On the trace problem for Lizorkin–Triebel spaces with mixed norms

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x₁ = 0 and x_n = 0 are applied to functions belonging to quasi‐homogeneous, mixed norm Lizorkin–Triebel spaces ; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised up to the borderline cases; these are also covered in case x₁ = 0. For x₁ the trace spaces are proved to be mixed‐norm Lizorkin–Triebel spaces with a specific sum exponent; for x_n they are similarly defined Besov spaces. The treatment includes continuous right‐inverses and higher order traces. The results rely on a sequence version of Nikol'skij's inequality, Marschall's inequality for pseudodifferential operators (and Fourier multiplier assertions), as well as dyadic ball criteria. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elliptic equations with nonzero boundary conditions in weighted Sobolev spaces

We present weighted Sobolev spaces along with a trace theorem and an interpolation theorem for the spaces. Then we solve nonzero boundary value problems for elliptic equations in .     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

On exposed functions in Bernstein spaces

For σ > 0, the Bernstein space {ie427-01} consists of those L ¹(ℝ) functions whose Fourier transforms are supported by [−σ, σ]. Since {ie427-02} is separable and dual to some Banach space, the closed unit ball {ie427-03} of {ie427-04} has sufficiently large sets of both exposed and strongly exposed points: {ie427-05} coincides with the closed convex hull of its strongly exposed points. We investigate some properties of exposed points, construct several examples, and obtain as corollaries relations between the sets of exposed, strongly exposed, weak* exposed, and weak* strongly exposed points of {ie427-06}.     

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.

     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be in  .

     

Remarks on the results by Koskela concerning the radial uniqueness for Sobolev functions

In this note we aim to complete the results by Koskela concerning the radial uniqueness for Sobolev functions.

Let be a positive nonincreasing function on the interval , and let denote the unit ball of . Consider a -precise function on such that

where . We give conditions on which assure that whenever has vanishing fine boundary limits on a set of positive -capacity.

We are also concerned with the sharpness.

     

On the well-posedness of a two-phase minimization problem

We prove a series of results concerning the emptiness and non-emptiness of a certain set of Sobolev functions related to the well-posedness of a two-phase minimization problem, involving both the p(x)-norm and the infinity norm. The results, although interesting in their own right, hold the promise of a wider applicability since they can be relevant in the context of other problems where minimization of the p-energy in a part of the domain is coupled with the more local minimization of the L^∞-norm on another region.     

New characterizations of Hajlasz-Sobolev spaces on metric spaces

This paper introduces the fractional Sobolev spaces on spaces of homogeneous type,includingmetric spaces and fractals. These Sobolev spaces include the well-known Hajfasz-Sobolev spaces as specialmodels.The author establishes varions chaaracterizations of(sharp)maximal functions for these spaces.Asapplications,the author identifies the fractional Sobolev spaces with some Lipscitz-type spaces.Moreover;some embedding theorems are also given.     

On a class of universal modular sequence spaces

The class of all subspaces of quotient spaces ofl _p ⊕l, contains all separable Orlicz sequence spaces. This is part of the author’s Ph.D. thesis prepared at University of California at Berkeley under the supervision of Professor H. P. Rosenthal.     

A capacitary weak type inequality for Sobolev functions and its applications

In this paper a capacitary weak type inequality for Sobolev functions is established and is applied to reprove some well-known results concerning Lebesgue points, Taylor expansions in the -sense, and the Lusin type approximation of Sobolev functions.

     

Holomorphic Sobolev spaces, Hermite and special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform

The images of Hermite and Laguerre-Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterized. These are used to characterize the image of Schwartz class of rapidly decreasing functions f on Rn and Cn under these semigroups. The image of the space of tempered distributions is also considered and a Paley-Wiener theorem for the windowed (short-time) Fourier transform is proved.     

On dualities between function spaces

By [6], the dualities between and , whereX andW are two sets and (i.e., the mappings satisfying for all and all index setsI), can be represented with the aid of functions . Here we show that they can be also represented with the aid of functions , whereR = (–, +). As an application, we show that every duality is completely determined by a suitable duality between 2 ^{X ×R} and 2 ^{W ×R} (i.e., a mapping 2 ^{X ×R} 2 ^{W ×R} satisfying for all {M _i} _iI 2 ^{X ×R} and all index setsI), applied to the epigraphs of the functions .     

Relative rearrangement and Lebesgue spaces with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L^p()(Ω) with variable exponent. In particular, we show that the maps uL^p()(Ω)→k(t)u_*L^p_*()(0,measΩ) and uL^p()(Ω)→u_*L^p*()(0,measΩ) are locally -Hölderian (u_* (resp. p^*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.