In this paper we generalize some results concerning bounded variation functions on sequence spaces.

We consider discrete versions of Morrey spaces introduced by Gunawan et al. in papers published in 2018 and 2019. We prove continuity and compactness of multiplication operators and commutators acting on them.

We consider a new Sobolev type function space called the space with multiweighted derivatives $W_{p,\overline{\alpha }}^n$, where $\overline{\alpha }=({\alpha }_0,{\alpha }_1,...,{\alpha }_n)$, ${\alpha }_i\in ℝ$, $i=0,1,...,n$, and ${\parallel f\parallel }_{W_{p,\overline{\alpha }}^n}=\parallel D_{\overline{\alpha }}^n{f\parallel }_p+{\sum }_{i=0}^{n-1}\left|D_{\overline{\alpha }}^{i}f\left(1\right)\right|$, $$D_{\overline{\alpha }}^{0}f\left(t\right)=t^{{\alpha }_0}f\left(t\right),D_{\overline{\alpha }}^{i}f\left(t\right)=t^{{\alpha }_i}\frac{\mathrm{d}}{\mathrm{d}t}{D}_{\overline{\alpha }}^{i-1}f\left(t\right),i=1,2,...,n.$$ We establish necessary and sufficient conditions for the boundedness and compactness of the embedding $W_{p,\overline{\alpha }}^n\hookrightarrow W_{q,\overline{\beta }}^m$, when $1\le q<p<\infty$, $0\le m<n$.

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.

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.

We are concerned with the boundedness of generalized fractional integral operators $I_{\rho ,\tau }$ from Orlicz spaces $L^{\Phi }\left(X\right)$ near $L^1\left(X\right)$ to Orlicz spaces $L^{\Psi }\left(X\right)$ over metric measure spaces equipped with lower Ahlfors $Q$-regular measures, where $\Phi$ is a function of the form $\Phi \left(r\right)=r\ell \left(r\right)$ and $\ell$ is of log-type. We give a generalization of paper by Mizuta et al. (2010), in the Euclidean setting. We deal with both generalized Riesz potentials and generalized logarithmic potentials.

We describe a class O of nonlinear operators which are bounded on the Lizorkin-Triebel spaces Fsp,q(Rn), for 0 &lt; s &lt; 1 and 1 &lt; p, q &lt; ∞. As a corollary, we prove that the Hardy-Littlewood maximal operator is bounded on Fsp,q(Rn), for 0 &lt; s &lt; 1 and 1 &lt; p, q &lt; ∞ ; this extends the result of Kinnunen (1997), valid for the Sobolev space H1p(Rn).

The family of block spaces with variable exponents is introduced. We obtain some fundamental properties of the family of block spaces with variable exponents. They are Banach lattices and they are generalizations of the Lebesgue spaces with variable exponents. Moreover, the block space with variable exponents is a pre-dual of the corresponding Morrey space with variable exponents. The main result of this paper is on the boundedness of the Hardy-Littlewood maximal operator on the block space with...

In this short note we consider necessary and sufficient conditions on normed linear spaces, that ensure the boundedness of any linear map whose adjoint maps extreme points of the unit ball of the domain space to continuous linear functionals.