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Bartlomiej Dyda - One of the best experts on this subject based on the ideXlab platform.
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characterizations for fractional Hardy Inequality
arXiv: Classical Analysis and ODEs, 2013Co-Authors: Bartlomiej Dyda, Antti V VahakangasAbstract:We provide a Maz'ya type characterization for a fractional Hardy Inequality. As an application, we show that a bounded open set $G$ admits a fractional Hardy Inequality if and only if the associated fractional capacity is quasiadditive with respect to Whitney cubes of $G$ and the zero extension operator acting on $C_c(G)$ is bounded in an appropriate manner.
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fractional Hardy sobolev maz ya Inequality for domains
Studia Mathematica, 2012Co-Authors: Bartlomiej Dyda, Rupert L FrankAbstract:We prove a fractional version of the Hardy–Sobolev–Maz’ya Inequality for arbitrary domains and Lp norms with p ≥ 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.
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fractional Hardy sobolev maz ya Inequality for domains
arXiv: Functional Analysis, 2011Co-Authors: Bartlomiej Dyda, Rupert L FrankAbstract: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.
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fractional Hardy Inequality with a remainder term
Colloquium Mathematicum, 2011Co-Authors: Bartlomiej DydaAbstract:We prove a Hardy Inequality for the fractional Laplacian on the interval with the optimal constant and additional lower order term. As a consequence, we also obtain a fractional Hardy Inequality with the best constant and an extra lower order term for general domains, following the method of M. Loss and C. Sloane [16]. 1. Main result and discussion Recently Loss and Sloane [16] have proven the following fractional Hardy Inequality (1.1) 1 2 ∫ D×D (u(x)− u(y)) |x− y|n+α dx dy ≥ κn,α ∫ D u(x) dist(x,Dc)α dx, u ∈ Cc(D), for convex domains D ⊂ R and 1 < α < 2, where (1.2) κn,α = π n−1 2 Γ( 2 ) Γ( 2 ) B ( 1+α 2 , 2−α 2 ) − 2 α2α is the optimal constant. Here B is the Euler beta function, and Cc(D) denotes the class of all continuous functions u : R → R with compact support inD. Inequality (1.1) with the optimal constant was earlier obtained for half-spaces and R \ {0}, see [10, 11, 5, 9]. In this note we will prove the following strengthening of (1.1) for the interval. Theorem 1.1. Let 1 < α < 2, −∞ < a < b <∞. For every u ∈ Cc(a, b), 1 2 ∫ b
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the fractional Hardy Inequality with a remainder term
arXiv: Analysis of PDEs, 2009Co-Authors: Bartlomiej DydaAbstract: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]]
Guofang Wang - One of the best experts on this subject based on the ideXlab platform.
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a Hardy moser trudinger Inequality
Advances in Mathematics, 2012Co-Authors: Guofang WangAbstract:Abstract In this paper we obtain an Inequality on the unit disk B in R 2 , which improves the classical Moser–Trudinger Inequality and the classical Hardy Inequality at the same time. Namely, there exists a constant C 0 > 0 such that ∫ B e 4 π u 2 H ( u ) d x ⩽ C 0 ∞ , ∀ u ∈ C 0 ∞ ( B ) ∖ { 0 } , where H ( u ) : = ∫ B | ∇ u | 2 d x − ∫ B u 2 ( 1 − | x | 2 ) 2 d x . This Inequality is a two-dimensional analog of the Hardy–Sobolev–Mazʼya Inequality in higher dimensions, which has been intensively studied recently. We also prove that the supremum is achieved in a suitable function space, which is an analog of the celebrated result of Carleson–Chang for the Moser–Trudinger Inequality.
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a Hardy moser trudinger Inequality
arXiv: Analysis of PDEs, 2010Co-Authors: Guofang WangAbstract:In this paper we obtain an Inequality on the unit disc $B$ in the plane, which improves the classical Moser-Trudinger Inequality and the classical Hardy Inequality at the same time. Namely, there exists a constant $C_0>0$ such that \[ \int_B e^{\frac {4\pi u^2}{H(u)}} dx \le C_0 < \infty, \quad \forall\; u\in C^\infty_0(B),\] where $$H(u) := \int_B |\n u|^2 dx - \int_B \frac {u^2}{(1-|x|^2)^2} dx.$$ This Inequality is a two dimensional analog of the Hardy-Sobolev-Maz'ya Inequality in higher dimensions, which was recently intensively studied. We also prove that the supremum is achieved in a suitable function space, which is an analog of the celebrated result of Carleson-Chang for the Moser-Trudinger Inequality.
Chuangxia Huang - One of the best experts on this subject based on the ideXlab platform.
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on pth moment exponential stability of stochastic cohen grossberg neural networks with time varying delays
Neurocomputing, 2010Co-Authors: Chuangxia HuangAbstract:With the help of Lyapunov function, stochastic analysis technique, generalized Halanay Inequality and Hardy Inequality, a set of novel sufficient conditions on pth moment exponential stability for non-autonomous stochastic Cohen-Grossberg neural networks is given, which modifies and generalizes some corresponding published results.
Durvudkhan Suragan - One of the best experts on this subject based on the ideXlab platform.
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euler semigroup Hardy sobolev and gagliardo nirenberg type inequalities on homogeneous groups
Semigroup Forum, 2020Co-Authors: Michael Ruzhansky, Durvudkhan Suragan, Nurgissa YessirkegenovAbstract:In this paper we describe the Euler semigroup $$\{e^{-t\mathbb {E}^{*}\mathbb {E}}\}_{t>0}$$ on homogeneous Lie groups, which allows us to obtain various types of the Hardy–Sobolev and Gagliardo–Nirenberg type inequalities for the Euler operator $$\mathbb {E}$$ . Moreover, the sharp remainder terms of the Sobolev type Inequality, maximal Hardy Inequality and $$|\cdot |$$ -radial weighted Hardy–Sobolev type Inequality are established.
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reverse integral Hardy Inequality on metric measure spaces
arXiv: Analysis of PDEs, 2019Co-Authors: Aidyn Kassymov, Michael Ruzhansky, Durvudkhan SuraganAbstract:In this note, we obtain a reverse version of the integral Hardy Inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy Inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski Inequality. Also, we present some consequences of the obtained reverse Hardy Inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds.
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layer potentials kac s problem and refined Hardy Inequality on homogeneous carnot groups
Advances in Mathematics, 2017Co-Authors: Michael Ruzhansky, Durvudkhan SuraganAbstract:Abstract We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's “principle of not feeling the boundary”. We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's Inequality and of the uncertainty principle.
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layer potentials green s formulae kac s problem and refined Hardy Inequality on homogeneous carnot groups
arXiv: Analysis of PDEs, 2015Co-Authors: Michael Ruzhansky, Durvudkhan SuraganAbstract:We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's "principle of not feeling the boundary". We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's Inequality and of the uncertainty principle.
Rupert L Frank - One of the best experts on this subject based on the ideXlab platform.
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fractional Hardy sobolev maz ya Inequality for domains
Studia Mathematica, 2012Co-Authors: Bartlomiej Dyda, Rupert L FrankAbstract:We prove a fractional version of the Hardy–Sobolev–Maz’ya Inequality for arbitrary domains and Lp norms with p ≥ 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.
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fractional Hardy sobolev maz ya Inequality for domains
arXiv: Functional Analysis, 2011Co-Authors: Bartlomiej Dyda, Rupert L FrankAbstract: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.
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sharp fractional Hardy inequalities in half spaces
arXiv: Functional Analysis, 2010Co-Authors: Rupert L Frank, Robert SeiringerAbstract:We determine the sharp constant in the Hardy Inequality for fractional Sobolev spaces on half-spaces. Our proof relies on a nonlinear and nonlocal version of the ground state representation.
