scholarly journals Riesz potentials of Radon measures associated to reflection groups

2018 ◽  
Vol 9 (2) ◽  
pp. 109-130 ◽  
Author(s):  
Léonard Gallardo ◽  
Chaabane Rejeb ◽  
Mohamed Sifi
Keyword(s):  
Heat Kernel ◽  
Root System ◽  
Radon Measure ◽  
Riesz Potential ◽  
Laplacian Operator ◽  
Reflection Groups ◽  
Riesz Potentials ◽  
Radon Measures ◽  
Dunkl Laplacian ◽  
Beta 2

AbstractFor a root systemRon{\mathbb{R}^{d}}and a nonnegative multiplicity functionkonR, we consider the heat kernel{p_{k}(t,x,y)}associated to the Dunkl Laplacian operator{\Delta_{k}}. For{\beta\in{]0,d+2\gamma[}}, where{\gamma=\frac{1}{2}\sum_{\alpha\in R}k(\alpha)}, we study the{\Delta_{k}}-Riesz kernel of index β, defined by{R_{k,\beta}(x,y)=\frac{1}{\Gamma(\beta/2)}\int_{0}^{+\infty}t^{{\beta/2}-1}p_% {k}(t,x,y)\,dt}, and the corresponding{\Delta_{k}}-Riesz potential{I_{k,\beta}[\mu]}of a Radon measure μ on{\mathbb{R}^{d}}. According to the values of β, we study the{\Delta_{k}}-superharmonicity of these functions, and we give some applications like the{\Delta_{k}}-Riesz measure of{I_{k,\beta}[\mu]}, the uniqueness principle and a pointwise Hedberg inequality.

scholarly journals B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces

2020 ◽  
Vol 18 (1) ◽  
pp. 715-730
Author(s):  
Javanshir J. Hasanov ◽  
Rabil Ayazoglu ◽  
Simten Bayrakci
Keyword(s):  
Differential Operator ◽  
Singular Integral ◽  
Riesz Potential ◽  
Type Theorem ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Singular Integral Operators ◽  
Sobolev Type ◽  
Riesz Potentials ◽  
Bessel Differential Operator

Abstract In this article, we consider the Laplace-Bessel differential operator {\Delta }_{{B}_{k,n}}=\mathop{\sum }\limits_{i=1}^{k}\left(\frac{{\partial }^{2}}{\partial {x}_{i}^{2}}+\frac{{\gamma }_{i}}{{x}_{i}}\frac{\partial }{\partial {x}_{i}}\right)+\mathop{\sum }\limits_{i=k+1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},{\gamma }_{1}\gt 0,\ldots ,{\gamma }_{k}\gt 0. Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator {M}_{b,\gamma } and the commutator {[}b,{A}_{\gamma }] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator {[}b,{I}_{\alpha ,\gamma }] of the B-Riesz potential on B-Morrey spaces {L}_{p,\lambda ,\gamma } , when b\in {\text{BMO}}_{\gamma } .

scholarly journals Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Pierre-Philippe Dechant
Keyword(s):  
Root System ◽  
Root Systems ◽  
Invariant Sets ◽  
Simple Construction ◽  
Reflection Groups ◽  
Computational Work ◽  
Algebraic Framework ◽  
Underlying Geometry ◽  
Coxeter Plane ◽  
Insight Into

AbstractRecent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

scholarly journals Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abderrazak Nabti ◽  
Ahmed Alsaedi ◽  
Mokhtar Kirane ◽  
Bashir Ahmad
Keyword(s):  
Diffusion Equation ◽  
Global Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Laplacian Operator ◽  
Nonlocal Source ◽  
Time Space ◽  
Nonexistence Of Solutions ◽  
Beta 2 ◽  
Fractional Laplacian Operator

Abstract We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source $$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^{ \gamma } \int _{0}^{t} (t-s)^{\alpha -1} \vert u \vert ^{p} \bigl\Vert \nu ^{ \frac{1}{q}}(x) u \bigr\Vert _{q}^{r} \,ds \end{aligned}$$ u t + ( − Δ ) β 2 u = ( 1 + | x | ) γ ∫ 0 t ( t − s ) α − 1 | u | p ∥ ν 1 q ( x ) u ∥ q r d s for $(x,t) \in \mathbb{R}^{N}\times (0,\infty )$ ( x , t ) ∈ R N × ( 0 , ∞ ) with initial data $u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$ u ( x , 0 ) = u 0 ( x ) ∈ L loc 1 ( R N ) , where $p,q,r>1$ p , q , r > 1 , $q(p+r)>q+r$ q ( p + r ) > q + r , $0<\gamma \leq 2 $ 0 < γ ≤ 2 , $0<\alpha <1$ 0 < α < 1 , $0<\beta \leq 2$ 0 < β ≤ 2 , $(-\Delta )^{\frac{\beta }{2}}$ ( − Δ ) β 2 stands for the fractional Laplacian operator of order β, the weight function $\nu (x)$ ν ( x ) is positive and singular at the origin, and $\Vert \cdot \Vert _{q}$ ∥ ⋅ ∥ q is the norm of $L^{q}$ L q space.

scholarly journals On Critical p-Laplacian Systems

2017 ◽  
Vol 17 (4) ◽  
pp. 641-659
Author(s):  
Zhenyu Guo ◽  
Kanishka Perera ◽  
Wenming Zou
Keyword(s):  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Existence And Multiplicity ◽  
Nonnegative Solutions ◽  
Least Energy Solution ◽  
Existence And Nonexistence ◽  
Alpha Beta ◽  
Beta 2 ◽  
Sobolev Exponent ◽  
Least Energy

AbstractWe consider the critical p-Laplacian system\left\{\begin{aligned} &\displaystyle{-}\Delta_{p}u-\frac{\lambda a}{p}\lvert u% \rvert^{a-2}u\lvert v\rvert^{b}=\mu_{1}\lvert u\rvert^{p^{\ast}-2}u+\frac{% \alpha\gamma}{p^{\ast}}\lvert u\rvert^{\alpha-2}u\lvert v\rvert^{\beta},&&% \displaystyle x\in\Omega,\\ &\displaystyle{-}\Delta_{p}v-\frac{\lambda b}{p}\lvert u\rvert^{a}\lvert v% \rvert^{b-2}v=\mu_{2}\lvert v\rvert^{p^{\ast}-2}v+\frac{\beta\gamma}{p^{\ast}}% \lvert u\rvert^{\alpha}\lvert v\rvert^{\beta-2}v,&&\displaystyle x\in\Omega,\\ &\displaystyle u,v\text{ in }D_{0}^{1,p}(\Omega),\end{aligned}\right.where {\Delta_{p}u:=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian operator defined onD^{1,p}(\mathbb{R}^{N}):=\bigl{\{}u\in L^{p^{\ast}}(\mathbb{R}^{N}):\lvert% \nabla u\rvert\in L^{p}(\mathbb{R}^{N})\bigr{\}},endowed with the norm {{\lVert u\rVert_{D^{1,p}}:=(\int_{\mathbb{R}^{N}}\lvert\nabla u\rvert^{p}\,dx% )^{\frac{1}{p}}}}, {N\geq 3}, {1<p<N}, {\lambda,\mu_{1},\mu_{2}\geq 0}, {\gamma\neq 0}, {a,b,\alpha,\beta>1} satisfy {a+b=p}, {\alpha+\beta=p^{\ast}:=\frac{Np}{N-p}}, the critical Sobolev exponent, Ω is {\mathbb{R}^{N}} or a bounded domain in {\mathbb{R}^{N}} and {D_{0}^{1,p}(\Omega)} is the closure of {C_{0}^{\infty}(\Omega)} in {D^{1,p}(\mathbb{R}^{N})}. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution of this system. We also consider the existence and multiplicity of the nontrivial nonnegative solutions.

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

2021 ◽  
pp. 1-28
Author(s):  
B. B. V. Maia ◽  
O. H. Miyagaki
Keyword(s):  
Riesz Potential ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Existence And Nonexistence ◽  
Exponential Critical Growth ◽  
Beta 2 ◽  
Image Position

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem \[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \] when $(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if $N\geq 3$ ) and $(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if $N=2$ ), where $I_{\alpha }$ and $I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem \[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \] where $\alpha ,\,\beta \in (0,\,2)$ and $f,\,g$ have exponential critical growth in the Trudinger–Moser sense.

scholarly journals Radon measure-valued solutions of first order scalar conservation laws

2018 ◽  
Vol 9 (1) ◽  
pp. 65-107
Author(s):  
Michiel Bertsch ◽  
Flavia Smarrazzo ◽  
Andrea Terracina ◽  
Alberto Tesei
Keyword(s):  
Radon Measure ◽  
Singular Part ◽  
Scalar Conservation Laws ◽  
Radon Measures ◽  
Lipschitz Continuous ◽  
First Order ◽  
Nonnegative Solutions ◽  
All Solutions ◽  
The Cauchy Problem ◽  
Scalar Conservation

AbstractWe study nonnegative solutions of the Cauchy problem\left\{\begin{aligned} &\displaystyle\partial_{t}u+\partial_{x}[\varphi(u)]=0&% &\displaystyle\phantom{}\text{in }\mathbb{R}\times(0,T),\\ &\displaystyle u=u_{0}\geq 0&&\displaystyle\phantom{}\text{in }\mathbb{R}% \times\{0\},\end{aligned}\right.where{u_{0}}is a Radon measure and{\varphi\colon[0,\infty)\mapsto\mathbb{R}}is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on φ, we prove their uniqueness if the singular part of{u_{0}}is a finite superposition of Dirac masses. Regarding the behavior of φ at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positivewaiting time(in the linear case{\varphi(u)=u}this happens for all times). In the latter case, we describe the evolution of the singular parts.

scholarly journals SPECTRUM AND HEAT KERNEL ASYMPTOTICS ON GENERAL LAAKSO SPACES

Fractals ◽  
2012 ◽  
Vol 20 (02) ◽  
pp. 149-162 ◽  
Author(s):  
MATTHEW BEGUÉ ◽  
LEVI DEVALVE ◽  
DAVID MILLER ◽  
BENJAMIN STEINHURST
Keyword(s):  
Heat Kernel ◽  
Spectral Dimension ◽  
Laplacian Operator ◽  
Heat Kernel Asymptotics ◽  
Leading Term

We introduce a method of constructing a general Laakso space while calculating the spectrum and multiplicities of the Laplacian operator. Using this information, we find the leading term of the trace of the heat kernel and the spectral dimension on an arbitrary Laakso space.

scholarly journals Ground state solutions of Nehari–Pohožaev type for a kind of nonlinear problem with general nonlinearity and nonlocal convolution term

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shuai Yuan ◽  
Fangfang Liao
Keyword(s):  
Nonlinear Problem ◽  
Ground State ◽  
Riesz Potential ◽  
Ground State Solution ◽  
State Solution ◽  
Ground State Solutions ◽  
Beta 2 ◽  
General Nonlinearity ◽  
New Inequalities

Abstract In this paper, we consider the following nonlinear problem with general nonlinearity and nonlocal convolution term: $$ \textstyle\begin{cases} -\Delta u+V(x)u+(I_{\alpha }\ast \vert u \vert ^{q}) \vert u \vert ^{q-2}u=f(u), \quad x\in {\mathbb{R}}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}), \quad \end{cases} $$ { − Δ u + V ( x ) u + ( I α ∗ | u | q ) | u | q − 2 u = f ( u ) , x ∈ R 3 , u ∈ H 1 ( R 3 ) , where $a\in (0,3)$ a ∈ ( 0 , 3 ) , $q\in [1+\frac{\alpha }{3},3+\alpha )$ q ∈ [ 1 + α 3 , 3 + α ) , $I_{\alpha }:\mathbb{R}^{3}\rightarrow \mathbb{R}$ I α : R 3 → R is the Riesz potential, $V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))$ V ∈ C ( R 3 , [ 0 , ∞ ) ) , $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ f ∈ C ( R , R ) and $F(t)=\int _{0}^{t}f(s)\,ds$ F ( t ) = ∫ 0 t f ( s ) d s satisfies $\lim_{|t|\to \infty }F(t)/|t|^{\sigma }=\infty $ lim | t | → ∞ F ( t ) / | t | σ = ∞ with $\sigma =\min \{2,\frac{2\beta +2}{\beta }\}$ σ = min { 2 , 2 β + 2 β } where $\beta =\frac{ \alpha +2}{2(q-1)}$ β = α + 2 2 ( q − 1 ) . By using new analytic techniques and new inequalities, we prove the above system admits a ground state solution under mild assumptions on V and f.

scholarly journals Twisted quadratic foldings of root systems

2021 ◽  
Vol 33 (1) ◽  
pp. 65-84
Author(s):  
M. Lanini ◽  
K. Zainoulline
Keyword(s):  
Root System ◽  
Equivariant Cohomology ◽  
Quaternion Algebra ◽  
Root Systems ◽  
Group Cohomology ◽  
Characteristic Classes ◽  
Reflection Groups ◽  
Content Type ◽  
Dynkin Diagrams ◽  
Projective Homogeneous Varieties

The present paper is devoted to twisted foldings of root systems that generalize the involutive foldings corresponding to automorphisms of Dynkin diagrams. A motivating example is Lusztig’s projection of the root system of type E 8 E_8 onto the subring of icosians of the quaternion algebra, which gives the root system of type H 4 H_4 . By using moment graph techniques for any such folding, a map at the equivariant cohomology level is constructed. It is shown that this map commutes with characteristic classes and Borel maps. Restrictions of this map to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups are also introduced and studied.

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