arXiv subjects
Publications and source records attributed to Ritva Hurri-Syrjänen.
We study integrability properties of the Wolff potential in context of Choquet integrals with respect to the Hausdorff content. As an application we give integrability results to the solutions of the $p$-Laplace equation in this context.
We study Poincare-Sobolev type inequalities for compactly supported smooth functions which are defined in the Euclidean $n$-space and whose absolute value of gradient are Choquet $δ/n$-integrable with respect to the $δ$-dimensional Hausdorff content, $n\geq 2$, $δ\in (0,n]$. In particular, our results imply a new Sobolev inequality for quasicontinuous functions defined in the Sobolev space $W^{1,1}_0(\mathbb{R}^n)$. As an application we extend a recently introduced superlevel Sobolev inequality into a context of the Hausdorff content.
We consider Choquet integrals with respect to dyadic Hausdorff content of non-negative functions which are not necessarily Lebesgue measurable. We study the theory of Lebesgue points. The studies yield convergence results and also a density result between function spaces. We provide examples which show sharpness of the main convergence theorem. These examples give additional information about the convergence in the norm also, namely the difference of the functions in this setting and continuous functions.
We consider integrals in the sense of Choquet with respect to the $δ$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<δ\le n$. In particular, for these functions we prove Sobolev inequalities in the limiting case $p=δ/n$ and in the case $p>δ$, here $p$ is the integrability exponent of the absolute value of the gradient of any given function. The results complement previously known Poincaré-Sobolev and Morrey inequalities.
We introduce Riesz potentials for non-Lebesgue measurable functions by taking the integrals in the sense of Choquet with respect to Hausdorff content and prove boundedness results for these operators. Some earlier results are recovered or extended now using integrals taken in the sense of Choquet with respect to Hausdorff content. Some earlier results also for maximal operators are considered, but now for non-measurable functions.
We consider inequalities where integrals are defined in the sense of Choquet with respect to Hausdorff content. We study cases where continuously differentiable functions are defined on open, connected sets with so much regularity that there exists a pointwise estimate between the values of a function and its gradient under the maximal operator or the Riesz potential, at every point of the set. We show that certain Hardy inequalities and Poincare-Sobolev inequalities are valid in this context.
We prove weighted inequalities between the Gagliardo and Sobolev seminorms and also between the Marcinkiewicz quasi-norm and the Sobolev seminorm. With $A_1$ weights we improve earlier results of Bourgain, Brezis, and Mironescu.
In this paper we unify and improve some of the results of Bourgain, Brezis and Mironescu and the weighted Poincaré-Sobolev estimate by Fabes, Kenig and Serapioni. More precisely, we get weighted counterparts of the Poincaré-Sobolev type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the fact that we keep track of the behaviour of the constants involved when the fractional parameter approaches to $1$. Our main method is based on techniques coming from harmonic analysis related to the self-improving property of generalized Poincaré inequalities.
We study weak-type estimates and exponential integrability for the variable order Riesz potential. As an application we prove an exponential integrability result with respect to the Hausdorff content for functions from variable exponent Sobolev spaces. In particular, the earlier exponential integrability results are improved to a corresponding one with respect to the Choque integral whenever John domains are considered. Moreover, new exponential integrability results also for domains with outward cusps are obtained.
In this article we obtain estimates of Neumann eigenvalues of $p$-Laplace operators in a large class of space domains satisfying quasihyperbolic boundary conditions. The suggested method is based on composition operators generated by quasiconformal mappings and their applications to Sobolev-Poincaré-inequalities. By using a sharp version of the inverse Hölder inequality we refine our estimates for quasi-balls, that is, images of balls under quasiconformal mappings of the whole space.
Weighted fractional Poincaré-type inequalities are proved on John domains whenever the weights defined on the domain are depending on the distance to the boundary and to an arbitrary compact set in the boundary of the domain.
We study the validity of (q,p)-Poincaré inequalities, q p_0, where p_0 is an explicit constant depending on q, on the logarithmic growth condition, and on the boundary of the domain.
We study point-wise estimates for the modified Riesz potential. We show that the point-wise estimates imply embeddings into Orlicz spaces from the L^1_p-space where the functions are defined in non-smooth domains. The Orlicz functions depend on the geometry of the domain. We show that the Orlicz function is optimal when p=1.
We establish local to global results for a function space which is larger than the well known BMO space, and was also introduced by John and Nirenberg.
We prove pointwise estimates to the modified Riesz potential. We show the boundedness of its Luxemburg norm. As an application we obtain Orlicz embedding results. We study the sharpness of the results.
We prove fractional Sobolev-Poincaré inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.
We prove capacitary strong type inequalities for functions belonging to Orlicz-Sobolev spaces. As an application we consider capacitary averages and their limits.
We establish fractional Hardy-type inequalities in a bounded domain with plump complement. In particular our results apply in bounded C^\infty domains and Lipschitz domains.