Boundedness and Compactness of Hankel Operators on Large Fock Space

We introduce the BMO spaces and use them to characterize complex-valued functions f such that the big Hankel operators H f and H f ¯ are both bounded or compact from a weighted large Fock space F p ϕ into a weighted Lebesgue space L p ϕ when 1 ≤ p < ∞ .

On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space

In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side.

Неравенство Бернштейна - Никольского в весовых пространствах Лебега

In this paper, we give some results concerning Bernstein--Nikol'skii inequality for weighted Lebesgue spaces. The main result is as follows: Let $1 < u,p < \infty$, $0<q+ 1/p <v + 1/u <1,$ $v-q\geq 0$, $\kappa >0$, $f \in L^u_v(\R)$ and $\supp\widehat{f} \subset [-\kappa, \kappa]$. Then $D^mf \in L^p_q(\R)$, $\supp\widehat{D^m f}=\supp\widehat{f}$ and there exists a~constant~$C$ independent of $f$, $m$, $\kappa$ such that $\|D^mf\|_{L^p_{q}} \leq C m^{-\varrho} \kappa^{m+\varrho} \|f\|_{ L^u_v}, $ for all $m = 1,2,\dots $, where $\varrho=v + \frac{1}{u} -\frac{1}{p} - q>0,$ and the weighted Lebesgue space $L^p_q$ consists of all measurable functions such that $\|f\|_{L^p_q} = \big(\int_{\R} |f(x)|^p |x|^{pq} dx\big)^{1/p} < \infty.$ Moreover, $ \lim_{m\to \infty}\|D^mf\|_{L^p_{q}}^{1/m}= \sup \big\{ |x|: \, x \in \textnormal{supp}\widehat{f}\big \}.$ The~advantage of our result is that $m^{-\varrho}$ appears on the right hand side of the inequality ($\varrho >0$), which has never appeared in related articles by other authors. The corresponding result for the $n$-dimensional case is also obtained.