Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.

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.

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).

A typical wavelet system constitutes an unconditional basis for various function spaces -Lebesgue, Besov, Triebel-Lizorkin, Hardy, BMO. One of the main reasons is the frequency localization of an element from such a basis. In this paper we study a wavelet-type system, called a brushlet system. In [3] it was noticed that brushlets constitute unconditional bases for classical function spaces such as the Triebel-Lizorkin and Besov spaces. In this paper we study brushlet expansions of functions in the...