Communications in Contemporary Mathematics

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz–Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.

We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case, it expresses an inversion principle of more limited scope, but still sufficient to compute the Möbius function as $\mu =\zeta \circ S$, just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors $S_{even}-S_{odd}$, and it is a refinement of the general Möbius inversion construction of Gálvez–Kock–Tonks, but exploiting the monoidal structure.

The discrete LS algebra over a totally ordered set is the homogeneous coordinate ring of an irreducible projective (normal) toric variety. We prove that this algebra is the ring of invariants of a finite abelian group containing no pseudo-reflection acting on a polynomial ring. This is used to study the Gorenstein property for LS algebras. Further we show that any LS algebra is Koszul.

We consider a class of Lévy-type processes derived via a Doob transform from Lévy processes conditioned by a control function called potential. These ground state transformed processes (also called $P{(\varphi )}_1$-processes) have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Lévy generators perturbed by the potential. We derive precise upper envelopes for the almost sure long-time behavior of these ground state-transformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.

A projective manifold $M$ is algebraically hyperbolic if there exists a positive constant $A$ such that the degree of any curve of genus $g$ on $M$ is bounded from above by $A(g-1)$. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here, we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

By $I$-method, the interaction Morawetz estimate, long-time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equation

Let $n\ge 3$, $\mathrm{\Omega }$ be a bounded (semi-)convex domain in ${ℝ}^n$ and the non-negative potential $V$ belong to the reverse Hölder class ${\mathrm{\text{RH}}}_n({ℝ}^n)$. Assume that $p\in (1,\infty )$ and $\omega \in A_p(\partial \mathrm{\Omega })$, where $A_p(\partial \mathrm{\Omega })$ denotes the Muckenhoupt weight class on $\partial \mathrm{\Omega }$, the boundary of $\mathrm{\Omega }$. In this paper, the authors show that, for any $p\in (1,\infty )$, the Neumann problem for the Schrödinger equation $-\mathrm{\Delta }u+Vu=0$ in $\mathrm{\Omega }$ with boundary data in (weighted) $L^p$ is uniquely solvable. The obtained results in this paper essentially improve the known results which are special cases of the results obtained by Shen [Indiana Univ. Math. J. 43 (1994) 143–176] and Tao and Wang [Canad. J. Math. 56 (2004) 655–672], via extending the range $p\in (1,2]$ of $p$ into $p\in (1,\infty )$.

We study bounds on the Riesz means of the mixed Steklov–Neumann and Steklov–Dirichlet eigenvalue problem on a bounded domain $\mathrm{\Omega }$ in ${ℝ}^n$. The Steklov–Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov–Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov–Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian, and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the Appendix which in particular show the asymptotic sharpness of the bounds we obtain.

In 1976, Brascamp and Lieb proved that the heat flow preserves logconcavity. In this paper, introducing a variation of concavity, we show that it preserves in fact a stronger property than logconcavity and we identify the strongest concavity preserved by the heat flow.

Let $L=-\mathrm{\text{div}}(A\nabla )$ be a second-order divergence form elliptic operator and $A$ an accretive, $n\times n$ matrix with bounded measurable complex coefficients in ${ℝ}^n.$ In this paper, we establish $L^p$ theory for the commutators generated by the fractional differential operators related to $L$ and bounded mean oscillation (BMO)–Sobolev functions.

We develop a constructive process which determines all extreme points of the unit ball in the space of $m$-linear forms, $m\ge 1.$ Our method provides a full characterization of the geometry of that space through finitely many elementary steps, and thus it can be extensively applied in both computational as well as theoretical problems; few consequences are also derived in this paper.

We prove that the Gibbs measures $\rho$ for a class of Hamiltonian equations written as