A remark on the Schrödinger propagator on Wiener amalgam spaces

In this paper, we study the boundedness of the Schrödinger propagator on Wiener amalgam spaces. In particular, we determine the necessary and sufficient conditions for the propagator to be bounded from to .     

Lindelöf theorems for monotone Sobolev functions in Orlicz spaces on uniform domains

In this paper, we are concerned with Lindelöf type theorems for monotone (in the sense of Lebesgue) Sobolev functions u on a uniform domain satisfying where ? denotes the gradient, denotes the distance from z to the boundary , φ is of log‐type and ω is a weight function satisfying the doubling condition.     

Multiple solutions for a class of quasilinear Schrödinger equations

In this paper, we study the following quasilinear Schrödinger equations of the form where , , . Some existence results for positive solutions, negative solutions and sequence of high energy solutions are obtained via a perturbation method.     

Weighted estimates for bilinear fractional integral operators

Moen (2016) proved weighted estimates for the bilinear fractional integrals where . We improve his results when and consider the case . As a corollary we obtain a bilinear Stein–Weiss inequality where .     

A T1 theorem and Calderón–Zygmund operators in Campanato spaces on domains

Given a Lipschitz domain , a Calderón–Zygmund operator T and a modulus of continuity , we solve the problem when the truncated operator sends the Campanato space into itself. The solution is a T1 type sufficient and necessary condition for the characteristic function of D: To check the hypotheses of T1 theorem we need extra restrictions on both the boundary of D and the operator T. It is proved that the truncated Calderón–Zygmund operator with an even kernel is bounded on , provided D is a ‐smooth domain.     

Scaling of spectra of a class of self‐similar measures on R

Let be two positive integers. For , let the self‐similar measure be defined by . It is known [18] that is a spectral measure with a spectrum where . In this paper, we give some conditions on under which the scaling set is also a spectrum of .     

Spectral statistics of random Schrödinger operator with growing potential

In this work we investigate the spectral statistics of random Schrödinger operators acting on where are i.i.d random variables distributed uniformly on [0,1].     

Singular spectral shift function for resolvent comparable operators

Let be a Schrödinger operator on , , or 3, where is a bounded measurable real‐valued function on . Let V be an operator of multiplication by a bounded integrable real‐valued function and put for real r. We show that the associated spectral shift function (SSF) ξ admits a natural decomposition into the sum of absolutely continuous and singular SSFs. In particular, the singular SSF is integer‐valued almost everywhere, even within the absolutely continuous spectrum where the same cannot be said of the SSF itself. This is a special case of an analogous result for resolvent comparable pairs of self‐adjoint operators, which generalises the case of a trace class perturbation appearing in [2] while also simplifying its proof. We present two proofs which demonstrate the equality of the singular SSF with two a priori different and intrinsically integer‐valued functions which can be associated with the pair H₀, V: the total resonance index [3] and the singular μ‐invariant [2].     

The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system

In this paper, we study the following critical fractional Schrödinger–Poisson system where is a small parameter, and , is the fractional critical exponent for 3‐dimension, has a positive global minimum, and are positive and have global maximums. We obtain the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials and Q as the concentration position of these ground state solutions as . Moreover, we consider some properties of these ground state solutions, such as convergence and decay estimate.     

Embedding Banach spaces into the space of bounded functions with countable support

We prove that a WLD subspace of the space consisting of all bounded, countably supported functions on a set Γ embeds isomorphically into if and only if it does not contain isometric copies of . Moreover, a subspace of is constructed that has an unconditional basis, does not embed into , and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of ).     

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 .     

Variable Hardy–Lorentz spaces

Let be a measurable function on with . We introduce the variable Hardy–Lorentz space for via the radial grand maximal function. Under the assumption that satisfies the log‐Hölder condition, we establish a version of Fefferman–Stein vector‐valued inequality in variable Lorentz space by interpolation. We also construct atomic decompositions for , and develop a theory of real interpolation and formulate the dual space of the variable Hardy–Lorentz space with and . As a byproduct, we obtain a new John–Nirenberg theorem. Furthermore, we get equivalent characterizations of the variable Hardy–Lorentz space by means of the Lusin area function, the Littlewood–Paley g‐function and the Littlewood–Paley ‐function. Finally, we investigate the boundedness of singular operators on for and .     

‐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.     

Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

We study solutions of the problem (0.1) where are open sets such that , , and f is a nonlinearity. Under different assumptions on f we prove that, if D₀ and D₁ are starshaped with respect to the same point , then the same occurs for every superlevel set of u.     

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L² restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived.     

A Szegő limit theorem for translation‐invariant operators on polygons

We prove Szeg?‐type trace asymptotics for translation‐invariant operators on polygons. More precisely, consider a Fourier multiplier on with a sufficiently decaying, smooth symbol . Let be the interior of a polygon and, for , define its scaled version . Then we study the spectral asymptotics for the operator , the spatial restriction of A onto : for entire functions h with we provide a complete asymptotic expansion of as . These trace asymptotics consist of three terms that reflect the geometry of the polygon. If P is replaced by a domain with smooth boundary, a complete asymptotic expansion of the trace has been known for more than 30 years. However, for polygons the formula for the constant order term in the asymptotics is new. In particular, we show that each corner of the polygon produces an extra contribution; as a consequence, the constant order term exhibits an anomaly similar to the heat trace asymptotics for the Dirichlet Laplacian.     

A positive quantization on type I locally compact groups

Let be a unimodular type I second countable locally compact group and let be its unitary dual. Motivated by a recent pseudo‐differential calculus, we develop a positive Berezin‐type quantization with operator‐valued symbols defined on .     

Keller–Osserman estimates and removability result for the anisotropic porous medium equation with gradient absorption term

We study “large” nonnegative solutions for a class of quasilinear equations model of which is We give a sufficient condition on the exponents and for the removability of isolated singularities.     

Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

We introduce a new scale of grand variable exponent Lebesgue spaces denoted by . These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in . The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.     

There are eight‐element orthogonal exponentials on the spatial Sierpinski gasket

The self‐affine measure corresponding to an expanding matrix and the digit set in the space is supported on the spatial Sierpinski gasket, where are the standard basis of unit column vectors in and . In the case and , it is conjectured that the cardinality of orthogonal exponentials in the Hilbert space is at most “4”, where the number 4 is the best upper bound. That is, all the four‐element sets of orthogonal exponentials are maximal. This conjecture has been proved to be false by giving a class of the five‐element orthogonal exponentials in . In the present paper, we construct a class of the eight‐element orthogonal exponentials in the corresponding Hilbert space to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.