Related papers: Necessary conditions for vector-valued operator in…

We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincar\'e inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincar\'e and weak Poincar\'e inequalities was studied in…

The paper is devoted to obtain first and second order necessary optimality conditions for continuous-time optimization problems with equality and inequality constraints. A full rank type regularity condition along with an uniform implicit…

We consider several harmonic analysis operators in the multi-dimensional context of the Dunkl Laplacian with the underlying group of reflections isomorphic to $\mathbb{Z}_2^n$ (also negative values of the multiplicity function are…

This article focuses on $L^p$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We introduce four critical…

Using Dunkl theory, we introduce into consideration some weighted $L_p$-spaces on $[-1,1]$ and on the unit Euclidean sphere $\mathbb{S}^{d-1}$, $d\geq 2$. Then we define a family of linear bounded operators $\{V_\kappa^p(x)\colon…

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yau's and Karp's…

Critical Sobolev-type inequality for a class of weighted Sobolev spaces on the entire space is established. We also investigate the existence of extremal function for the associated variational problem. As an application, we prove the…

We establish Sobolev-Poincar\'e inequalities for piecewise $W^{1,p}$ functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of…

Although Ornstein's nonestimate entails the impossibility to control in general all the $L^1$-norm of derivatives of a function by the $L^1$-norm of a constant coefficient homogeneous vector differential operator, the corresponding endpoint…

We consider perturbed nonlinear ill-posed equations in Hilbert spaces, with operators that are monotone on a given closed convex subset. A simple stable approach is Lavrentiev regularization, but existence of solutions of the regularized…

Making use of the method of subordination chains, we obtain some sufficient conditions for the univalence of an integral operator. In particular, as special cases, our results imply certain known univalence criteria. A refinement to a…

In this note we consider weighted conditional type operators between different Orlicz spaces and generalized conditional type Holder inequality that we defined in [2]. Then we give some necessary and sufficient conditions for boundedness of…

Oscillatory integral operators with $1$-homogeneous phase functions satisfying a convexity condition are considered. For these we show the $L^p - L^p$-estimates for the Fourier extension operator of the cone due to Ou--Wang via polynomial…

We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the…

We obtain a complete characterization of $L^p-L^q$ Carleman estimates with weight $e^{v\cdot x}$ for the polyharmonic operators. Our result extends the Carleman inequalities for the Laplacian due to Kenig--Ruiz--Sogge. Consequently, we…

This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction…

In this paper we prove and discuss some new $\left(H_{p},weak-L_{p}\right) $ type inequalities of maximal operators of Vilenkin-N\"orlund means with monotone coefficients. We also apply these results to prove a.e. convergence of such…

We prove invariant Harnack inequalities for certain classes of non-divergence form equations of Kolmogorov type. The operators we consider exhibit invariance properties with respect to a homogeneous Lie group structure. The coefficient…

We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential…

Let $1\leq q\leq (n-1)$. We first show that a necessary condition for a Hankel operator on $(0,q-1)$-forms on a convex domain to be compact is that its symbol is holomorphic along $q$-dimensional analytic varieties in the boundary. Because…