Fractional calculus for power functions
We calculate the fractional Laplacian for functions of the form $u(x)=(1-|x|^2)_+^p$ and $v(x)=x_d u(x)$. As an application, we estimate the first eigenvalues of the fractional Laplacian in a ball.
math.AP↗
arXiv subjects
Bartłomiej Dyda
Publications and source records attributed to Bartłomiej Dyda.
Comparability and regularity estimates for symmetric nonlocal Dirichlet forms
The aim of this work is to study comparability of nonlocal Dirichlet forms. We provide sufficient conditions on the kernel for local and global comparability. As an application we prove a-priori estimates in Hölder spaces for solutions to integrodifferential equations. These solutions are defined with the help of symmetric nonlocal Dirichlet forms.
math.AP↗
Fractional Hardy-Sobolev-Maz'ya inequality for domains
We prove a fractional version of the Hardy--Sobolev--Maz'ya inequality for arbitrary domains and $L^p$ norms with $p\geq 2$. This inequality combines the fractional Sobolev and the fractional Hardy inequality into a single inequality, while keeping the sharp constant in the Hardy inequality.
math.FA↗
Fractional Hardy-Sobolev-Maz'ya inequality on balls and halfspaces
We prove fractional Hardy--Sobolev--Maz'ya inequality for balls and a half-space, partially answering the open problem posed by Frank and Seiringer [arXiv:0906.1561v1 [math.FA], 2009] We note that for half-spaces this inequality has been recently obtained by Sloane [arXiv:1004.4828v1 [math.FA], 2010].
math.FA↗
The fractional Hardy inequality with a remainder term
We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]
math.AP↗
The best constant in a fractional Hardy inequality
We prove an optimal Hardy inequality for the fractional Laplacian on the half-space.
math.AP↗