We study when a weighted composition operator acting between different weighted Bergman spaces is bounded, resp. compact.

The purpose of this article is to obtain a multidimensional extension of Lacey and Thiele's result on the boundedness of a model sum which plays a crucial role in the boundedness of the bilinear Hilbert transform in one dimension. This proof is a simplification of the original proof of Lacey and Thiele modeled after the presentation of Bilyk and Grafakos.

The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, $T_b^m$, on Herz spaces $K{̇}_q^{\alpha ,p}\left(ℝⁿ\right)$ and $K_q^{\alpha ,p}\left(ℝⁿ\right)$, and on a new class of Herz-type Hardy spaces $HK{̇}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$ and $H{K}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$, where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

We study boundedness in Orlicz norms of convolution operators with integrable kernels satisfying a generalized Lipschitz condition with respect to normal quasi-distances of ℝⁿ and continuity moduli given by growth functions.

In this paper, the boundedness of the Riesz potential generated by generalized shift operator $I_{B_k}^{\alpha }$ from the spaces $a=(L_{p_m,\nu }\left({ℝ}_n^k\right),a_m)$ to the spaces $a^{\text{'}}=(L_{q_m,\nu }\left({ℝ}_n^k\right),a_m^{\text{'}})$ is examined.

Some boundedness results are established for sublinear operators on the homogeneous Herz spaces. As applications, some new theorems about the boundedness on homogeneous Herz spaces for commutators of singular integral operators are obtained.

The main purpose of this paper is to investigate the behavior of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non-doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on Euclidean space and their known extensions for doubling...

Let $f$ be a non-zero positive vector of a Banach lattice $L$, and let $T$ be a positive linear operator on $L$ with the spectral radius $r\left(T\right)$. We find some groups of assumptions on $L$, $T$ and $f$ under which the inequalities $$sup\{c\ge 0:Tf\ge cf\}\le r\left(T\right)\le inf\{c\ge 0:Tf\le cf\}$$ hold. An application of our results gives simple upper and lower bounds for the spectral radius of a product of positive operators in terms of positive eigenvectors corresponding to the spectral radii of given operators. We thus extend the matrix result obtained by Johnson and Bru which...