eta ₁,...,{\eta }_N)}\in Lip(\alpha ₁,...,{\alpha }_N)$ for each choice of $(\eta ₁,...,{\eta }_N)$ with ${\eta }_l=0$ or 1 for 1 ≤ l ≤ N.

Maximal functions written as convolution with a multiparametric family of positive measures, and singular integrals whose kernel is decomposed as a multiple series of measures, are shown to be bounded in $L^p$, $1\&lt;p\&lt;\infty$. The proofs are based on the decomposition of the operators according to the size of the Fourier transform of the measures, assuming some regularity at zero and decay at infinity of these Fourier transforms. Applications are given to homogeneous singular integrals in product spaces with size...

We study regularity properties of a positive measure in the euclidean space in terms of two square functions which are the multiplicative analogues of the usual martingale square function and of the Lusin area function of a harmonic function. The size of ...

Extension by integer translates of compactly supported function for multiplier spaces on periodic Hardy spaces to multiplier spaces on Hardy spaces is given. Shannon sampling theorem is extended to Hardy spaces.

The author proves the boundedness for a class of multiplier operators on product spaces. This extends a result obtained by Lung-Kee Chen in 1994.

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)$$(\lambda \in C(...$

We study multipliers M (bounded operators commuting with translations) on weighted spaces L ω p (ℝ), and establish the existence of a symbol µM for M, and some spectral results for translations S t and multipliers. We also study operators T on the weighted space L ω p (ℝ+) commuting either with the right translations S t , t ∈ ℝ+, or left translations P +S −t , t ∈ ℝ+, and establish the existence of a symbol µ of T. We characterize completely the spectrum σ(S t ) of the operator S t proving that...