diff --git "a/B9E1T4oBgHgl3EQfpgVg/content/tmp_files/load_file.txt" "b/B9E1T4oBgHgl3EQfpgVg/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/B9E1T4oBgHgl3EQfpgVg/content/tmp_files/load_file.txt" @@ -0,0 +1,561 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf,len=560 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='03332v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='AP] 9 Jan 2023 THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY ON THE H-TYPE GROUP QIAOHUA YANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We determine the optimal constant in the L2 Folland-Stein in- equality on the H-type group, which partially confirms the conjecture given by Garofalo and Vassilev (Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof is inspired by the work of Frank and Lieb (Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=', 2012) and Hang and Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Introduction Let G be a stratified, simply connected nilpotent Lie group (in short a Carnot group) of step r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Denote by g the Lie algebra of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It is known that g = �r i=1 Vi satisfying (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' [10]) [V1, Vj] = Vj+1, 1 ≤ j ≤ r − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' [V1, Vr] = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' As a simply connected nilpotent group, G is differential with RN, N = �r i=1 dim Vi, via the exponential map exp : g → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' There is a natural family of nonisotropic dilations δλ : g → g for λ > 0 and we define it as follows: δλ(X1 + · · · + Xr) = λX1 + · · · + λrXr, Xj ∈ Vj, 1 ≤ j ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The homogeneous dimension of G, associated with δλ, is Q = �r j=1 j dim Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Via the exponential map exp : g → G, we define the group of dilations on G as follows: δλ(g) = exp ◦δλ ◦ exp−1(g), g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Set nj = dim Vj, 1 ≤ j ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let {X1, · · · , Xn1} be a basis of V1 and denote by ∇G = (X1, · · · , Xn1) the horizontal gradient of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The sub-Laplacian on G is ∆G = �n1 i=1 X2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The Sobolev space W 1,p 0 (G) is the closure of C∞ 0 (G) with respect to the norm ∥u∥W 1,p 0 (G) = �� G |∇Gu|pdg � 1 2 , where dg is the Haar measure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We remark that the Haar measure on G, induced by the exponential mapping from the Lebesgue measure on g = RN, coin- cides the Lebesgue measure on RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The Folland-Stein inequality on G reads that there exits some constant C > 0 such that for each u ∈ W 1,p 0 (G) (see [8, 9]), �� G |u| pQ Q−p dg � Q−p pQ ≤ C �� G |∇Gu|pdg � 1 p , 1 < p < Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Primary: 43A80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 46E35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 22E25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Folland-Stein inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Heisenberg group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' H-type group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' best constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The work was partially supported by the National Natural Science Foundation of China(No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='11201346).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 1 2 QIAOHUA YANG For the existence and regularity of minimizers of the Folland-Stein inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we refer to [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The Heisenberg group is the simplest example of Carnot group of step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We denote it by Hn = (Cn × R, ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The group law on Hn is given by (z, t) ◦ (z′, t′) = (z + z′, t + t′ + 2Imz · z′), where z · z′ = �n j=1 zj¯z′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The homogeneous norm on Hn is given by |(z, t)| = (|z|4 + t2) 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In a series of papers [18, 19, 20], Jerison and Lee, among other results, determined the explicit computation of the extremal functions in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) in the case p = 2 and G = Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In fact, the extremal functions are, up to group translations and dilations, c((1 + |z|2)2 + t2)− Q−2 4 , c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Such inequalities play an important role in the study of CR Yamabe problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Later, in a celebrated paper [11], Frank and Lieb established sharp Hardy-Littlewood- Sobolev inequalities on Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We state the result as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1 (Frank-Lieb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 0 < λ < Q and p = 2Q Q−λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then for any f, g ∈ Lp(Hn), ����� � � Hn×Hn f(z, t)g(z′, t′) |(z, t)−1 ◦ (z′, t′)|λ dzdtdz′dt′ ����� ≤ � πn+1 2n−1n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' �λ/Q n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='Γ( Q−λ 2 ) Γ2( Q−2λ 4 ) ∥f∥p∥g∥p, with equality if and only if, up to group translations and dilations, f = c((1 + |z|2)2 + t2)− 2Q−λ 4 , g = c′((1 + |z|2)2 + t2)− 2Q−λ 4 for some c, c′ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In particular, choosing λ = Q−2 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1 yields the Jerison and Lee’s in- equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Using the method in [11], Frank and Lieb [12] also gave a new, rearrangement- free proof of sharp Hardy-Littlewood-Sobolev inequalities on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Recently, Hang and Wang [15] present a shorter proof of the Frank-Lieb inequality, in which they bypasses the subtle proof of existence and the Hersch-type argument via subcritical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Some of the results of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1 have been generalized to the cases of quater- nionic Heisenberg group and octonionic Heisenberg group (see [4, 5, 16, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We note that Heisenberg group, quaternionic Heisenberg group and octonionic Heisen- berg group are known as the groups of Iwasawa type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=', the nilpotent component in the Iwasawa decomposition of simple groups of rank one (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The aim of this paper is to look for the optimal constant of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) when p = 2 and G is a group of Heisenberg type (in short a H-type group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Recall that a H-type group G is a Carnot group of step two with the following properties (see Kaplan [21]): the Lie algebra g of G is endowed with an inner product ⟨, ⟩ such that, if z is the center of g, then [z⊥, z⊥] = z and moreover, for every fixed z ∈ z, the map Jz : z⊥ → z⊥ defined by ⟨Jz(v), ω⟩ = ⟨z, [v, ω]⟩, ∀ω ∈ z⊥ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2) is an orthogonal map whenever ⟨z, z⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It is known (see [6]) that a H-type group G is the group of Iwasawa type if and only if its Lie algebra satisfies the following THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 3 J2-condition: for any v ∈ z⊥ and z, z′ ∈ z such that ⟨z, z′⟩ = 0, there exists z′′ ∈ z such that JzJz′v = Jz′′v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, most of H-type groups are not groups of Iwasawa type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Set m = dim z⊥ and n = dim z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Since G has step two, we can fix on G a system of coordinates (x, t) such that the group law on G has the form (see [2]) (x, t) ◦ (x′, t′) = � xi + x′ i, i = 1, 2, · · · , m tj + t′ j + 1 2⟨x, U (j)x′⟩, j = 1, 2, · · · , n � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) for suitable skew-symmetric matrices U (j)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Nextly, we set U(ξ) = �� 1 + |x|2 4 �2 + |t|2 �− Q−2 4 , ξ = (x, t) ∈ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) Uλ,η(ξ) =λ Q−2 2 U(δλ(η−1 ◦ ξ)), η ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5) It has been shown that [m(Q − 2)] Q−2 4 Uλ,η(ξ) satisfies the Yamabe-type equation (see [13, 14]) ∆G[m(Q − 2)] Q−2 4 Uλ,η + {[m(Q − 2)] Q−2 4 Uλ,η} Q+2 Q−2 = 0, or equivalently, ∆GUλ,η + m(Q − 2)U Q+2 Q−2 λ,η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) In the paper [13], Garofalo and Vassilev gave the following conjecture: Conjecture (Garofalo-Vassilev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In a H-type group G, the functions [m(Q − 2)] Q−2 4 Uλ,η(ξ) are the only nontrivial entire solutions to � ∆Gu + u Q+2 Q−2 = 0, u ∈ W 1,2 0 (G), u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' If the conjecture is true, then one can obtain the optimal constant of L2 Folland- Stein inequality on H-type groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In this paper we shall use the method given by Frank and Lieb [11, 12] and Hang and Wang [15] to determine the optimal constant, instead of proving the conjecture directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To this end, we have Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It holds that � G |∇Gu|2dxdt ≥ Sm,n �� G |u| 2Q Q−2 dxdt � Q−2 Q , u ∈ W 1,2 0 (G), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7) where Sm,n = 4− 2n Q m(Q − 2)π m+n Q � Γ( m+n 2 ) Γ(m + n) �1/Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The inequality is sharp and an extremal function is U(x, t) = �� 1 + |x|2 4 �2 + |t|2 �− Q−2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 4 QIAOHUA YANG By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, it is easy to see that the functions cUλ,η(ξ)(c ∈ R) are also extremal functions of inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' As an application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, we study the eigenvalues of −∆Gv = µU 4 Q−2 λ,η v, v ∈ W 1,2 0 (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) We note that the eigenvalues of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) play an important role in the study of stability for the Folland-Stein inequality (see [1, 3, 7] for the case of Euclidean space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 we show that the embedding map W 1,2 0 (G) ֒→ L2(G, U(x, t) 4 Q−2 dxdt) is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So the spectrum of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore, we have the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let µi, i = 1, 2, · · · be the eigenvalues of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) given in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then (1) µ1 = m(Q − 2) is simple with eigenfunction Uλ,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2) µ2 = m(Q + 2) and {∂λUλ,η, ∇ηUλ,η} are eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore, the eigenvalues do not depend on λ and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It seems that µ2 has multiplicity m + n + 1 with corresponding eigenspace spanned by {∂λUλ,η, ∇ηUλ,η}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' However, we fail to prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Once it has been proven, it would provide a generalization of the results of Bianchi and Egnell ([1], Lemma A1) to the setting of H-type groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' preliminaries on H-type groups In the rest of paper, we let G be a H-type group with group law given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The nonisotropic dilations δλ on G is δλ(x, t) = (λx, λ2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' For (x, t) ∈ G, the homogeneous norm of (x, t) is ρ(x, t) = �|x|4 16 + |t|2 � 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' With this norm ρ, we can define the ball centered at origin with radius R BR(0) = {(x, t) ∈ G : ρ(x, t) < R} and the unit sphere Σ = ∂B1(0) = {(x, t) ∈ G : ρ(x, t) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Given any (x, t) ∈ G with ρ(x, t) ̸= 0, we set x∗ = x ρ(x,t) and t∗ = t ρ(x,t)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The polar coordinates on G associated with ρ are the following (see [10]): � G f(x, t)dxdt = � ∞ 0 � Σ f(ρx∗, ρ2t∗)ρQ−1dσdρ, f ��� L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The following theorem was proved in [2], Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' : Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' G is a H-type group if and only if G is (isomorphic to) Rm+n with the group law in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) and the matrices U (1), U (2), · · · , U (n) have the following properties: (1) U (j) is a m × m skew symmetric and orthogonal matrix, for every j = 1, 2, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2) U (i)U (j) + U (j)U (i) = 0 for every i, j ∈ {1, 2, · · · , n} with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 5 The vector field in the Lie algebra g that agrees at the origin with ∂ ∂xj (j = 1, · · · , m) is given by Xj = ∂ ∂xj + 1 2 n � k=1 � m � i=1 U (k) i,j xi � ∂ ∂tk and g is spanned by the left-invariant vector fields X1, · · · , Xm, T1 = ∂ ∂t1 , · · · , Tn = ∂ ∂tn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore (see [2], Page 200, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) ), [Xi, Xj] = n � r=1 U (r) i,j Tr, i, j ∈ {1, 2, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) The exponential map exp : g → G is exp : g → Rm+n, m � i=1 xiXi + n � j=1 tjTj �→ (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We note that by exponential mapping, the group law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) is nothing but the Baker-Campbell-Hausdorff formula (see [2], the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2) exp X ◦ exp Y = exp(X + Y + 1 2[X, Y ]), X, Y ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we have that for t = (t1, · · · , tn) = t1T1 + · · · + tnTn and x = (x1, · · · , xm) = x1X1 + · · · + xmXm, the map Jt, defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2), is (see also [2], Page 201) Jtx = n � r=1 m � i=1 trxiJTr(Xi) = n � r=1 m � i=1 trxi \uf8eb \uf8ed m � j=1 U (r) i,j Xj \uf8f6 \uf8f8 = m � j=1 � n � r=1 m � i=1 trxiU (r) i,j � Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Since Jt is an orthogonal map whenever |t| = 1, we obtain |Jtx|2 = |t|2|x|2 = m � j=1 � n � r=1 m � i=1 trxiU (r) i,j �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2) The horizontal gradient on G is ∇G = (X1, · · · , Xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The sub-Laplacian on G is given by (see [2], Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=') ∆G = m � j=1 X2 j = m � j=1 � ∂ ∂xj + 1 2 n � k=1 � m � i=1 U (k) i,j xi � ∂ ∂tk �2 = ∆x + 1 4|x|2∆t + n � k=1 ⟨x, U (k)∇x⟩ ∂ ∂tk , where ∆x = m � j=1 � ∂ ∂xj �2 , ∆t = n � k=1 � ∂ ∂tk �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We remark that ∆G is homogeneous of degree two with respect to δλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6), we have the following Hardy inequality (see [22], Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4 for Hardy inequality of fractional powers of the sublaplacian on G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 6 QIAOHUA YANG Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It holds that, for u ∈ W 1,2 0 (G), � G |∇Gu|2dxdt ≥ m(Q − 2) � G u2 (1 + |x|2 4 )2 + |t|2 dxdt, with equality if and only if u = cU(x, t), where c ∈ R and U(x, t) is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We have, through integration by parts, 0 ≤ � G U 2|∇G(U(x, t)−1u)|2dxdt = � G ���∇Gu − u U ∇GU ��� 2 dxdt = � G |∇Gu|2dxdt + � G |∇GU|2 U 2 u2dxdt − � G 1 U ⟨∇Gu2, ∇GU⟩dxdt = � G |∇Gu|2dxdt + � G u2 1 U ∆GUdxdt = � G |∇Gu|2dxdt − m(Q − 2) � G u2 (1 + |x|2 4 )2 + |t|2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) To get the last equality, we use (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The desired result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ Set η = (y1, · · · , ym, w1, · · · , wn) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6), we have ∆G ∂Uλ,η ∂yj + m(Q + 2)U 4 Q−2 λ,η ∂Uλ,η ∂yj = 0, j = 1, , · · · , m, ∆G ∂Uλ,η ∂wr + m(Q + 2)U 4 Q−2 λ,η ∂Uλ,η ∂wr = 0, r = 1, , · · · , n, ∆G ∂Uλ,η ∂λ + m(Q + 2)U 4 Q−2 λ,η ∂Uλ,η ∂λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) Furthermore, we have the following lemma: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It holds that m � j=1 ���� ∂Uλ,η ∂yj |λ=1,η=0 ���� 2 + n � r=1 ���� ∂Uλ,η ∂wr |λ=1,η=0 ���� 2 + 1 4 ���� ∂Uλ,η ∂λ |λ=1,η=0 ���� 2 = (Q − 2)2 16 U(ξ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It is easy to see η−1 = −η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5), we have Uλ,η(x, t) =λ Q−2 2 \uf8ee \uf8f0 � 1 + λ2 4 m � i=1 (xi − yi)2 �2 + λ4 n � r=1 � tr − wr − ⟨y, U (r)x⟩ 2 �2\uf8f9 \uf8fb − Q−2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 7 We compute ∂Uλ,η ∂yj |λ=1,η=0 = − Q − 2 4 U(ξ) Q+2 Q−2 � 2 � 1 + |x|2 4 � � −xj 2 � + n � r=1 tr � − m � i=1 U (r) j,i xi �� =Q − 2 4 U(ξ) Q+2 Q−2 �� 1 + |x|2 4 � xj + n � r=1 m � i=1 trxiU (r) j,i � , j = 1, · · · , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ∂Uλ,η ∂wr |λ=1,η=0 = − Q − 2 4 U(ξ) Q+2 Q−2 (−2tr) =Q − 2 2 U(ξ) Q+2 Q−2 tr, r = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ∂Uλ,η ∂λ |λ=1,η=0 =Q − 2 2 U(ξ) − Q − 2 4 U(ξ) Q+2 Q−2 � 2 � 1 + |x|2 4 � |x|2 2 + 4|t|2 � = − Q − 2 2 U(ξ) Q+2 Q−2 � −1 + |x|4 16 + |t|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Since each U (j)(1 ≤ j ≤ n) is a m × m skew symmetric matrix, we have, by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2), m � j=1 ���� ∂Uλ,η ∂yj |λ=1,η=0 ���� 2 =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 m � j=1 �� 1 + |x|2 4 � xj − n � r=1 m � i=1 trxiU (r) i,j �2 =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 |x|2 + |t|2|x|2− 2 � 1 + |x|2 4 � n � r=1 tr \uf8eb \uf8ed m � i=1 m � j=1 U (r) i,j xixj \uf8f6 \uf8f8 \uf8f9 \uf8fb =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 |x|2 + |t|2|x|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the last equality, we use the fact m � i=1 m � j=1 U (r) i,j xixj = 0 8 QIAOHUA YANG since U (r)(1 ≤ r ≤ n) is a m × m skew symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' we have m � j=1 ���� ∂Uλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η ∂yj |λ=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η=0 ���� 2 + n � r=1 ���� ∂Uλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η ∂wr |λ=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η=0 ���� 2 + 1 4 ���� ∂Uλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η ∂λ |λ=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η=0 ���� 2 =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 |x|2 + |t|2|x|2 � + (Q − 2)2 4 U(ξ)2 Q+2 Q−2 |t|2+ (Q − 2)2 16 U(ξ)2 Q+2 Q−2 � −1 + |x|4 16 + |t|2 �2 =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 |x|2 + |t|2|x|2 + 4|t|2 + � −1 + |x|4 16 + |t|2 �2� =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 + |t|2 �2 =(Q − 2)2 16 U(ξ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the third equality, we use the fact � −1 + |x|4 16 + |t|2 �2 = �� 1 + |x|2 4 �2 + |t|2 − 2 � 1 + |x|2 4 ��2 = �� 1 + |x|2 4 �2 + |t|2 �2 + 4 � 1 + |x|2 4 �2 − 4 � 1 + |x|2 4 � �� 1 + |x|2 4 �2 + |t|2 � = �� 1 + |x|2 4 �2 + |t|2 �2 − � 1 + |x|2 4 �2 |x|2 − |t|2|x|2 − 4|t|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ For simplicity, we set ωj = 4 Q − 2U(ξ)−1 ∂Uλ,η ∂yj |λ=1,η=0, j = 1, · · · , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ωj+r = 4 Q − 2U(ξ)−1 ∂Uλ,η ∂wr |λ=1,η=0, r = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ωm+n+1 = 2 Q − 2U(ξ)−1 ∂Uλ,η ∂λ |λ=1,η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4), we have m+n+1 � j=1 ω2 j =1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) ∆G(U(ξ)ωj) + m(Q + 2)U(ξ) Q+2 Q−2 ωj =0, 1 ≤ j ≤ m + n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7) THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3 In this section, we shall prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof depends on a scheme of subcritical approximation due to Hang and Wang [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We first establish the following subcritical Sobolev inequality on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 2 ≤ p < 2Q Q−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' There exists C > 0 such that for each u ∈ W 1,2 0 (G), � G |∇Gu|2dxdt ≥ C �� G |u|pU(x, t) 2Q Q−2 −pdxdt � 2 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By H¨older’s inequality, we have � G |u|pU(x, t) 2Q Q−2 −pdxdt = � G � |u|U(x, t) 2 Q−2 �Q− Q−2 2 p |u| Q 2 (p−2)dxdt ≤ �� G |u|2U(x, t) 4 Q−2 dxdt � 2Q−(Q−2)p 4 �� G |u| 2Q Q−2 dxdt � (Q−2)(p−2) 4 = �� G u2 (1 + |x|2 4 )2 + |t|2 dxdt � 2Q−(Q−2)p 4 �� G |u| 2Q Q−2 dxdt � (Q−2)(p−2) 4 ≤C � G |∇Gu|2dxdt, where C is a positive constant independent of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the last inequality above, we use Folland-Stein inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1, we have W 1,2 0 (G) ֒→ Lp(G, U(x, t) 2Q Q−2 −pdxdt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore, the embedding map is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 2 ≤ p < 2Q Q−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The embedding map W 1,2 0 (G) ֒→ Lp(G, U(x, t) 2Q Q−2 −pdxdt) is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof is similar to that given by Schneider (see [23], section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let φ : G → [0, 1] be a cut-off function that is equal to one in B1(0) and zero outside of B2(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Consider the operator IR : W 1,2 0 (G) ֒→ Lp(G, U(x, t) 2Q Q−2 −pdxdt) 10 QIAOHUA YANG defined by IR(u) = u(x, t)φ � x R, t R2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Since the imbedding map W 1,2 0 (B2(0)) ֒→ Lp(B2(0)) is compact, so is IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Moreover, by H¨older’s inequality, � G |u − IR(u)|pU(x, t) 2Q Q−2 −pdxdt ≤ � G\\BR(0) |u|pU(x, t) 2Q Q−2 −pdxdt ≤ �� G\\BR(0) |u| 2Q Q−2 dxdt � Q−2 2Q p �� G\\BR(0) U(x, t) 2Q Q−2 dxdt �1− Q−2 2Q p ≤C �� G |∇Gu|2dxdt � p 2 �� G\\BR(0) U(x, t) 2Q Q−2 dxdt �1− Q−2 2Q p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the last inequality above, we use the Folland-Stein inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By polar coordinates, we have � G\\BR(0) U(x, t) 2Q Q−2 dxdt = � G\\BR(0) �� 1 + |x|2 4 �2 + |t|2 �− Q 2 dxdt ≤ � G\\BR(0) 1 ( |x|2 4 + |t|2) Q 2 dxdt = � ∞ R � Σ 1 ρ2Q ρQ−1dρdσ =|Σ| 1 QRQ → 0, R → ∞, where |Σ| is the volume of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, the embedding map W 1,2 0 (G) ֒→ Lp(G, U(x, t) 2Q Q−2 −pdxdt) is a limit of compact operators and thus it is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 is thereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, the minimization problem Λp = inf �� G |∇Gu|2dxdt : � G |u|pU(x, t) 2Q Q−2 −pdxdt = 1 � , 2 ≤ p < 2Q Q − 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) has a positive solution u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We shall show that such u satisfies a moment zero condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The main result is the following lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 2 ≤ p < 2Q Q−2 and u be a positive solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then we have � G upU(x, t) 2Q Q−2 −pωidxdt = 0, i = 1, 2, · · · , m + n + 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2) where ωi(1 ≤ i ≤ m + n + 1) is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' For simplicity, we set Fp(u) = � G |∇Gu|2dxdt �� G |u|pU(x, t) 2Q Q−2 −pdxdt � 2 p THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 11 and uλ−1,η−1(ξ) =λ− Q−2 2 u(δλ−1(η ◦ ξ)), λ > 0, η = (y1, · · · , ym, w1, · · · , wn) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' A simple calculation shows � G |∇Guλ−1,η−1|2dxdt = � G |∇Gu|2dxdt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' � G up λ−1,η−1U(x, t) 2Q Q−2 −pdxdt = � G upUλ,η(x, t) 2Q Q−2 −pdxdt, where Uλ,η is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, Fp(uλ−1,η−1(ξ)) = � G |∇Gu|2dxdt �� G |u|pUλ,η(x, t) 2Q Q−2 −pdxdt � 2 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) Since u is a positive solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we have ∂ ∂yj Fp(uλ−1,η−1(ξ))|λ=1,η=0 =0, j = 1, · · · , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ∂ ∂wr Fp(uλ−1,η−1(ξ))|λ=1,η=0 =0, r = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ∂ ∂λFp(uλ−1,η−1(ξ))|λ=1,η=0 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We remark that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3 is also valid for u > 0 satisfying the Yamabe-type equation ∆Gu + ΛpupU(x, t) 2Q Q−2 −p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof is same and we omit it (see [15], Corollary 1 for the case of CR sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It holds that, for any u ∈ W 1,2 0 (G), m+n+1 � i=1 � G |∇G(uωi)|2dxdt = � G |∇Gu|2dxdt + 4m � G u2 (1 + |x|2 4 )2 + |t|2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 12 QIAOHUA YANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let u = U(ξ)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We compute, through integration by parts, m+n+1 � i=1 � G |∇G(uωi)|2dxdt = m+n+1 � i=1 � G |∇G(vUωi)|2dxdt = m+n+1 � i=1 � G |Uωi∇Gv + v∇G(Uωi)|2dxdt = m+n+1 � i=1 �� G |∇Gv|2U 2ω2 i dxdt + � G |∇G(Uωi)|2v2dxdt+ 1 2 � G ⟨∇G(Uωi)2, ∇Gv2⟩dxdt � = m+n+1 � i=1 �� G |∇Gv|2U 2ω2 i dxdt − � G v2Uωi∆G(Uωi)dxdt � = � G |∇Gv|2U 2dxdt + m(Q + 2) � G v2U 2Q Q−2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5) To get the last equality, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3), we have � G |∇Gv|2U 2dxdt = � G ���∇G u U ��� 2 U 2dxdt = � G |∇Gu|2dxdt − m(Q − 2) � G u2 (1 + |x|2 4 )2 + |t|2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5), we obtain m+n+1 � i=1 � G |∇G(uωi)|2dxdt = � G |∇Gu|2dxdt + 4m � G u2 (1 + |x|2 4 )2 + |t|2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ Now we can give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The idea is due to Frank and Lieb [11, 12] and Hang and Wang [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 2 ≤ p < 2Q Q−2 and up be a positive solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The 2nd variation of the functional Fp around up shows that � G |∇Gf|2dxdt � G up pU(x, t) 2Q Q−2 −pdxdt− (p − 1) � G |∇Gup|2dxdt � G up−2 p Uλ,η(x, t) 2Q Q−2 −pf 2dxdt ≥ 0 for any f with � G up pUλ,η(x, t) 2Q Q−2 −pfdxdt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 13 By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3, we may choose f = upωi, i = 1, 2, · · · , m + n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Summing the corresponding inequalities for all such f’s yields, in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5, 0 ≤ m+n+1 � i=1 � G |∇G(upωi)|2dxdt − (p − 1) � G |∇Gup|2dxdt =4m � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt − (p − 2) � G |∇Gup|2dxdt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (p − 2) �� G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt � ≤m(Q − 2) � 2Q Q − 2 − p � � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt ≤m(Q − 2) � 2Q Q − 2 − p � �� G up pU(x, t) 2Q Q−2 −pdxdt � 2 p �� G U(x, t) 2Q Q−2 dxdt �1− 2 p =m(Q − 2) � 2Q Q − 2 − p � �� G U(x, t) 2Q Q−2 dxdt �1− 2 p → 0, p ր 2Q Q − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the last equality, we use the fact � G up pU(x, t) 2Q Q−2 −pdxdt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, we obtain � G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt → 0, p ր 2Q Q − 2, or equivalently, � G |∇G(U −1up)|2U 2dxdt → 0, p ր 2Q Q − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So we can choose a sequence {pk : k = 1, 2, · · ·} such that pk ր 2Q Q−2 and upk converges to a nonzero function c0U (for reader’s convenience, we prove it in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Thus c0U is an extremal function of Λ = inf �� G |∇Gu|2dxdt : � G |u| 2Q Q−2 dxdt = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The value Sm,n has been calculated in [13], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 is thereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let up(2 ≤ p < 2Q Q−2) be a positive solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' If � G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt → 0, p ր 2Q Q − 2, then there exists c0 > 0 and a sequence {pk : k = 1, 2, · · ·} such that pk ր 2Q Q−2 and � G |∇G(upk − c0U)|2dxdt → 0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 14 QIAOHUA YANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, µ1 = m(Q − 2) is simple with eigenfunction U of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) with λ = 1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Decompose up as up = λpU + vp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7) with λp = � G U Q+2 Q−2 updxdt � G U 2Q Q−2 dxdt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then vp ⊥ U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' � G U 4 Q−2 · Uvpdx = � G U Q+2 Q−2 vpdxdt = 0, � G ⟨∇GU, ∇Gvp⟩dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) Therefore, we have � G |∇Gvp|2dxdt ≥ µ2 � G U 4 Q−2 · v2 pdx = µ2 � G v2 p (1 + |x|2 4 )2 + |t|2 dxdt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='9) where µ2 is the second eigenvalue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) with with λ = 1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We compute, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='9), � G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt = � G � λ2 p|∇GU|2 + |∇Gvp|2� dxdt − µ1 � G λ2 pU 2 + v2 p (1 + |x|2 4 )2 + |t|2 dxdt = � G |∇Gvp|2dxdt − µ1 � G v2 p (1 + |x|2 4 )2 + |t|2 dxdt =µ1 µ2 �� G |∇Gvp|2dxdt − µ2 � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt � + µ2 − µ1 µ2 � G |∇Gvp|2dxdt ≥µ2 − µ1 µ2 � G |∇Gvp|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, � G |∇Gvp|2dxdt ≤ µ2 µ2 − µ1 �� G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt � → 0, p ր 2Q Q − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='10) THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 15 On the other hand, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7), Minkowski’s inequalities, H¨older’s inequality and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we have λp �� G U(x, t) 2Q Q−2 dxdt � 1 p = �� G (up − vp)pU(x, t) 2Q Q−2 −pdxdt � 1 p ≤ �� G up pU(x, t) 2Q Q−2 −pdxdt � 1 p + �� G |vp|pU(x, t) 2Q Q−2 −pdxdt � 1 p =1 + �� G |vp|pU(x, t) 2Q Q−2 −pdxdt � 1 p ≤1 + �� G |vp| 2Q Q−2 dxdt � Q−2 2Q �� G U 2Q Q−2 dxdt � 1 p − Q−2 2Q ≤1 + C �� G |∇Gvp|2dxdt � 1 2 �� G U 2Q Q−2 dxdt � 1 p − Q−2 2Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='11) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='11), we obtain lim sup pր 2Q Q−2 λp ≤ �� G U(x, t) 2Q Q−2 dxdt �− Q−2 2Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, there exists c0 ≥ 0 and a sequence {pk : k = 1, 2, · · · } such that pk ր 2Q Q−2 and λpk → c0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='12) We claim that � G |∇G(upk − c0U)|2dxdt → 0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='13) In fact, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='12), we obtain � G |∇G(upk − c0U)|2dxdt = � G |∇G(vpk + (λpk − c0)U)|2dxdt = � G |∇Gvpk|2dxdt + (λpk − c0)2 � G |∇GU|2dxdt → 0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Finally, we show that c0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In fact, if c0 = 0, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='13), � G |∇Gupk|2dxdt → 0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 16 QIAOHUA YANG On the other hand, by H¨older’s inequality and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we obtain 1 = �� G upk pkU(x, t) 2Q Q−2 −pkdxdt � 1 pk ≤ �� G |upk| 2Q Q−2 dxdt � Q−2 2Q �� G U 2Q Q−2 dxdt � 1 pk − Q−2 2Q ≤C �� G |∇Gupk|2dxdt � 1 2 �� G U 2Q Q−2 dxdt � 1 pk − Q−2 2Q → 0, k → ∞, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So c0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6 is thereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ Finally, we give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3 A simple scaling argument shows that the eigenvalues do not depend on λ and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So we may assume λ = 1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 we know that µ1 = m(Q − 2) is simple with eigenfunction U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Nextly, we show µ2 ≥ m(Q + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let V ̸= 0 be a eigenfunction of µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='14) µ2 = � G |∇GV |2dxdt � G U 4 Q−2 V 2dxdt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore, since V ⊥ U, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='15) � G ⟨∇GU, ∇GV ⟩dxdt = 0, � G U 4 Q−2 · UV dxdt = � G U Q+2 Q−2 V dxdt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Set Φ(ǫ) = � G |∇G(U + ǫV )|2dxdt �� G |U + ǫV | 2Q Q−2 dxdt � Q−2 Q , ǫ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, U is an extremal function of Folland-Stein inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So we have Φ′(0) = 0 and Φ′′(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We compute Φ′(ǫ) =2 � G⟨∇G(U + ǫV ), ∇GV ⟩dxdt �� G |U + ǫV | 2Q Q−2 dxdt � Q−2 Q − 2 � G |∇G(U + ǫV )|2dxdt �� G |U + ǫV | 2Q Q−2 dxdt � 2Q−2 Q � G |U + ǫV | 4 Q−2 (U + ǫV )V dxdt =Φ1(ǫ) − Φ2(ǫ), where Φ1(ǫ) =2 � G⟨∇G(U + ǫV ), ∇GV ⟩dxdt �� G |U + ǫV | 2Q Q−2 dxdt � Q−2 Q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Φ2(ǫ) =2 � G |∇G(U + ǫV )|2dxdt �� G |U + ǫV | 2Q Q−2 dxdt � 2Q−2 Q � G |U + ǫV | 4 Q−2 (U + ǫV )V dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 17 By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='15), we have Φ′ 1(0) =2 � G |∇GV |2dxdt �� G |U| 2Q Q−2 dxdt � Q−2 Q − 4 � G⟨∇GU, ∇GV ⟩dxdt �� G |U| 2Q Q−2 dxdt � 2Q−2 Q � G U Q+2 Q−2 V dxdt =2 � G |∇GV |2dxdt �� G |U| 2Q Q−2 dxdt � Q−2 Q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Φ′ 2(0) =4 � G⟨∇GU, ∇GV ⟩dxdt �� G |U| 2Q Q−2 dxdt � 2Q−2 Q � G U Q+2 Q−2 V dxdt − 8(Q − 1) Q − 2 � G |∇GU|2dxdt �� G |U| 2Q Q−2 dxdt � 3Q−2 Q �� G U 4 Q−2 V 2dxdt �2 + 2(Q + 2) Q − 2 � G |∇GU|2dxdt �� G U 2Q Q−2 dxdt � 2Q−2 Q � G U 4 Q−2 V 2dxdt =2(Q + 2) Q − 2 � G |∇GU|2dxdt �� G U 2Q Q−2 dxdt � 2Q−2 Q � G U 4 Q−2 V 2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, 0 ≤ Φ′′(0) =Φ′ 1(0) − Φ′ 2(0) =2 � G |∇GV |2dxdt �� G |U| 2Q Q−2 dxdt � Q−2 Q − 2(Q + 2) Q − 2 � G |∇GU|2dxdt �� G U 2Q Q−2 dxdt � 2Q−2 Q � G U 4 Q−2 V 2dxdt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' � G |∇GV |2dxdt � G |U| 4 Q−2 V 2dxdt ≥ Q + 2 Q − 2 � G |∇GU|2dxdt � G |U| 2Q Q−2 dxdt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='16) Combing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='16) yields µ2 ≥ Q + 2 Q − 2 � G |∇GU|2dxdt � G |U| 2Q Q−2 dxdt = Q + 2 Q − 2µ1 = m(Q + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4), {∂λUλ,η|λ=1,η=0, ∇ηUλ,η|λ=1,η=0} are eigenfunctions of m(Q + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So µ2 = m(Q + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Bianchi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Egnell, A note on the Sobolev inequality, J.' metadata={'source': 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