Maximal inequalities and Riesz transform estimates on L p spaces for magnetic Schrödinger operators II

The paper concerns the magnetic Schrödinger operator ${H({\bf a},V)=\sum_{j=1}^{n} (\frac{1}{i}\frac{\partial}{\partial x_{j}}-a_{j})^{2}+V }$ on ${\mathbb{R}^n}$ . We prove some L ^p estimates on the Riesz transforms of H and we establish some related maximal inequalities. The conditions that we arrive at, are essentially based on the control of the magnetic field by the electric potential.     

L p Brunn–Minkowski type inequalities for Blaschke–Minkowski homomorphisms

In this article, some Brunn–Minkowski type inequalities for (radial) Blaschke–Minkowski homomorphisms with respect to (radial) L _p Minkowski addition are established.     

Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces

Some Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications in relation with the celebrated Hölder–McCarthy’s inequality for positive operators and Ky Fan’s inequality for real numbers are given as well.     

Riesz–Kantorovich formulas for operators on multi-wedged spaces

We introduce the notions of multi-suprema and multi-infima for vector spaces equipped with a collection of wedges, generalizing the notions of suprema and infima in ordered vector spaces. Multi-lattices are vector spaces that are closed under multi-suprema and multi-infima and are thus an abstraction of vector lattices. The Riesz decomposition property in the multi-wedged setting is also introduced, leading to Riesz–Kantorovich formulas for multi-suprema and multi-infima in certain spaces of operators.     

Two-weight weak-type inequalities for potential type operators

For the potential type operator
TФf（x）=∫RnФ（x-y）f（y）dy,
where Ф is a non-negative locally integrable function on R^n and satisfies weak growth condition, a two-weight weak-type （p,q） inequality for TФ is obtained.     

Fejér–Riesz type inequalities for Bergman spaces

We obtain Fejér?CRiesz type inequalities for the weighted Bergman spaces on the unit disk of the complex plane. We show that the Fejér?CRiesz inequalities can be expressed as boundedness and compactness problems for certain Toeplitz operators.     

On Hardy-Littlewood-Pólya and Taikov type inequalities for multiple operators in Hilbert spaces

We present a unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Pólya and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemannian manifolds and derive the well-known Taikov and Hardy-Littlewood-Pólya inequalities for functions defined on the d-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of another class. In addition, we establish sharp Solyar type inequalities for unbounded closed operators with closed range.

     

Čebyšev's type inequalities for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given.     

Weighted sobolev-poincaré inequalities for grushin type operators

Abstract

We show global existence and uniqueness for a system of partial differential equations that is a model for chalcopyrite disease within sphalerite. Using direct methods in the calculus of variations, we proof existence of a solution to an implicit time discretisation, derive uniform bounds and pass to the limit. By considering a regularised problem, it is possible to extend the existence results to logarithmic free energies. Furthermore, by an integration in time method we can show uniqueness of the solution. Additionally a free energy inequality affirms thermodynamical correctness of the model.     

Weighted LΦ ntergral inequalities for maximal operators

Sufficient (almost necessary) conditions are given on the weight funotiousu(·),v(·) for $$\Phi _2^{ - 1} \left[ {\int\limits_{\mathbb{R}^n } {\Phi _2 (C_2 (M_s f)(x))u(x)dx} } \right] \leqslant \Phi _1^{ - 1} \left[ {C_1 \int\limits_{\mathbb{R}^n } {\Phi _1 (|f(x)|)} v(x)dx} \right]$$ to hold when Φ₁, Φ₂ are ?-functions with subadditive Φ₁Φ ₂ ^?1 , andM _s (0≤s<n), is the usual fractional maximal operator.     

Some trace inequalities of Čebyšev type for functions of operators in hilbert spaces

Some trace operator inequalities for synchronous functions that are related to the ?eby?ev inequality for sequences of real numbers are given.     

Global L p estimates for degenerate Ornstein–Uhlenbeck operators

We consider a class of degenerate Ornstein–Uhlenbeck operators in ${\mathbb{R}^{N}}We consider a class of degenerate Ornstein–Uhlenbeck operators in \mathbbR^N{\mathbb{R}^{N}} , of the kind

A o ?i, j=1p0aij?xixj2 + ?i, j=1Nbijxi?xj\mathcal{A}\equiv\sum_{i, j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} + \sum_{i, j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}      14. Domains of type 1,1 operators: a case for Triebel–Lizorkin spaces      Jon Johnsen 《Comptes Rendus Mathematique》2004,339(2):115-118 Pseudo-differential operators of type $1\text{,}1$ are proved continuous from the Triebel–Lizorkin space $F_{p\text{,}1}^d$ to $L_p$, $1?p<\infty$, when of order d, and this is, in general, the largest possible domain among the Besov and Triebel–Lizorkin spaces. Hörmander's condition on the twisted diagonal is extended to this framework, using a general support rule for Fourier transformed pseudo-differential operators. To cite this article: J. Johnsen, C. R. Acad. Sci. Paris, Ser. I 339 (2004).      15. Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces      Tuomas P. Hyt?nen  Michael T. Lacey  Henri Martikainen  Tuomas Orponen  Maria Carmen Reguera  Eric T. Sawyer  Ignacio Uriarte-Tuero 《Journal d'Analyse Mathématique》2012,118(1):177-220 We show that for 1 < p < ??, weight w ?? A p , and any L 2-bounded Calderón-Zygmund operator T, there is a constant C T,p such that the weak- and strong-type inequalities $${\left\| {{T_\natural}f} \right\|_{{L^{p,\infty }}(w)}} \le {C_{T,p}}{\left\| w \right\|_{{A_p}}}{\left\| f \right\|_{{L^p}(w )}}$$ $${\left\| {{T_\natural}f} \right\|_{{L^p}(w)}} \le {C_{T,p}}\left\| w \right\|_{{A_p}}^{\max \{ 1,{{(p - 1)}^{ - 1}}}{\left\| f \right\|_{{L^p}(w)}}$$ hold, where T ? denotes the maximal truncations of T and ${\left\| w \right\|_{{A_p}}}$ denotes the Muckenhoupt A p characteristic of w. These estimates are not improvable in the power of ${\left\| w \right\|_{{A_p}}}$ . Our argument follows the outlines of those of Lacey-Petermichl-Reguera (Math. Ann. 2010) and Hyt?nen-Pérez-Treil-Volberg (arXiv, 2010) and contains new ingredients, including a weak-type estimate for certain duals of T ? and sufficient conditions for two-weight inequalities in L p for T ?. Our proof does not rely upon extrapolation.      16. On Trudinger–Moser type inequalities involving Sobolev–Lorentz spaces      Bernhard Ruf  Cristina Tarsi 《Annali di Matematica Pura ed Applicata》2009,188(3):369-397 Generalizations of the Trudinger–Moser inequality to Sobolev–Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of concentration–compactness.       17. Lieb–Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators      Diomba Sambou 《Journal of Functional Analysis》2014 Let H:=H0+V$H:=H_0+V$ and H⊥:=H0,⊥+V$H_{\perp }:=H_{0,\perp }+V$ be respectively perturbations of the unperturbed Schrödinger operators H0$H_0$ on L2(R3)$L^2({\mathbb{R}}^3)$ and H0,⊥$H_{0,\perp }$ on L2(R2)$L^2({\mathbb{R}}^2)$ with constant magnetic field of strength b>0$b>0$, and V a complex relatively compact perturbation. We prove Lieb–Thirring type inequalities on the discrete spectrum of H   and H⊥$H_{\perp }$. In particular, these estimates give a priori information on the distribution of eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.      18. Composition operators on μ-Bloch type spaces      José Giménez  Renny Malavé  Julio C. Ramos Fernández 《Rendiconti del Circolo Matematico di Palermo》2010,59(1):107-119 We characterize boundedness, closedness of the range and compactness for composition operators acting on μ-Bloch spaces, where μ is a positive continuous function defined on the interval 0 < t ≤ 1, that satisfy certain holomorphic extension properties. This extends results that appear in [15],[17],[8],[3].      19. Fefferman–Stein inequalities for the dyadic-like maximal operators      Rapicki  Mateusz 《Archiv der Mathematik》2019,113(1):81-93 The paper contains a transference theorem which allows to extend a large class of unweighted inequalities for the dyadic maximal operator to their weighted Fefferman–Stein counterparts on general probability spaces.       20. Generalised weighted composition operators on Maeda–Ogasawara spaces      《Indagationes Mathematicae》2017,28(4):854-862 Every Archimedean Riesz space can be embedded as an order dense subspace of some $C^{\infty }\left(X\right)$, the Riesz space of all extended continuous functions on a Stonean space $X$, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.

Domains of type 1,1 operators: a case for Triebel–Lizorkin spaces

Pseudo-differential operators of type $1\text{,}1$ are proved continuous from the Triebel–Lizorkin space $F_{p\text{,}1}^d$ to $L_p$, $1?p<\infty$, when of order d, and this is, in general, the largest possible domain among the Besov and Triebel–Lizorkin spaces. Hörmander's condition on the twisted diagonal is extended to this framework, using a general support rule for Fourier transformed pseudo-differential operators. To cite this article: J. Johnsen, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces

We show that for 1 < p < ??, weight w ?? A _p , and any L ²-bounded Calderón-Zygmund operator T, there is a constant C _T,p such that the weak- and strong-type inequalities $${\left\| {{T_\natural}f} \right\|_{{L^{p,\infty }}(w)}} \le {C_{T,p}}{\left\| w \right\|_{{A_p}}}{\left\| f \right\|_{{L^p}(w )}}$$ $${\left\| {{T_\natural}f} \right\|_{{L^p}(w)}} \le {C_{T,p}}\left\| w \right\|_{{A_p}}^{\max \{ 1,{{(p - 1)}^{ - 1}}}{\left\| f \right\|_{{L^p}(w)}}$$ hold, where T _? denotes the maximal truncations of T and ${\left\| w \right\|_{{A_p}}}$ denotes the Muckenhoupt A _p characteristic of w. These estimates are not improvable in the power of ${\left\| w \right\|_{{A_p}}}$ . Our argument follows the outlines of those of Lacey-Petermichl-Reguera (Math. Ann. 2010) and Hyt?nen-Pérez-Treil-Volberg (arXiv, 2010) and contains new ingredients, including a weak-type estimate for certain duals of T _? and sufficient conditions for two-weight inequalities in L ^p for T _?. Our proof does not rely upon extrapolation.     

On Trudinger–Moser type inequalities involving Sobolev–Lorentz spaces

Generalizations of the Trudinger–Moser inequality to Sobolev–Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of concentration–compactness.      

Lieb–Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators

Let H:=_H0+V$H:=H_0+V$ and _H⊥:=_H0,⊥+V$H_{\perp }:=H_{0,\perp }+V$ be respectively perturbations of the unperturbed Schrödinger operators _H0$H_0$ on L²(R³)$L^2({\mathbb{R}}^3)$ and _H0,⊥$H_{0,\perp }$ on L²(R²)$L^2({\mathbb{R}}^2)$ with constant magnetic field of strength b>0$b>0$, and V a complex relatively compact perturbation. We prove Lieb–Thirring type inequalities on the discrete spectrum of H   and _H⊥$H_{\perp }$. In particular, these estimates give a priori information on the distribution of eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.     

Composition operators on μ-Bloch type spaces

We characterize boundedness, closedness of the range and compactness for composition operators acting on μ-Bloch spaces, where μ is a positive continuous function defined on the interval 0 < t ≤ 1, that satisfy certain holomorphic extension properties. This extends results that appear in [15],[17],[8],[3].     

Fefferman–Stein inequalities for the dyadic-like maximal operators

The paper contains a transference theorem which allows to extend a large class of unweighted inequalities for the dyadic maximal operator to their weighted Fefferman–Stein counterparts on general probability spaces.

     

Generalised weighted composition operators on Maeda–Ogasawara spaces

Every Archimedean Riesz space can be embedded as an order dense subspace of some $C^{\infty }\left(X\right)$, the Riesz space of all extended continuous functions on a Stonean space $X$, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.