First-order Sobolev Research Articles

Optimal Sobolev trace embeddings

Optimal target spaces are exhibited in arbitrary-order Sobolev type embeddings for traces of n n -dimensional functions on lower dimensional subspaces. Sobolev spaces built upon any rearrangement-invariant norm are allowed. A key step in our approach consists of showing that any trace embedding can be reduced to a one-dimensional inequality for a Hardy type operator depending only on n n and on the dimension of the relevant subspace. This can be regarded as an analogue for trace embeddings of a well-known symmetrization principle for first-order Sobolev embeddings for compactly supported functions. The stability of the optimal target space under iterations of Sobolev trace embeddings is also established and is part of the proof of our reduction principle. As a consequence, we derive new trace embeddings, with improved (optimal) target spaces, for classical Sobolev, Lorentz-Sobolev and Orlicz-Sobolev spaces.

On the Regularity of the Multisublinear Maximal Functions

AbstractThis paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.

Stochastic Lagrangian flows on the group of volume-preserving homeomorphisms of the spheres

We consider stochastic differential equations on the group of volume-preserving homeomorphisms of the sphere . The diffusion part is given by the divergence-free eigenvector fields of the Laplacian acting on -vector fields, while the drift is some other divergence-free vector field. We show that the equation generates a unique flow of measure-preserving homeomorphisms when the drift has first-order Sobolev regularity, and derive a formula for the distance between two Lagrangian flows. We also compute the rotation process of two particles on the sphere when they are close to each other.

Higher-order Sobolev embeddings and isoperimetric inequalities

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in Rn, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, our results are applied to any-order Sobolev embeddings in regular (John) domains of the Euclidean space, in Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and in families of product probability spaces, of which the Gauss space is a classical instance.

Geometric analysis on Cantor sets and trees

Abstract Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.

L 2-theory for non-symmetric Ornstein–Uhlenbeck semigroups on domains

We prove that the mild solution of the stochastic evolution equation \({{d}X(t) = AX(t)\,{d}t + {d}W(t)}\) on a Banach space E has a continuous modification if the associated Ornstein–Uhlenbeck semigroup is analytic on L2 with respect to the invariant measure. This result is used to extend recent work of Da Prato and Lunardi for Ornstein–Uhlenbeck semigroups on domains \({\mathcal{O} \subseteq E}\) to the non-symmetric case. Denoting the generator of the Ornstein–Uhlenbeck semigroup by \({L_\mathcal{O}}\), we obtain sufficient conditions in order that the domain of \({\sqrt{-L_\mathcal{O}}}\) be a first-order Sobolev space.

WELL-POSEDNESS OF FOKKER–PLANCK TYPE EQUATIONS ON THE WIENER SPACE

We consider Fokker–Planck type equations on the abstract Wiener space. Under the assumptions that the coefficients have a certain Sobolev regularity and they, together with their gradients and divergences, are exponentially integrable, we establish the existence of solutions to these equations, based on the estimates for solutions to Fokker–Planck equations in the finite-dimensional case. Moreover, the solution is unique if it belongs to the first-order Sobolev space.

Reconstruction from free‐breathing cardiac MRI data using reproducing kernel Hilbert spaces

This paper describes a rigorous framework for reconstructing MR images of the heart, acquired continuously over the cardiac and respiratory cycle. The framework generalizes existing techniques, commonly referred to as retrospective gating, and is based on the properties of reproducing kernel Hilbert spaces. The reconstruction problem is formulated as a moment problem in a multidimensional reproducing kernel Hilbert spaces (a two-dimensional space for cardiac and respiratory resolved imaging). Several reproducing kernel Hilbert spaces were tested and compared, including those corresponding to commonly used interpolation techniques (sinc-based and splines kernels) and a more specific kernel allowed by the framework (based on a first-order Sobolev RKHS). The Sobolev reproducing kernel Hilbert spaces was shown to allow improved reconstructions in both simulated and real data from healthy volunteers, acquired in free breathing.

On extensions of Sobolev functions defined on regular subsets of metric measure spaces

We characterize the restrictions of first-order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator.

Homogeneous sharp Sobolev inequalities on product manifolds

Let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. For p-homogeneous convex functions f(s, t) on [0,∞) × [0, ∞), we study the validity and non-validity of the first-order optimal Sobolev inequality on H1, p(M × N) where and Kf = Kf (m, n, p) is the best constant of the homogeneous Sobolev inequality on D1, p (Rm+n), The proof of the non-validity relies on the knowledge of extremal functions associated with the Sobolev inequality above. In order to obtain such extremals we use mass transportation and convex analysis results. Since variational arguments do not work for general functions f, we investigate the validity in a uniform sense on f and argue with suitable approximations of f which are also essential in the non-validity. Homogeneous Sobolev inequalities on product manifolds are connected to elliptic problems involving a general class of operators.

Approximate real inversion formulas of the gaussian convolution

We shall give very and surprisingly simple approximate real inversion formulas of the Gaussian convolution (the Weierstrass transform) for the first-order Sobolev Hilbert space on the whole real line by using best approximations and the theory of reproducing kernels and by using a good connection with the Tikhonov regularization.

A strong Lebesgue point property for Sobolev functions

We show that first-order Sobolev functions fulfill a Wiener integral type Lebesgue point property outside a set of Sobolev capacity zero. Our condition is stronger than the standard Lebesgue point property, but the exceptional set is slightly larger.

REGULARITY OF THE FRACTIONAL MAXIMAL FUNCTION

The purpose of this work is to show that the fractional maximal operator has somewhat unexpected regularity properties. The main result shows that the fractional maximal operator maps -spaces boundedly into certain first-order Sobolev spaces. It is also proved that the fractional maximal operator preserves first-order Sobolev spaces. This extends known results for the Hardy–Littlewood maximal operator.