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Abstract: Abstract We consider a class of hypoelliptic operators of the following type $$\begin{aligned} {\mathcal {L}}=\sum \limits _{i,j=1}^{p_0} a_{ij} \partial _{x_i x_j}^2+\sum \limits _{i,j=1}^{N} b_{ij} x_i \partial _{x_j}-\partial _t, \end{aligned}$$ where \((a_{ij})\) , \((b_{ij})\) are constant matrices and \((a_{ij})\) is symmetric positive definite on \({\mathbb {R}}^{p_0}\) \((p_0\le N)\) . We obtain generalized Hölder estimates for \({\mathcal {L}}\) on \({\mathbb {R}}^{N+1}\) by establishing several estimates of singular integrals in generalized Morrey spaces. PubDate: 2024-07-10

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Abstract: Abstract This work addresses the resolvent convergence of generalized MIT bag operators to Dirac operators with zigzag type boundary conditions. We prove that the convergence holds in strong but not in norm resolvent sense. Moreover, we show that the only obstruction for having norm resolvent convergence is the existence of an eigenvalue of infinite multiplicity for the limiting operator. More precisely, we prove the convergence of the resolvents in operator norm once projected into the orthogonal of the corresponding eigenspace. PubDate: 2024-07-08

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Abstract: Abstract For G an open set in \({\mathbb {C}}\) and W a non-vanishing holomorphic function in G, in the late 1990’s, Pritsker and Varga (Constr Approx 14, 475-492 1998) characterized pairs (G, W) having the property that any f holomorphic in G can be locally uniformly approximated in G by weighted holomorphic polynomials \(\{W(z)^np_n(z)\}, \ deg(p_n)\le n\) . We further develop their theory in first proving a quantitative Bernstein-Walsh type theorem for certain pairs (G, W). Then we consider the special case where \(W(z)=1/(1+z)\) and G is a loop of the lemniscate \(\{z\in {\mathbb {C}}: z(z+1) =1/4\}\) . We show the normalized measures associated to the zeros of the \(n-th\) order Taylor polynomial about 0 of the function \((1+z)^{-n}\) converge to the weighted equilibrium measure of \({\overline{G}}\) with weight W as \(n\rightarrow \infty \) . This mimics the motivational case of Pritsker and Varga (Trans Amer Math Soc 349, 4085-4105 1997) where G is the inside of the Szegő curve and \(W(z)=e^{-z}\) . Lastly, we initiate a study of weighted holomorphic polynomial approximation in \({\mathbb {C}}^n, \ n>1\) . PubDate: 2024-07-08

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Abstract: Abstract We are interested in four-dimensional Dirac–Klein–Gordon equations, a fundamental model in particle physics. The main goal of this paper is to establish global existence of solutions to the coupled system and to explore their long-time behavior. The results are valid uniformly for mass parameters varying in the interval [0, 1]. PubDate: 2024-07-06

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Abstract: Abstract We exhaustively classify the Lie reductions of the real dispersionless Nizhnik equation to partial differential equations in two independent variables and to ordinary differential equations. Lie and point symmetries of reduced equations are comprehensively studied, including the analysis of which of them correspond to hidden symmetries of the original equation. If necessary, associated Lie reductions of a nonlinear Lax representation of the dispersionless Nizhnik equation are carried out as well. As a result, we construct wide families of new invariant solutions of this equation in explicit form in terms of elementary, Lambert and hypergeometric functions as well as in parametric or implicit form. We show that Lie reductions to algebraic equations lead to no new solutions of this equation in addition to the constructed ones. Multiplicative separation of variables is used for illustrative construction of non-invariant solutions. PubDate: 2024-07-05

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Abstract: Abstract We obtain a Lie theoretic intrinsic characterization of the connected and simply connected solvable Lie groups whose regular representation is a factor representation. When this is the case, the corresponding von Neumann algebras are isomorphic to the hyperfinite \(\textrm{II}_\infty \) factor, and every Casimir function is constant. We thus obtain a family of geometric models for the standard representation of that factor. Finally, we show that the regular representation of any connected and simply connected solvable Lie group with open coadjoint orbits is always of type \(\textrm{I}\) , though the group needs not be of type \(\textrm{I}\) , and include some relevant examples. PubDate: 2024-07-04

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Abstract: Abstract Let \(1<q\le p \le r\le \infty \) and \(\tau \in (0,\infty ]\) . Besov–Bourgain–Morrey spaces \({\mathcal {M}}\dot{B}^{p,\tau }_{q,r}({\mathbb {R}}^n)\) in the special case where \(\tau =r\) , extending what was introduced by J. Bourgain, have proved useful in the study related to the Strichartz estimate and the non-linear Schrödinger equation. In this article, by cleverly mixing the norm structures of grand Lebesgue spaces and Besov–Bourgain–Morrey spaces and adding an extra exponent \(\theta \in [0,\infty )\) , the authors introduce a new class of function spaces, called generalized grand Besov–Bourgain–Morrey spaces \({\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)\) . The authors explore their various real-variable properties including pre-dual spaces and the Gagliardo–Peetre and the ± interpolation theorems. Via establishing some equivalent quasi-norms of \({\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)\) related to Muckenhoupt \(A_1({\mathbb {R}}^n)\) -weights, the authors then obtain an extrapolation theorem of \({\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)\) . Applying this extrapolation theorem, the Calderón product, and the sparse family of dyadic grids of \({\mathbb {R}}^n\) , the authors establish the sharp boundedness on \({\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)\) of the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator. PubDate: 2024-06-20

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Abstract: Abstract This paper is concerned with a class of periodic Schrödinger lattice systems with spectrum 0 and saturable nonlinearities. The existence of ground state solitons of the systems under weak assumptions is obtained. The main novelties are as follows. (1) Some new sufficient conditions for the existence of ground state solitons under the “spectral endpoint” assumption are constructed. (2) Our “non-monotonic” conditions make the proofs of the boundedness of the (PS) sequences to be easier. (3) Our result extends and improves the related results in the literature. Besides, some examples are given to illuminate our result. PubDate: 2024-06-20

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Abstract: Abstract This paper explores strongly parabolic-elliptic systems within Orlicz–Sobolev spaces. It introduces the concept of capacity solutions and emphasizes the establishment of existence and regularity of solutions through rigorous proofs. Specifically, it addresses the existence of capacity solutions for a strongly nonlinear coupled system without reliance on the \(\Delta _2\) -condition for the N-function. This system, akin to a modified thermistor problem, concerns the determination of variables representing the temperature within a conductor and the associated electrical potential. PubDate: 2024-06-15

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Abstract: Abstract In the setting of Carnot groups, we propose an approach of taming singularities to get coercive inequalities. To this end, we develop a technique to introduce natural singularities in the energy function U in order to force one of the coercivity conditions. In particular, we explore explicit constructions of probability measures on Carnot groups which secure Poincaré and even Logarithmic Sobolev inequalities. As applications, we get analogues of the Dyson–Ornstein–Uhlenbeck model on the Heisenberg group and obtain results on the discreteness of the spectrum of related Markov generators. PubDate: 2024-06-12

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Abstract: Abstract This paper mainly deals with the Sturm–Liouville operator $$\begin{aligned} \textbf{H}=\frac{1}{w(x)}\left( -\frac{\textrm{d}}{\textrm{d}x}p(x)\frac{ \textrm{d}}{\textrm{d}x}+q(x)\right) ,\text { }x\in \Gamma \end{aligned}$$ acting in \(L_{w}^{2}\left( \Gamma \right) ,\) where \(\Gamma \) is a metric graph. We establish a relationship between the bottom of the spectrum and the positive solutions of quantum graphs, which is a generalization of the classical Allegretto–Piepenbrink theorem. Moreover, we prove the Persson-type theorem, which characterizes the infimum of the essential spectrum. PubDate: 2024-06-11

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Abstract: Abstract We consider discrete Schrödinger operators with periodic potentials on periodic graphs. Their spectra consist of a finite number of bands. We perturb a periodic graph by adding edges in a periodic way (without changing the vertex set) and show that if the added edges are long enough, then the perturbed graph is asymptotically isospectral to some periodic graph of a higher dimension but without long edges. We also obtain a criterion for the perturbed graph to be not only asymptotically isospectral but just isospectral to this higher dimensional periodic graph. One of the simplest examples of such asymptotically isospectral periodic graphs is the square lattice perturbed by long edges and the cubic lattice. We also get asymptotics of the endpoints of the spectral bands for the Schrödinger operator on the perturbed graph as the length of the added edges tends to infinity. PubDate: 2024-06-08

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Abstract: Abstract Consider a finite Blaschke product f with \(f(0) = 0\) which is not a rotation and denote by \(f^n\) its n-th iterate. Given a sequence \(\{a_n\}\) of complex numbers, consider the series \(F(z) = \sum _n a_n f^n(z).\) We show that for any \(w \in \mathbb {C},\) if \(\{a_n\}\) tends to zero but \(\sum _n a_n = \infty ,\) then the set of points \(\xi \) in the unit circle for which the series \(F(\xi )\) converges to w has Hausdorff dimension 1. Moreover, we prove that this result is optimal in the sense that the conclusion does not hold in general if one considers Hausdorff measures given by any measure function more restrictive than the power functions \(t^\delta ,\) \(0< \delta < 1.\) PubDate: 2024-06-06

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Abstract: Abstract For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range. PubDate: 2024-05-30

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Abstract: Abstract We establish the monotonicity of positive solutions to the problem $$\begin{aligned} -\Delta _p u + a(u) \nabla u ^q = f(u) \text { in } \mathbb {R}^N_+, \quad u=0 \text { on } \partial \mathbb {R}^N_+, \end{aligned}$$ where \(p>2\) , \(q\ge p-1\) and a, f are locally Lipschitz continuous functions such that f is positive on \((0,+\infty )\) and it is either sublinear or superlinear near 0. The main tool we use is the refined method of moving planes for quasilinear elliptic problems in half-spaces. PubDate: 2024-05-25

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Abstract: Abstract In this paper we initiate the study of the forward and backward shifts on the discrete generalized Hardy space of a tree and the discrete generalized little Hardy space of a tree. In particular, we investigate when these shifts are bounded, find the norm of the shifts if they are bounded, characterize the trees in which they are an isometry, compute the spectrum in some concrete examples, and completely determine when they are hypercyclic. PubDate: 2024-05-24

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Abstract: Abstract Let \(\mathbf {E_{\mathbb {X}}}\) be a unit ball on complex Banach space \(\mathbb {X}\) and \(\Phi \) be a convex function such that \(\Phi (0)=1\) and \(\Re \Phi (\xi )>0\) on \(\mathbb {D}=\{z\in \mathbb {C}: z <1\}\) . In this paper, we continue the work related to the class \(Q_\textbf{B}^{\Phi }(\mathbf {E_{\mathbb {X}}})\) of quasi-convex mappings of type \(\textbf{B}\) which have a \(\Phi \) -parametric representation on \(\mathbf {E_{\mathbb {X}}}\) , where the mappings \(f\in Q_\textbf{B}^{\Phi }(\mathbf {E_{\mathbb {X}}})\) are k-fold symmetric, \(k\in \mathbb {N}.\) We give the improved Fekete-Szegö inequalities for the class \(Q_\textbf{B}^{\Phi }(\mathbf {E_{\mathbb {X}}})\) and establish the sharp bounds of all terms of homogeneous polynomial expansions for some subclasses of \(Q_\textbf{B}^{\Phi }(\mathbf {E_{\mathbb {X}}})\) . Our main results are closely related to the Bieberbach conjecture in higher dimensions. PubDate: 2024-05-22

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Abstract: Abstract We describe a general method for constructing Heisenberg uniqueness pairs \((\Gamma ,\Lambda )\) in the euclidean space \(\mathbb {R}^{n}\) based on the study of boundary value problems for partial differential equations. As a result, we show, for instance, that any pair made of the boundary \(\Gamma \) of a bounded convex set \(\Omega \) and a sphere \(\Lambda \) is an Heisenberg uniqueness pair if and only if the square of the radius of \(\Lambda \) is not an eigenvalue of the Laplacian on \(\Omega \) . The main ingredients for the proofs are the Paley–Wiener theorem, the uniqueness of a solution to a homogeneous Dirichlet or initial boundary value problem, the continuity of single layer potentials, and some complex analysis in \(\mathbb {C}^{n}\) . Denjoy’s theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful. PubDate: 2024-05-20