Related papers: Martingale interpretation of weakly cancelling dif…

We prove a martingale analog of van Schaftingen's theorem and give sharp estimates on the lower Hausdorff dimension of measures in martingale shift invariant spaces. We also provide martingale analogs of trace theorems for Sobolev…

We suggest an elementary Harmonic Analysis approach to canceling and weakly canceling differential operators, which allows to extend these notions to anisotropic setting and also replace differential operators with Fourier multiplies with…

We establish foundational properties of fractional operators on Lie groups of homogeneous type. We prove embedding theorems for the associated Sobolev-type spaces.

We study extension operators on Sobolev spaces with decreasing integrability on the base of set functions associated with the operator norms. Sharp necessary conditions in the terms of the generalized density condition and the terms of weak…

Sobolev type inequalities involving homogeneous elliptic canceling differential operators and rearrangement-invariant norms on the Euclidean space are considered. They are characterized via considerably simpler one-dimensional Hardy type…

In this paper, several weak Orlicz-Hardy martingale spaces associated with concave functions are introduced, and some weak atomic decomposition theorems for them are established. With the help of weak atomic decompositions, a sufficient…

We propose a functional framework of fractional Sobolev spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak H\"ormander condition. We characterize these spaces as real interpolation of natural order intrinic…

Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a…

In this paper we prove analogues of Korovkin's theorem in the context of weakly nonlinear and monotone operators acting on Banach lattices of functions of several variables. Our results concern the convergence almost everywhere, the…

We prove a weak-type estimate for a class of operators extending some of the almost orthogonality issues involved in the study of the bilinear Hilbert transform by Lacey and Thiele.

We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the $A_1$ conjecture of Muckenhoupt, which stayed open after Muckenhoupt--Wheeden's conjecture was disproved by Reguera--Thiele.

We propose a new weak convergence theorem for martingales, under gentler conditions than the usual convergence in probability of the sequence of associated quadratic variations. Its proof requires the combined use of Skorohod's…

We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak H\"ormander condition. We prove continuous embeddings into Lorentz and intrinsic H\"older spaces. We also prove…

This paper is concerned at the minimization fundamental gap problem for a class of two-dimensional degenerate sub-elliptic operators. We establish existence results for weak solutions, Sobolev embedding theorem and spectral theory of…

This paper considers the martingale problem for a class of weakly coupled L\'{e}vy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process…

In an abstract framework, a new concept on time operator, ultra-weak time operator, is introduced, which is a concept weaker than that of weak time operator. Theorems on the existence of an ultra-weak time operator are established. As an…

Supermartingales are here defined on a non-probabilistic setting and can be interpreted solely in terms of superhedging operations. The classical expectation operator is replaced by a pair of subadditive operators one of them providing a…

In this paper we extend Korovkin's theorem to the context of sequences of weakly nonlinear and monotone operators defined on certain Banach function spaces. Several examples illustrating the theory are included.

The injectively elliptic vector differential operators $A (\mathrm{D})$ from $V$ to $E$ on $\mathbb{R}^n$ such that the estimate \[ \Vert D^\ell u\Vert_{L^{n/(n - \ell)} (\mathbb{R}^n)} \le \Vert A (\mathrm{D}) u\Vert_{L^1 (\mathbb{R}^n)}…

We prove collective versions of semi-duality theorems for sets of almost (limitedly, order) L-weakly compact operators.