Related papers: Correct Solvability, Embedding Theorems and Separa…

In this concise article, we compute the weight decomposition of $\mathfrak{sl}_d(\mathbb R)$ with respect to the adjoint representation of $\mathfrak{so}(p,q)$, where $d=p+q$ and demonstrate in detail that $\mathfrak{sl}_d(\mathbb R)$…

We study the Sturm-Liouville problem $-y''-\rho y=0$, $y(0)=y(1)=0$. $\rho$ is a generalized derivative of function $P\in L_2[0,1]$. For self-similar $P$ asymptotic formulas for eigenvalues are obtained. In this paper we consider two cases…

In this paper we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations in weighted Sobolev spaces. We also provide uniqueness and existence results for second-order elliptic and parabolic partial…

The aim of this work is to prove a compact embedding for a weighted fractional Sobolev spaces. As an application, we use this embedding to prove, via variational methods, the existence of solutions for the following Schr\"odinger equation…

We found hermitian realizations of the position vector $\vec{r}$, angular momentum $\vec{\Lambda}$ and linear momentum $\vec{p}$ behaving like vectors with respect to the $SU_q(2)$ algebra, generated by $L_0$ and $L_\pm$. They are used to…

In this paper we consider the recurrent equation $$\Lambda_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)\Lambda_{q}\Lambda_{p+1-q}$$ for $p\ge 1$ with $f\in C[0,1]$ and $\Lambda_1=y>0$ given. We give conditions on $f$ that guarantee…

In this paper, we for the first time prove local solvability and stability of the inverse Sturm-Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The proof…

In this study, the theorem on necessary and sufficient conditions for the solvability of inverse problem for Sturm-Liouville operator with discontinuous coefficient is proved and the algorithm of reconstruction of potential from spectral…

Given any $1 < q \le 2$, we use new free probability techniques to construct a completely isomorphic embedding of $\ell_q$ (equipped with its natural operator space structure) into the predual of a sufficiently large QWEP von Neumann…

We revisit basics of classical Sturm-Liouville theory and, as an application, recover Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions by a new argument. We also outline how a wider class…

We obtain necessary and sufficient conditions on weights for a wide class of integral transforms to be bounded between weighted $L^p-L^q$ spaces, with $1\leq p\leq q\leq \infty$. The kernels $K(x,y)$ of such transforms are only assumed to…

Let $L^{m,p}(\R^n)$ be the Sobolev space of functions with $m^{th}$ derivatives lying in $L^p(\R^n)$. Assume that $n< p < \infty$. For $E \subset \R^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in…

The paper describes the splines $Q_{n,k}(x,a)$, which for an arbitrary point $a\in(0;1)$ and an arbitrary function $y\in\mathring{W}^n_p[0;1]$ set the relations $y^{(k)}(a)=\int_0^1 y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx$. The relation of the…

We consider a Sturm-Liouville equation $\ell y:=-y'' + qy = \lambda y$ on the intervals $(-a,0)$ and $(0,b)$ with $a,b>0$ and $q \in L^2(-a,b)$. We impose boundary conditions $y(-a)\cos\alpha = y'(-a)\sin\alpha$, $y(b)\cos\beta =…

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem \begin{equation}\label{ab eqn} \begin{cases} \partial_t u(t,x)=\psi(t,-i\nabla)u(t,x)+f(t,x),\quad &(t,x)\in(0,T)\times\mathbb{R}^d,\\…

We introduce and investigate symmetric operators $L_0$ associated in the complex Hilbert space $L^2(\mathbb{R})$ with a formal differential expression \[l[u] :=-(pu')'+qu + i((ru)'+ru') \] under minimal conditions on the regularity of the…

Let $V$ be a finite tree with radially decaying weights. We show that there exists a set $E \subset \mathbb{R}^2$ for which the following two problems are equivalent: (1) Given a (real-valued) function $\phi$ on the leaves of $V$, extend it…

In this work we consider the general functional-integral equation: \begin{equation*} y(t) = f\left(t, \int_{a}^{b} k(t,s)g(s,y(s))ds\right), \qquad t\in [a,b], \end{equation*} and give conditions that guarantee existence and uniqueness of…

In this paper we characterize the trace spaces of a class of weighted function spaces of intersection type with mixed regularities. To a large extent we can overcome the difficulty of mixed scales by employing a microscopic improvement in…

We study Liouville type of theorems for the Navier-Stokes and the Euler equations on $\Bbb R^N$, $N\geq 2$. Specifically, we prove that if a weak solution $(v,p)$ satisfies $|v|^2 +|p| \in L^1 (0,T; L^1(\Bbb R^N, w_1(x)dx))$ and $\int_{\Bbb…