Let v be a standard weight on the upper half-plane , i.e. v: → ]0,∞[ is continuous and satisfies v(w) = v(i Im w), w ∈ , v(it) ≥ v(is) if t ≥ s > 0 and $li{m}_{t\to 0}v\left(it\right)=0$. Put v₁(w) = Im wv(w), w ∈ . We characterize boundedness and surjectivity of the differentiation operator D: Hv() → Hv₁(). For example we show that D is bounded if and only if v is at most of moderate growth. We also study composition operators on Hv().

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 $1\le q<p<\infty$, $0\le m<n$.

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...

We consider Littlewood-Paley functions associated with a non-isotropic dilation group on ${ℝ}^n$. We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted $L^p$ spaces, $1<p<\infty$, with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ to a quasi-Banach space ℬ if and only if sup${\left|\right|T\left(a\right)\left|\right|}_{ℬ}$: a is an infinitely differentiable (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ < ∞, where the (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ is as defined by Han, Paluszyński and Weiss.