Refined Sobolev Inequalities in Lorentz Spaces

We establish refined Sobolev inequalities between the Lorentz spaces and homogeneous Besov spaces. The sharpness of these inequalities is illustrated on several examples, in particular based on non-uniformly oscillating functions known as chirps. These results are also used to derive refined Hardy inequalities.     

Moser Type Inequalities for Higher-Order Derivatives in Lorentz Spaces

Sharp constants are exhibited in exponential inequalities corresponding to the limiting case of the Sobolev inequalities in Lorentz-Sobolev spaces of arbitrary order.      

Convolution Inequalities in Lorentz Spaces

In this paper we study boundedness of the convolution operator in different Lorentz spaces. We obtain the limit case of the Young-O’Neil inequality in the classical Lorentz spaces. We also investigate the convolution operator in the weighted Lorentz spaces.     

Operators on Sobolev Type Spaces

In this paper, we introduce the work done on Hardy-Sobolev spaces and Fock-Sobolev spaces and their operators and operator algebras, including the study of boundedness, compactness, Fredholm property, index theory, spectrum and essential spectrum, norm and essential norm, Schatten-p classes, and the $C^∗$ algebras generated by them.     

Optimal Rearrangement Invariant Sobolev Embeddings in Mixed Norm Spaces

We improve the Sobolev-type embeddings due to Gagliardo (Ric Mat 7:102–137, 1958) and Nirenberg (Ann Sc Norm Sup Pisa 13:115–162, 1959) in the setting of rearrangement invariant (r.i.) spaces. In particular, we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between r.i. spaces and mixed norm spaces. As a consequence, we prove that the classical estimate for the standard Sobolev space \(W^{1}L^{p}\) by Poornima (Bull Sci Math 107(3):253–259,  1983), O’Neil (Duke Math J 30:129–142,  1963) and Peetre (Ann Inst Fourier 16(1):279–317,  1966) (\(1 \le p < n\)), and by Hansson (Math Scand 45(1):77–102,  1979, Brezis and Wainger (Commun Partial Differ Equ 5(7):773–789,  1980) and Maz’ya (Sobolev spaces,  1985) (\(p=n\)) can be further strengthened by considering mixed norms on the target spaces.     

Optimal Weyl-type Inequalities for Operators in Banach Spaces

Let (s_n) be an s-number sequence. We show for each k = 1, 2, . . . and n ≥ k + 1 the inequality between the eigenvalues and s-numbers of a compact operator T in a Banach space. Furthermore, the constant (k + 1)^1/2 is optimal for n = k + 1 and k = 1, 2, . . .. This inequality seems to be an appropriate tool for estimating the first single eigenvalues. On the other hand we prove that the Weyl numbers form a minimal multiplicative s-number sequence and by a well-known inequality between eigenvalues and Weyl numbers due to A. Pietsch they are very good quantities for investigating the optimal asymptotic behavior of eigenvalues. Research of the second author was supported by the DFG Emmy-Noether grant Hi 584/2-3.     

Two-Weighted Inequalities for Integral Operators in Lorentz Spaces Defined on Homogeneous Groups

The optimal sufficient conditions are found for weights, which guarantee the validity of two-weighted inequalities for singular integrals in the Lorentz spaces defined on homogeneous groups. In some particular case, the found conditions are necessary for the corresponding inequalities to be valid. Also, the necessary and sufficient conditions are found for pairs of weights, which provide the validity of two-weighted inequalities for the generalized Hardy operator in the Lorentz spaces defined on homogeneous groups.     

On Sobolev and Capacitary Inequalities for Contractive Besov Spaces over d-sets

We derive Sobolev inequalities for Besov spaces B ^p,p (F), 0<<1, 1p< on d-sets F in R ⁿ , dn, from a metric property of the Bessel capacity on R ⁿ . We first extend Kaimanovitch's result on the equivalence of Sobolev and capacitary inequalites for contractive p-norms in a general setting allowing unbounded Lévy kernels. A simple part of the Jonsson–Wallin trace theorem for Besov spaces and some basic properties of Bessel and Besov capacities on R ⁿ are then utilized in getting the desired inequalities. When p=2, the Besov space being considered is a non-local regular Dirichlet space and gives rise to a jump type symmetric Markov process M over the d-set. The upper bound of the transition function of M and metric properties of M -polar sets are then exhibited.     

A Property of Sobolev Spaces and Existence in Optimal Design

Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W ₀ ^l,p (Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary value problems of arbitrary order, in arbitrary dimension and with general cost functionals.     

A Property of Sobolev Spaces and Existence in Optimal Design

   Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W ₀ ^l,p (Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary value problems of arbitrary order, in arbitrary dimension and with general cost functionals.     

Cheeger Type Sobolev Spaces for Metric Space Targets

In this paper, we consider the natural generalization of Cheeger type Sobolev spaces to maps into a metric space. We solve Dirichlet problem for CAT(0)-space targets, and obtain some results about the relation between Cheeger type Sobolev spaces for maps into a Banach space and those for maps into a subset of that Banach space. We also prove the minimality of upper pointwise Lipschitz constant functions for locally Lipschitz maps into an Alexandrov space of curvature bounded above.     

Trace Inequalities of Sobolev Type in the Upper Triangle Case

We give a new non-capacitary characterization of positive Borelmeasures µ on Rⁿ such that the potential space I*L^p isimbedded in L^q(dµ) for $1qp+, that is, the trace inequality holds, for Riesz potentials I = (- )². A weak-type trace inequality is also characterized. A non-isotropic version on the unit sphere Sⁿ is studied,as well as the holomorphic case for Hardy–Sobolev spaces in the ball. 1991 MathematicsSubject Classification: primary 31C15, 42B20; secondary 32A35.     

Variational Inequalities in Hilbert Spaces with Measures and Optimal Stopping Problems

We study the existence theory for parabolic variational inequalities in weighted L ² spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L ² setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.     

Distribution of Points and Hardy Type Inequalities in Spaces of Homogeneous Type

In the setting of spaces of homogeneous type, we study some Hardy type inequalities, which notably appeared in the proofs of local T(b) theorems. We give some sufficient conditions ensuring their validity, related to the geometry and distribution of points in the homogeneous space. We study the relationships between these conditions and give some examples and counterexamples in the complex plane.     

Weak Type Inequalities for Best Simultaneous Approximation in Banach Spaces

For arbitrary Banach spaces Butzer and Scherer in 1968 showed that the approximation order of best approximation can characterized by the order of certain K-functionals. This general theorem has many applications such as the characterization of the best approximation of algebraic polynomials by moduli of smoothness involving the Legendre, Chebyshev, or more general the Jacobi transform. In this paper we introduce a family of seminorms on the underlying approximation space which leads to a generalization of the Butzer–Scherer theorems. Now the characterization of the weighted best algebraic approximation in terms of the so-called main part modulus of Ditzian and Totik is included in our frame as another particular application. The goal of the paper is to show that for the characterization of the orders of best approximation, simultaneous approximation (in different spaces), reduction theorems, and K-functionals one has (essentially) only to verify three types of inequalities, namely inequalities of Jackson-, Bernstein-type and an equivalence condition which guarantees the equivalence of the seminorm and the underlying norm on certain subspaces. All the results are given in weak-type estimates for almost arbitrary approximation orders, the proofs use only functional analytic methods.     

Sobolev Type Spaces on the Dual of the Laguerre Hypergroup

Sobolev type spaces E ^s,p (0, sR, p[1,+]) are defined on R×N by using the Fourier transform and its inverse on the Laguerre hypergroup. An analogue of H ^s (R ⁿ ), denoted by H ^s is investigated in this paper. Some properties including completeness and imbedding results for these spaces are given, Reillich-type theorem and Poincaré's inequality are proved. Also, global regularity results for certain differential operators are obtained.     

Equivalence of Sobolev Spaces

We repair the proof of equivalence of certain L²-Sobolev spaces on manifolds with bounded curvature of all orders from [4]. The results are extended to generalized compatible Dirac operators, fractional order Sobolev spaces and weighted Sobolev spaces. A certain way of doing coordinate free computations is presented.