Integral operators commuting with dilations and rotations in generalized Morrey-type spaces

We find conditions for the boundedness of integral operators $K$ commuting with dilations and rotations in a local generalized Morrey space. We also show that under the same conditions, these operators preserve the subspace of such Morrey space, known as vanishing Morrey space. We also give necessary conditions for the boundedness when the kernel is non-negative. In the case of classical Morrey spaces, the obtained sufficient and necessary conditions coincide with each other. In the one-dimensional case, we also obtain similar results for global Morrey spaces. In the case of radial kernels, we also obtain stronger estimates of $Kf$ via spherical means of $f$. We demonstrate the efficiency of the obtained conditions for a variety of examples such as weighted Hardy operators, weighted Hilbert operator, their multidimensional versions, and others.     

‐Theory for Schrödinger systems

In this article, we study for the ‐realization of the vector‐valued Schrödinger operator . Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, we prove that the ‐realization of , defined on the intersection of the natural domains of the differential and multiplication operators which form , generates a strongly continuous contraction semigroup on . We also study additional properties of the semigroup such as extension to L¹, positivity, ultracontractivity and prove that the generator has compact resolvent.     

A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

In this article, we study the eigenvalues of p‐fractional Hardy operator where , , , and Ω is an unbounded domain in with Lipschitz boundary containing 0. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is uniquely associated to a nonnegative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues as .     

Non‐self‐adjoint singular second‐order dynamic operators on time scale

In this study, maximal dissipative second‐order dynamic operators on semi‐infinite time scale are studied in the Hilbert space , that the extensions of a minimal symmetric operator in limit‐point case. We construct a self‐adjoint dilation of the dissipative operator together with its incoming and outgoing spectral representations so that we can determine the scattering function of the dilation as stated in the scheme of Lax‐Phillips. Moreover, we construct a functional model of the dissipative operator and identify its characteristic function in terms of the Weyl‐Titchmarsh function of a self‐adjoint second‐order dynamic operator. Finally, we prove the theorems on completeness of the system of root functions of the dissipative and accumulative dynamic operators.     

Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds

Let M be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form ${\mathbb{R}}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by $L_{T}f=-\text{div}(T\mathrm{\nabla }f)$, where T is a general symmetric, positive definite and divergence-free $(1,1)$-tensor on M. The upper bound is given in terms of an integration involving tr T and $|H_T{|}^2$, where tr T is the trace of the tensor T and $H_T={\sum }_{i=1}^{n}A(T{e}_i,e_i)$ is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator $L_T$ can be regarded as a natural generalization of the well-known operator $L_r$ which is the linearized operator of the first variation of the $(r+1)$-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension.     

The differentiation operator in the space of uniformly convergent Dirichlet series

Continuity, compactness, the spectrum and ergodic properties of the differentiation operator are investigated, when it acts in the Fréchet space of all Dirichlet series that are uniformly convergent in all half-planes for each . The properties of the formal inverse of the differentiation are also investigated.     

A generalization of Hardy’s operator and an asymptotic Müntz–Szász Theorem

We prove an asymptotic version of the Muntz–Szasz theorem, and use it to prove that all monomial operators are vanishing preserving.