Schr\"{o}dinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter

We study various direct and inverse spectral problems for the one-dimensional Schr\"{o}dinger equation with distributional potential and boundary conditions containing the eigenvalue parameter.Comment: 28 pages, very minor corrections, published version. arXiv admin note: text overlap with arXiv:1708.0749

Boundedness of fractional maximal operator and its commutators on generalized Orlicz-Morrey spaces

We consider generalized Orlicz-Morrey spaces $M_{\Phi,\varphi}(\mathbb{R}^{n})$ including their weak versions $WM_{\Phi,\varphi}(\mathbb{R}^{n})$. We find the sufficient conditions on the pairs $(\varphi_{1},\varphi_{2})$ and $(\Phi, \Psi)$ which ensures the boundedness of the fractional maximal operator $M_{\alpha}$ from $M_{\Phi,\varphi_1}(\mathbb{R}^{n})$ to $M_{\Psi,\varphi_2}(\mathbb{R}^{n})$ and from $M_{\Phi,\varphi_1}(\mathbb{R}^{n})$ to $WM_{\Psi,\varphi_2}(\mathbb{R}^{n})$. As applications of those results, the boundedness of the commutators of the fractional maximal operator $M_{b,\alpha}$ with $b \in BMO(\mathbb{R}^{n})$ on the spaces $M_{\Phi,\varphi}(\mathbb{R}^{n})$ is also obtained. In all the cases the conditions for the boundedness are given in terms of supremal-type inequalities on weights $\varphi(x,r)$, which do not assume any assumption on monotonicity of $\varphi(x,r)$ on $r$.Comment: 23 pages. Complex Anal. Oper. Theory (to appear). arXiv admin note: substantial text overlap with arXiv:1310.660

On the Riesz potential and its commutators on generalized Orlicz-Morrey spaces

We consider generalized Orlicz-Morrey spaces M_{\Phi,\varphi}(\Rn) including their weak versions WM_{\Phi,\varphi}(\Rn). In these spaces we prove the boundedness of the Riesz potential from M_{\Phi,\varphi_1}(\Rn) to M_{\Psi,\varphi_2}(\Rn) and from M_{\Phi,\varphi_1}(\Rn) to WM_{\Psi,\varphi_2}(\Rn). As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $(\varphi_{1},\varphi_{2})$, which do not assume any assumption on monotonicity of $\varphi_{1}(x,r)$, $\varphi_{2}(x,r)$ in r.Comment: 23 pages. J. Funct. Spaces Appl.(to appear

Maximal, potential and singular operators in the local "complementary" variable exponent Morrey type spaces

We consider local "complementary" generalized Morrey spaces M-c({x0})p(.).omega (Omega) in which the p-means of function are controlled over Omega \ B(x(0), r) instead of B(x(0), r), where Omega subset of R-n is a bounded open set, p(x) is a variable exponent, and no monotonicity type condition is imposed onto the function omega(r) defining the "complementary" Morrey-type norm. In the case where omega is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev type M-c({x0})p(.).omega (Omega) -> M-c({x0})p(.).omega (Omega)-theorem for the potential operators I-alpha(.), also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities-on omega(r), which do not assume any assumption on monotonicity of omega(r).Science Development Foundation under the President of the Republic of Azerbaijan [EIF-2010-1(1)-40/06-1]; Scientific and Technological Research Council of Turkey (TUBITAK) [110T695]info:eu-repo/semantics/publishedVersio