We determine the norm in $L^p\left(ℝ₊\right)$, 1 < p < ∞, of the operator $I-{ℱ}_s{ℱ}_c$, where ${ℱ}_c$ and ${ℱ}_s$ are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the $L^p$-norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real a,b. Best...

We investigate best constants for inequalities between the Orlicz norm and Luxemburg norm in Orlicz spaces.

Let $S^{\text{'}}$ be the class of tempered distributions. For $f\in S^{\text{'}}$ we denote by $J^{-\alpha }f$ the Bessel potential of $f$ of order $\alpha$. We prove that if $J^{-\alpha }f\in \mathrm{B}MO$, then for any $\lambda \in (0,1)$, $J^{-\alpha }{\left(f\right)}_{\lambda }\in \mathrm{B}MO$, where ${\left(f\right)}_{\lambda }={\lambda }^{-n}f\left(\phi ({\lambda }^{-1}·)\right)$, $\phi \in S$. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order $\alpha >0$ belongs to the $\mathrm{V}MO$ space.

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.