Sharp Embedding Research Articles

On the modulus of continuity of fractional Orlicz-Sobolev functions

AbstractNecessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on $${\mathbb {R}}^n$$ R n to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.

Sharp embedding between Wiener amalgam and some classical spaces

Sharp embedding between Wiener amalgam and some classical spaces

Newton–Okounkov bodies and symplectic embeddings into nontoric rational surfaces

AbstractWe develop new methods of both constructing and obstructing symplectic embeddings into nontoric rational surfaces using the theory of Newton–Okoukov bodies. Applications include sharp embedding results for concave toric domains into nontoric rational surfaces, and new cases of nonexistence for infinite staircases in the nontoric setting.

Sobolev-Type Embeddings for Neural Network Approximation Spaces

We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in L^p) by ReLU neural networks with an increasing number of coefficients, subject to bounds on the magnitude of the coefficients and the number of hidden layers. We prove embedding theorems between these spaces for different values of p. Furthermore, we derive sharp embeddings of these approximation spaces into Hölder spaces. We find that, analogous to the case of classical function spaces (such as Sobolev spaces, or Besov spaces) it is possible to trade “smoothness” (i.e., approximation rate) for increased integrability. Combined with our earlier results in Grohs and Voigtlaender (Proof of the theory-to-practice gap in deep learning via sampling complexity bounds for neural network approximation spaces, 2021. arXiv preprint arXiv:2104.02746), our embedding theorems imply a somewhat surprising fact related to “learning” functions from a given neural network space based on point samples: if accuracy is measured with respect to the uniform norm, then an optimal “learning” algorithm for reconstructing functions that are well approximable by ReLU neural networks is simply given by piecewise constant interpolation on a tensor product grid.

Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

Hilbert spaces and low-lying zeros of L-functions

Generalizing previous work of Iwaniec, Luo, and Sarnak (2000), we use information from one-level density theorems to estimate the proportion of non-vanishing of L-functions in a family at a low-lying height on the critical line (measured by the analytic conductor). To solve the Fourier optimization problems that arise, we provide a unified framework based on the theory of reproducing kernel Hilbert spaces of entire functions (there is one such space associated to each symmetry type). Explicit expressions for the reproducing kernels are given. We also revisit the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). We solve the associated Fourier optimization problem in this setting by establishing a connection to the theory of de Branges spaces of entire functions and using the explicit reproducing kernels. In an appendix, we study the related problem of determining the sharp embeddings between the Hilbert spaces associated to the five symmetry types and the classical Paley-Wiener space.

The Modulus of p-Variation and Its Applications

Let \(\nu \) be a nondecreasing concave sequence of positive real numbers and \(1\le p<\infty \). In this note, we introduce the notion of modulus of p-variation for a function of a real variable, and show that it serves in at least two important problems, namely, the uniform convergence of Fourier series and computation of certain K-functionals. Using this new tool, we first define a Banach space, denoted \(V_p[\nu ]\), that is a natural unification of the Wiener class \(BV_p\) and the Chanturiya class \(V[\nu ]\). Then we prove that \(V_p[\nu ]\) satisfies a Helly-type selection principle which enables us to characterize continuous functions in \(V_p[\nu ]\) in terms of their Fejér means. We also prove that a certain K-functional for the couple \((C,BV_p)\) can be expressed in terms of the modulus of p-variation, where C denotes the space of continuous functions. Next, we obtain equivalent optimal conditions for the uniform convergence of the Fourier series of all functions in each of the classes \(C\cap V_p[\nu ]\) and \(H^\omega \cap V_p[\nu ]\), where \(\omega \) is a modulus of continuity and \(H^\omega \) denotes its associated Lipschitz class. Finally, we establish sharp embeddings into \(V_p[\nu ]\) of various spaces of functions of generalized bounded variation. As a by-product of these latter results, we infer embedding results for certain symmetric sequence spaces.

The logarithmic Sobolev capacity

This paper presents various functional-geometric aspects of the logarithmic Sobolev capacity Caplog⁡,γ,p - a capacity generated by the logarithmic Sobolev space Wplog⁡,γ, where (γ,p)∈(0,∞)×[1,∞). This new nonlinear-inhomogeneous-nontrivial capacity is closely related to the logarithmic Hausdorff capacity. Not only continuously sharp embedding property of Wplog⁡,γ into certain logarithmic weighted Lebesgue space and the tracing inequalities are established, but also the logarithmic perimeter and the Lebesgue measure are utilized to: reformulate Caplog⁡,γ,p; find the first variation of the logarithmic perimeter via the corresponding logarithmic mean curvature; characterize the hypersurface of a constant logarithmic mean curvature.

On Variations of the Neumann Eigenvalues of p-Laplacian Generated by Measure Preserving Quasiconformal Mappings

We study variations of the first nontrivial eigenvalue of the two-dimensional p-Laplace operator, p > 2, generated by measure preserving quasiconformal mappings. The study is based on the geometric theory of composition operators in Sobolev spaces and sharp embedding theorems. Using a sharp version of the reverse Holder inequality, we obtain a lower estimate for the first nontrivial eigenvalue in the case of Ahlfors type domains.

On some new sharp embedding theorems in area Nevanlinna spaces and related problems

We provide some new sharp embedding theorems for analytic area Nevanlinna spaces in the unit disk extending some previously known assertions in various directions.

On some new sharp embedding theorems for multifunctional Herz-type and Bergman-type spaces in tubular domains over symmetric cones

We introduce new multifunctional mixed norm analytic Herz-type spaces in tubular domains over symmetric cones and provide new sharp embedding theorems for them. Some results are new even in case of onefunctional holomorphic spaces. Some new related sharp results for new multifunctional Bergman-type spaces will be also provided under one condition on Bergman kernel.

On some sharp embedding theorems in area Nevanlinna spaces and related problems

We provide some new sharp embedding theorems for analytic area Nevanlinna spaces in the unit disk extending some previously known assertions in various directions. Various related results (embedding theorems for area Nevanlinna type spaces) in some other general domains and polydomains will be discussed and provided by us.

Embeddings and characterizations of Lipschitz spaces

In this paper we give a thorough study of Lipschitz spaces. We obtain the following new results:(i)Sharp Jawerth-Franke-type embeddings between the Besov and Lipschitz spaces extending the classical results for Besov and Sobolev spaces.(ii)Sharp embeddings between Lipschitz spaces with different parameters extending the Brézis-Wainger result.(iii)Characterizations for Lipschitz norms via wavelets.

Domains of elliptic operators on sets in Wiener space

Let [Formula: see text] be a separable Banach space endowed with a non-degenerate centered Gaussian measure [Formula: see text]. The associated Cameron–Martin space is denoted by [Formula: see text]. Consider two sufficiently regular convex functions [Formula: see text] and [Formula: see text]. We let [Formula: see text] and [Formula: see text]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form [Formula: see text] and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with [Formula: see text] and on half-spaces, namely if [Formula: see text] and [Formula: see text] is an affine function, then the domain of the operator defined via (0.1) is the space [Formula: see text] where [Formula: see text] is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.

Local well and ill posedness for the modified KdV equations in subcritical modulation spaces

We consider the Cauchy problem of the modified KdV equation (mKdV). Local well-posedness of this problem is obtained in modulation spaces $M^{1/4}_{2,q}(\mathbb{{R}})$ $(2\leq q\leq\infty)$. Moreover, we show that the data-to-solution map fails to be $C^3$ continuous in $M^{s}_{2,q}(\mathbb{{R}})$ when $s<1/4$. It is well-known that $H^{1/4}$ is a critical Sobolev space of mKdV so that it is well-posedness in $H^s$ for $s\geq 1/4$ and ill-posed (in the sense of uniform continuity) in $H^{s'}$ with $s'<1/4$. Noticing that $M^{1/4}_{2,q} \subset B^{1/q-1/4}_{2,q}$ is a sharp embedding and $H^{-1/4}\subset B^{-1/4}_{2,\infty}$, our results contains all of the subcritical data in $M^{1/4}_{2,q}$, which contains a class of functions in $H^{-1/4}\setminus H^{1/4}$.

On some new sharp embedding theorems for multifunctional Herz-type and Bergman-type spaces in pseudoconvex domains

We introduce new multifunctional mixed norm analytic Herz-type spaces in strongly pseudoconvex domains and provide new sharp embedding theorems for them. Some results are new even in case of onefunctional holomorphic spaces. Some new related sharp results for new multifunctional Bergman-type spaces will be also provided under one condition on Bergman kernel. Similar results with similar proofs in unbounded tubular domains over symmetric cones and bounded symmetric domains will be also shortly mentioned.

Subcritical approach to conformally invariant extension operators on the upper half space

In this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators. This family includes (among others) the classical Poisson extension operator and the extension operator with Riesz kernel. We show that the sharp constants in these inequalities are attained and classify the corresponding extremal functions. We also compute the limiting behavior at the boundary of the extensions of the extremal functions.

English

We obtain new sharp embedding theorems for mixed-norm Herz-type analytic spaces in tubular domains over symmetric cones. These results enlarge the list of recent sharp theorems in analytic spaces obtained by Nana and Sehba (2015).

Risk analysis and decision theory: A bridge

The risk-triplet approach pioneered by Kaplan and Garrick is the keystone of operational risk analysis. We perform a sharp embedding of the elements of this framework into the one of formal decision theory, which is mainly concerned with the methodological and modeling issues of decision making. The aim of this exercise is twofold: on the one hand, it gives operational risk analysis a direct access to the rich toolbox that decision theory has developed, in the last decades, in order to deal with complex layers of uncertainty; on the other, it exposes decision theory to the challenges of operational risk analysis, thus providing it with broader scope and new stimuli.

On some new sharp embedding theorems in minimal and pseudoconvex domains

We present new sharp embedding theorems for mixed-norm analytic spaces in pseudoconvex domains with smooth boundary. New related sharp results in minimal bounded homogeneous domains in higher dimension are also provided. Last domains we consider are domains which are direct generalizations of the well-studied so-called bounded symmetric domains in Cn. Our results were known before only in the very particular case of domains of such type in the unit ball. As in the unit ball case, all our proofs are heavily based on nice properties of the r-lattice. Some results of this paper can be also obtained in some unbounded domains, namely tubular domains over symmetric cones.