Analysis and Mathematical Physics最新文献

Let (nge 1,0<rho <1, max {rho ,1-rho }le delta le 1) and

If the amplitude a belongs to the Hörmander class (S^{m_1}_{rho ,delta }) and (phi in Phi ^{2}) satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator (T_{phi ,a}) defined by

is bounded from the local Hardy space (h^1({mathbb {R}}^n)) to (L^1({mathbb {R}}^n)). As a corollary, we can also obtain the corresponding (L^p({mathbb {R}}^n))-boundedness when (1<p<2). These theorems are rigorous improvements on the recent works of Staubach and his collaborators. When (0le rho le 1,delta le max {rho ,1-rho }), by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results.

We prove the boundedness of the fractional maximal operator and the Riesz potential operator on variable exponent Stummel spaces. The main results rely on refined uniform weighted inequalities involving special weights with non-standard growth.

We find new limits of the Lie product formula type in Banach algebras with unit. Some sample results: Let X, Y, Z be Banach algebras with unit, ( left( x_{n},y_{n}right) _{nin mathbb {N}}subset Xtimes Y) convergent sequences with (lim nolimits _{nrightarrow infty }x_{n}=x), ( lim nolimits _{nrightarrow infty }y_{n}=y) and (T:Xtimes Yrightarrow Z) a continuous bilinear operator with (Tleft( textbf{1},textbf{1}right) = textbf{1}). Then for all sequences of natural numbers (left( a_{n}right) _{nin mathbb {N}}) with (lim nolimits _{nrightarrow infty }a_{n}=infty ) we have

We consider the magnetic Dirichlet Laplacian with constant magnetic field on domains of finite measure. First, in the case of a disk, we prove that the eigenvalue branches with respect to the field strength behave asymptotically linear with an exponentially small remainder term as the field strength goes to infinity. We compute the asymptotic expression for this remainder term. Second, we show that for sufficiently large magnetic field strengths, the spectral bound corresponding to the Pólya conjecture for the non-magnetic Dirichlet Laplacian is violated up to a sharp excess factor which is independent of the domain.

In this paper, we introduce a notion of multiplicative unimodularity for a coisotropic Poisson homogeneous space. Then, we discuss the unimodularity and the multiplicative unimodularity for these spaces and the existence of an invariant volume form for explicit Hamiltonian systems on such spaces. Several interesting examples illustrating the theoretical results are also presented.

The purpose of this paper is to study the boundedness of solutions of the Chern-Simons-Higgs equation

and system

where (gamma >0), (beta ) is a bounded continuous function and (Delta _{lambda }) is the strongly degenerate operator defined by

Under some general hypotheses of (lambda _i), we shall establish some boundedness properties of solutions of the equation and system above. Our result can be seen as an extension of that in [Li, Yayun; Lei, Yutian, Boundedness for solutions of equations of the Chern-Simons-Higgs type. Appl. Math. Lett.88(2019), 8-12.]. In addition, we provide a simple proof of the boundedness of solutions.

We consider the motion of charged test particles in the presence of a Dirac magnetic monopole. We use an extension of Noether’s theorem for systems with magnetic forces and integrate explicitly the corresponding equations of motion.

We study the magnetic Laplacian in a two-dimensional exterior domain with Neumann boundary condition and uniform magnetic field. For the exterior of the disk we establish accurate asymptotics of the low-lying eigenvalues in the weak magnetic field limit. For the exterior of a star-shaped domain, we obtain an asymptotic upper bound on the lowest eigenvalue in the weak field limit, involving the (4)-moment, and optimal for the case of the disk. Moreover, we prove that, for moderate magnetic fields, the exterior of the disk is a local maximizer for the lowest eigenvalue under a (p)-moment constraint.

Properties of Riesz capacity are developed with respect to the kernel exponent (p in (-infty ,n)), namely that capacity is strictly monotonic as a function of p, that its endpoint limits recover the diameter and volume of the set, and that capacity is left-continuous with respect to p and is right-continuous provided (when (p ge 0)) that an additional hypothesis holds. Left and right continuity properties of the equilibrium measure are obtained too.

In the theory of (g_alpha )-potentials on a domain (Dsubset mathbb R^n), (ngeqslant 2), (g_alpha ) being the (alpha )-Green kernel associated with the (alpha )-Riesz kernel (|x-y|^{alpha -n}) of order (alpha in (0,n)), (alpha leqslant 2), we establish the existence and uniqueness of the (g_alpha )-balayage (mu ^F) of a positive Radon measure (mu ) onto a relatively closed set (Fsubset D), we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for (mu ^F(D)=mu (D)) to hold, given in terms of the (alpha )-harmonic measure of suitable Borel subsets of (overline{mathbb R^n}), the one-point compactification of (mathbb R^n). As a by-product, we find necessary and/or sufficient conditions for the existence of the (g_alpha )-equilibrium measure (gamma _F), (gamma _F) being understood in an extended sense where (gamma _F(D)) might be infinite. We also discover quite a surprising version of Deny’s principle of positivity of mass for (g_alpha )-potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann Acad Sci Fenn Math 43:121–145, 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed.