Sharp Embedding Research Articles

CB-Norm Estimates for Maps Between NoncommutativeLp-Spaces and Quantum Channel Theory

In the first part of this work, we show how certain techniques from quantum information theory can be used in order to obtain very sharp embeddings between noncommutative Lp-spaces. Then, we use these estimates to study the classical capacity with restricted assisted entanglement of the quantum erasure channel and the quantum depolarizing channel. In particular, we exactly compute the capacity of the first one and we show that certain nonmultiplicative results hold for the second one.

Sharp Embedding of Sobolev Spaces Involving General Kernels and its Application

The purpose of this paper is to extend the embedding theorem of Sobolev spaces involving general kernels and we provide a sharp critical exponent in the embedding. As an application, solutions for equations driven by a general integro-differential operator, with homogeneous Dirichlet boundary conditions, is established by using the Mountain Pass Theorem.

Limiting Sobolev inequalities and the 1-biharmonic operator

Abstract In this article we present recent results on optimal embeddings, and associated PDEs, of the space of functions whose distributional Laplacian belongs to L1 . We discuss sharp embedding inequalities which allow to improve the optimal summability results for solutions of Poisson equations with L1 -data by Maz'ya (N ≥ 3) and Brezis–Merle (N = 2). Then, we consider optimal embeddings of the mentioned space into L1 , for the simply supported and the clamped case, which yield corresponding eigenvalue problems for the 1-biharmonic operator (a higher order analogue of the 1-Laplacian). We derive some properties of the corresponding eigenfunctions, and prove some Faber–Krahn type inequalities.

Estimation of the uniform modulus of continuity of the generalized Bessel potential

We study generalized Bessel potentials constructed by means of convolutions of functions with kernels that generalize the classical Bessel-Macdonald kernels. In contrast to the classical case, nonpower singularities of kernels in a neighborhood of the origin are admitted. The integral properties of functions are characterized in terms of decreasing rearrangements. The differential properties of potentials are described by the kth-order moduli of continuity in the uniform norm. An order-sharp upper estimate is established for the modulus of continuity of a potential. Such estimates play an important role in the theory of function spaces. They allow one to establish sharp embedding theorems for potentials, find majorants of the moduli of continuity, and estimate the approximation numbers of embedding operators.

Optimal embedding and sharp estimates of the continuity envelope for generalized Bessel potentials

In the theory of function spaces it is an important problem to describe the differential properties for the convolution u = G * f in terms of the behavior of kernel near the origin, and at the infinity. In our paper the differential properties of convolution are characterized by their modulus of continuity of order k ∈ N in the uniform norm. The kernels of convolution generalize the classical kernels determining the Bessel and Riesz potential. They admit non-power behavior near the origin. The order-sharp estimates are obtained for moduli of continuity of the convolution in the uniform norm as well as for continuity envelope function of generalized Bessel potentials. Such estimates admit sharp embedding theorems into a Calderon space and imply estimates for the approximation numbers of the embedding operator.

Optimal limiting embeddings for Δ-reduced Sobolev spaces in [formula omitted

We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with L1 data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson–Brezis–Wainger spaces, for the optimal embeddings of borderline Sobolev spaces W0k,n/k.

New results on p-Carleson measures and some related measures in the unit disk

We provide some new sharp embeddings for p-Carleson mea- sures and some related measures in the unit disk of the complex plane.

Sharp embedding results for spaces of smooth functions with power weights

We consider function spaces of Besov, Triebel–Lizorkin, Bessel-potential and Sobolev type on $\mathbb R^d$, equipped with power weights $w(x) = |x|^\gamma$, $\gamma>-d$. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely

Adams’ Inequality and Limiting Sobolev Embeddings into Zygmund Spaces

We exhibit sharp embedding constants for Sobolev spaces of any order into Zygmund spaces, obtained as the product of sharp embedding constants for second order Sobolev space into Lorentz spaces. As a consequence, we derive a new proof of Adams’ inequality, which holds in the larger hypotheses of homogenoeous Navier boundary contidions.

Characterization of a rearrangement-invariant hull of a Besov space via interpolation

Let X be a rearrangement-invariant Banach function space over a Lipschitz domain Ω⊂ℝn. We characterize the K-functionals for the pairs (X,V1X) and (X,SX), where V1X is the reduced Sobolev space built upon X and SX is the class of measurable functions on Ω such that \(\|t\sp{-\frac{1}{n}}(f^{**}(t)-f^{*}(t))\|_{\overline{X}}<\infty\), \(\overline{X}\) being the representation space of X. Using this result, we obtain an estimate of rearrangements of a function in terms of moduli of continuity and prove its sharpness. Finally we establish sharp embeddings of general Besov spaces into Lorentz spaces and characterize the rearrangement-invariant hull of a general Besov space.

Embedding relations and boundedness of the multifunctional operators in tube domains over symmetric cones

We obtain a new general sufficient condition for the continuity of the Bergman projection in tube domains over symmetric cones using multifunctional embeddings. We also obtain some sharp embedding relations between the generalized Hilbert-Hardy spaces and the mixed-norm Bergman spaces in this setting.

Best approximation with wavelets in weighted Orlicz spaces

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L Φ(w) in terms of the fundamental function of L Φ(w). In particular, we prove that these bases are greedy in L Φ(w) if and only if L Φ(w) = L p(w), 1 < p < ∞. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L p(w) the approximation spaces are identified with weighted Besov spaces.

Sharp Sobolev Embeddings and Related Hardy Inequalities: The Sub -- Critical Case

The paper deals with sharp embeddings of the spaces B and F into rearrangement-variant spaces and related Hardy inequalities. Here (1/p, s) belongs to the interior of the shaded invariant spaces region in the Figure

A Note on Sharp Embedding Theorems for Holomorphic Classes Based on Lorentz Spaces on A Unit Circle

We provide new sharp embedding theorems for holomorphic classes in the unit disk based on Lorentz classes on the unit circle.

ON SOME NEW EMBEDDING THEOREMS FOR SOME ANALYTIC CLASSES IN THE UNIT BALL

We provide new sharp embedding theorems for analytic classes in unit ball expanding at the same time some previously known assertions.

On sharp embeddings of Besov and Triebel-Lizorkin spaces in the subcritical case

We discuss the growth envelopes of Fourier-analytically defined Besov and Triebel-Lizorkin spaces B s p,q (ℝ n ) and F s p,q (ℝ n ) in the limiting case s = σ p := n max( 1 / p - 1, 0). These results may also be reformulated as optimal embeddings into the scale of Lorentz spaces L p,q (ℝ n ). We close several open problems outlined already in [H. Triebel, The structure of functions, Birkhauser, Basel, 2001] and explicitly stated in [D. D. Haroske, Envelopes and sharp embeddings of function spaces, Chapman & Hall/CRC, Boca Raton, FL, 2007].

Best constants in a borderline case of second-order Moser type inequalities

We study optimal embeddings for the space of functions whose Laplacian Δu belongs to L1(Ω), where Ω⊂RN is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W2,1(Ω) in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when N=2, we establish a sharp embedding inequality into the Zygmund space Lexp(Ω). On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem Δu=f(x)∈L1(Ω), u=0 on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger–Moser type inequalities to the case of higher-order derivatives. Extensions to dimension N⩾3 are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions.

Optimal Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving k-Modulus of Smoothness

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H σ X(IR n ) with order of smoothness σ ∈ (0, n), modelled upon rearrangement invariant Banach function spaces X(IR n ), into generalized Holder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR n ) is the Lorentz-Karamata space \(L_{p,q;b}({{\rm I\kern-.17em R}}^n)\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \(H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)\) into generalized Holder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces W k + 1 L n/k(logL) α (IR n ) and W k L n/k(logL) α (IR n ) into generalized Holder spaces.

Continuity Envelopes of Spaces of Generalized Smoothness in the Critical Case

The continuity envelope for the Besov and Triebel-Lizorkin spaces of generalized smoothness Bpq(s,Ψ)(ℝn) and Fpq(s,Ψ)(ℝn), respectively, are computed in the critical case s=n/p, provided that Ψ satisfies an appropriate critical condition. Surprisingly, in this critical situation, the corresponding optimal index is ∞, when compared with all the known results. Moreover, in the particular case of the classical spaces, we solve an open problem posed by Haroske in Envelopes and Sharp Embeddings of Function Spaces, Research Notes in Mathematics, vol. 437, Chapman & Hall, Boca Raton, 2007. As an immediate application of our results we give an upper estimate for approximation numbers of related embeddings.

Sharp Estimates of the Constants of Equivalence Between Integral Moduli of Smoothness and K-functionals in the Multivariate Case

The equivalence between various types of moduli of smoothness and respective Peetre K-functionals has been actively explored since the 1960s, in view of the importance of this topic for revealing connections among approximation theory, functional analysis and operator theory. The existence of the embedding constants in this equivalence relation (together with one-sided estimates for these constants) has been established in great generality, but the derived one-sided bounds are rather coarse (see, e.g., Johnen and Scherer, in Constructive Theory of Functions of Several Variables. Proc. Conf., Math. Res. Inst. Oberwolfach, 1976, pp. 119–140, Springer, Berlin, 1977 and the references therein). The problem of finding the sharp embedding constants for this equivalence was posed in Dechevsky, C. R. Acad. Bulg. 42(2), 21–24, 1989 and Int. J. Pure Appl. Math. 33(2), 157–186, 2006, where this problem was solved in the particular case of L2-metric, for real-valued and complex-valued functions of one real variable, with definition domain Ω=ℝ or \(\varOmega =\mathbb{T}\) (the periodic case). In the present paper we extend the results of Dechevsky to the case of several real variables: Ω=ℝn or \(\varOmega =\mathbb{T}^{n}\) , n∈ℕ. We consider two different types of equivalent norms for the Sobolev spaces involved in the K-functional (with and without intermediate mixed partial derivatives) and obtain a separate set of sharp two-sided bounds for the embedding constants in each of these two cases. We also briefly outline how the approach of the present study can be extended to the case of n-dimensional Lie (semi)groups.