Morrey–Sobolev Extension Domains

We show that every uniform domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 2\) is a Morrey–Sobolev \({\mathscr {W}}^{1,\,p}\)-extension domain for all \(p\in [1,\,n)\), and moreover, that this result is essentially the best possible for each \(p\in [1,\,n)\) in the sense that, given a simply connected planar domain or a domain of \({{{\mathbb {R}}}^n}\) with \(n\ge 3\) that is quasiconformal equivalent to a uniform domain, if it is a \({\mathscr {W}}^{1,\,p} \)-extension domain, then it must be uniform.     

Sharp Hardy–Sobolev inequalities

Let Ω be a smooth bounded domain in ${\mathbb{R}}^N$, $N?3$. We show that Hardy's inequality involving the distance to the boundary, with best constant ($\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$), may still be improved by adding a multiple of the critical Sobolev norm. To cite this article: S. Filippas et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Fractional Sobolev,Moser–Trudinger,Morrey–Sobolev inequalities under Lorentz norms

We consider the Sobolev type inequalities under Lorentz norms on bounded open domains for fractional derivatives (−∆) ^s/2 in the following three cases: n > ps, n = ps, and n < ps, whence establishing the weak type Sobolev inequalities, Moser–Trudinger and Morrey–Sobolev inequalities for fractional derivatives in Lorentz norms. Applying these inequalities, we obtain the trace forms of six related functional inequalities. Bibliography: 44 titles.     

Sharp convex Lorentz–Sobolev inequalities

New sharp Lorentz–Sobolev inequalities are obtained by convexifying level sets in Lorentz integrals via the L ^p Minkowski problem. New L ^p isocapacitary and isoperimetric inequalities are proved for Lipschitz star bodies. It is shown that the sharp convex Lorentz–Sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities.     

Nondiagonal Hermite–Sobolev Orthogonal Polynomials

In this work, we study algebraic and analytic properties for the polynomials { Q _n } _{n 0}, which are orthogonal with respect to the inner product where , R such that – ² > 0.     

Affine Moser–Trudinger and Morrey–Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L ⁿ energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L ⁿ Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L ^p energy, with p > n, is replaced by the affine energy.     

Interpolation operators in Orlicz–Sobolev spaces

We study classical interpolation operators for finite elements, like the Scott–Zhang operator, in the context of Orlicz–Sobolev spaces. Furthermore, we show estimates for these operators with respect to quasi-norms which appear in the study of systems of p-Laplace type.     

Pointwise behaviour of Orlicz–Sobolev functions

We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded in the Orlicz space $L^{\Psi}(X)We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded in the Orlicz space , then each Orlicz–Sobolev function can be approximated by a H?lder continuous function both in the Lusin sense and in norm. The research is supported by the Centre of Excellence Geometric Analysis and Mathematical Physics of the Academy of Finland.     

Composition operators in Orlicz–Sobolev spaces

We obtain necessary and sufficient conditions for a homeomorphism of domains in a Euclidean space to generate a bounded embedding operator of the Orlicz–Sobolev spaces defined by a special class of N-functions.     

Hardy–Littlewood–Sobolev inequality on the parabolic biangle

The Ramanujan Journal - We define the heat semigroup associated with a system of bivariate Jacobi polynomials which are orthogonal with respect to a probability measure on the parabolic biangle...     

On Trudinger–Moser type inequalities involving Sobolev–Lorentz spaces

Generalizations of the Trudinger–Moser inequality to Sobolev–Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of concentration–compactness.      

A trace law for the Hardy–Morrey–Sobolev space

This article shows that if $H{L}^{p,\kappa }$ and $H{L}_{\mu }^{q,\lambda }$ are $(p,\kappa )$-Hardy–Morrey space based on the standard Lebesgue measure and $(q,\lambda )$-Hardy–Morrey space induced by the non-negative Radon measure μ respectively then one has the following trace law for the Hardy–Morrey–Sobolev space $I_{\alpha }(H{L}^{p,\kappa })$ of Riesz potentials of order α of $H{L}^{p,\kappa }$-functions:

$\begin{array}{cc}\hfill & \underset{0<{6f6}_{H{L}^{p,\kappa }}<\infty }{\sup }?{6I_{\alpha }f6}_{H{L}_{\mu }^{q,\lambda }}{6f6}_{H{L}^{p,\kappa }}^{?1}<\infty \hfill \\ \hfill & ?\underset{(x,r)\in {\mathbb{R}}^n\times (0,\infty )}{\sup }?r^{?\beta }\mu (B(x,r))<\infty \hfill \end{array}$

under

$\{\begin{array}{c}0<\lambda \le \kappa \le n;\hfill \\ 0<\alpha <n;\hfill \\ 0<p<\kappa /\alpha ;\hfill \\ n?\alpha p<\beta \le n;\hfill \\ 0<q=p(\beta +\lambda ?n)/(\kappa ?\alpha p).\hfill \end{array}$

     

Nonhomogeneous multiparameter problems in Orlicz–Sobolev spaces

This paper is concerned with the existence and multiplicity of solutions for a class of problems involving the Φ-Laplacian operator with general assumptions on the nonlinearities, which include both semipositone cases and critical concave convex problems. The research is based on the subsupersolution technique combined with a truncation argument and an application of the Mountain Pass Theorem. The results in this paper improve and complement some recent contributions to this field.     

Fredholmness of multipliers on Hardy–Sobolev spaces

This paper discusses the Fredholmness of multipliers on Hardy–Sobolev Spaces and obtains an index formula for the multipliers with some special symbols. Our results show that Hardy–Sobolev spaces have richer properties than classical holomorphic function spaces, and the behavior of the operators on these spaces is complex. Some methods of Hardy or Bergman spaces fail in the case of the Hardy–Sobolev space.     

Carleson Type Measures for Fock–Sobolev Spaces

We describe the \((p,q)\) Fock–Carleson measures for weighted Fock–Sobolev spaces in terms of the objects \((s,t)\) -Berezin transforms, averaging functions, and averaging sequences on the complex space \(\mathbb{C }^n\) . The main results show that while these objects may have growth not faster than polynomials to induce the \((p,q)\) measures for \(q\ge p\) , they should be of \(L^{p/(p-q)}\) integrable against a weight of polynomial growth for \(q<p\) . As an application, we characterize the bounded and compact weighted composition operators on the Fock–Sobolev spaces in terms of certain Berezin type integral transforms on \(\mathbb{C }^n\) . We also obtained estimation results for the norms and essential norms of the operators in terms of the integral transforms. The results obtained unify and extend a number of other results in the area.