Rearrangement-invariant Norms Research Articles

Symmetric gradient Sobolev spaces endowed with rearrangement-invariant norms

A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangement-invariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into spaces of uniformly continuous functions, and their optimal targets, are also exhibited. By contrast, these embeddings may be weaker than the corresponding ones for the full gradient. Related results, of independent interest in the theory of symmetric gradient Sobolev spaces, are established. They include global approximation and extension theorems under minimal assumptions on the domain. A formula for the K-functional, which is pivotal for our method based on reduction to one-dimensional inequalities, is provided as well. The case of symmetric gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum mechanics driven by nonlinearities of non-power type, is especially focused.

Poincaré-Sobolev inequalities with rearrangement-invariant norms on the entire space

Poincar\'{e}-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $\mathbb{R}^n$ are provided. Namely, inequalities of the type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$ are either rearrangement-invariant spaces over $\mathbb{R}^n$ or Orlicz spaces over $\mathbb{R}^n$, $u$ is a $m-$times weakly differentiable function whose gradient is in $X$, $P$ is a polynomial of order at most $m-1$, depending on $u$, and $C$ is a constant independent of $u$, are studied. In a sense optimal rearrangement-invariant spaces or Orlicz spaces $Y$ in these inequalities when the space $X$ is fixed are found. A variety of particular examples for customary function spaces are also provided.

Banach algebras of weakly differentiable functions

The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an n-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also established. The special cases of Orlicz-Sobolev and Lorentz-Sobolev spaces are discussed in detail. New results for classical Sobolev spaces on possibly irregular domains follow as well.

Extremal distributions of discrepancy functions

The irregularities of a distribution of N points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form [0,t)⊂[0,1) or arbitrary subintervals of the unit interval. In the former case, it is a well known fact in discrepancy theory that the N-element point set in the unit interval with the lowest L2 or L∞ norm of the discrepancy function is the centered regular grid ΓN≔2n+12N:n=0,1,…,N−1. We show a stronger result on the distribution of discrepancy functions of point sets in [0,1], which basically says that the distribution of the discrepancy function of ΓN is in some sense minimal among all N-element point sets. As a consequence, we can extend the above result to rearrangement-invariant norms, including Lp, Orlicz and Lorentz norms.We study the same problem for the discrepancy notions with respect to arbitrary subintervals. In this case, we will observe that we have to deal with integrals of convolutions of functions. To this end, we prove a general upper bound on such expressions, which might be of independent interest as well.

Compactness for Sobolev-type trace operators

Compactness of arbitrary-order Sobolev type embeddings for traces of n-dimensional functions on lower dimensional subspaces is investigated. Sobolev spaces built upon any rearrangement-invariant norm are allowed. In particular, we characterize compactness of trace embeddings for classical Sobolev, Lorentz–Sobolev and Orlicz–Sobolev type spaces.

Sharp Sobolev type embeddings on the entire Euclidean space

A comprehensive approach to Sobolev type embeddings, involving arbitrary rearrangement-invariant norms on the entire Euclidean space \begin{document}${\mathbb R^n}$\end{document} , is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

Norms supporting the Lebesgue differentiation theorem

A version of the Lebesgue differentiation theorem is offered, where the [Formula: see text] norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation theorem are established. In particular, Lorentz, Orlicz and other customary norms for which Lebesgue’s theorem holds are characterized.

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.

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.

THE MULTIPLICATIVE PROPERTY CHARACTERIZES ℓpAND LpNORMS

We show that $\ell_p$ norms are characterized as the unique norms which are both invariant under coordinate permutation and multiplicative with respect to tensor products. Similarly, the $L_p$ norms are the unique rearrangement-invariant norms on a probability space such that $\|X Y\|=\|X\|\cdot\|Y\|$ for every pair $X,Y$ of independent random variables. Our proof relies on Cram\'er's large deviation theorem.

Optimal embeddings of generalized inhomogeneous Sobolev spaces on R^n

We prove optimal embeddings in the subcritical case of the inhomogeneous Sobolev spaces built-up over function spaces in Rn with K−monotone and rearrangement invariant norm into another rearrangement invariant function spaces. The investigation is based on pointwise and integral estimates of the rearrangement or the oscillation of the rearrangement of f in terms of the rearrangement of the derivatives of f . Mathematics subject classification (2010): 46E35.

Optimal embeddings of generalized homogeneous Sobolev spaces

We prove optimal embeddings of homogeneous Sobolev spaces built over function spaces in ${\mathbb R}^n$ with $K$-monotone and rearrangement invariant norm into other rearrangement invariant function spaces. The investigation is based on pointwise and in

Optimal Gaussian Sobolev embeddings

A reduction theorem is established, showing that any Sobolev inequality, involving arbitrary rearrangement-invariant norms with respect to the Gauss measure in R n , is equivalent to a one-dimensional inequality, for a suitable Hardy-type operator, involving the same norms with respect to the standard Lebesgue measure on the unit interval. This result is exploited to provide a general characterization of optimal range and domain norms in Gaussian Sobolev inequalities. Applications to special instances yield optimal Gaussian Sobolev inequalities in Orlicz and Lorentz(–Zygmund) spaces, point out new phenomena, such as the existence of self-optimal spaces, and provide further insight into classical results.

Boundary trace inequalities and rearrangements

A new approach to boundary trace inequalities for Sobolev functions is presented, which reduces any trace inequality involving general rearrangement-invariant norms to an equivalent, considerably simpler, one-dimensional inequality for a Hardy-type operator. In particular, improvements of classical boundary trace embeddings and new optimal trace embeddings are derived.

Compactness of Sobolev imbeddings involving rearrangement-invariant norms

We nd necessary and sucient conditions on a pair of rearrangement- invariant norms,% and , in order that the Sobolev spaceW m;% ( ) be compactly imbedded into the rearrangement-invariant space L ( ), where is a bounded domain in R n with Lipschitz boundary and 1 m n 1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from L%(0;j j) into L (0;j j). The results are illustrated with examples in which % and are both Orlicz norms or both Lorentz Gamma norms.

Optimal Sobolev embeddings on $\mathbb{R}^n$

We study Sobolev-type embeddings involving rearrangement-invariant norms. In particular, we focus on the question when such embeddings are optimal. We concentrate on the case when the functions involved are defined on $\mathbb R\sp n$. This subject has been studied before, but only on bounded domains. We first establish the equivalence of the Sobolev embedding to a new type of inequality involving two integral operators. Next, we show this inequality to be equivalent to the boundedness of a certain Hardy operator on a specific new type of cone of positive functions. This Hardy operator is then used to provide optimal domain and range rearrangement-invariant norm in the embedding inequality. Finally, the limiting case of the Sobolev embedding on Rn is studied in detail.

Compactness properties of Sobolev imbeddings for rearrangement invariant norms

Compactness properties of Sobolev imbeddings are studied within the context of rearrangement invariant norms. Attention is focused on the extremal situation, namely, when the imbedding is considered as defined on its optimal Sobolev domain (with the range space fixed). The techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator (of just one variable).

Discretization and anti-discretization of rearrangement-invariant norms

We develop a new method of discretization and anti-discretization of weighted inequalities which we apply to norms in classical Lorentz spaces and to spaces endowed with the so-called Hilbert norm. Main applications of our results include new integral conditions characterizing embeddings Γp(v)
→ Γq(w) and Γp(v)
→ Λq(w) and an integral characterization of the associate space to Γp(v), where p, q ∈ (0, ∞), v, w are weights on [0, ∞) and fΛp(v) = ∞ 0 f∗(t) pv(t) dt1/p, fΓp(v) = ∞ 0 f∗∗(t) pv(t) dt1/p.

Rearrangement invariant norms of symmetric sequence norms of independent sequences of random variables

LetX 1,X 2, …,X n be a sequence of independent random variables, letM be a rearrangement invariant space on the underlying probability space, and letN be a symmetric sequence space. This paper gives an approximate formula for the quantity ‖‖(X i )‖ N ‖ M wheneverL q embeds intoM for some 1≤q<∞. This extends work of Johnson and Schechtman who tackled the case whenN=l p , and recent work of Gordon, Litvak, Schutt and Werner who obtained similar results for Orlicz spaces.

Measuring the magnitude of sums of independent random variables

This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Lévy property. We then give a connection between the tail distribution and the pth moment, and between the pth moment and the rearrangement invariant norms.