Sharp constants for weighted Moser–Trudinger inequalities on groups of Heisenberg type

Sharp constants for weighted Moser–Trudinger inequalities on groups of Heisenberg type

Nonlinear Analysis 89 (2013) 95–109 Contents lists available at SciVerse ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Sharp constants for weighted Moser–Trudinger inequalities on groups of Heisenberg type Nguyen Lam a , Hanli Tang b,∗ a Department of Mathematics, Wayne State University, Detroit, MI 48202, USA b School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China article abstract info Let G be a group of Heisenberg type, Q = m + 2q be its homogeneous dimension, ′ a For u ∈ G, we write u = (z (u), t (u)) ∈ G, where t (u) Qa = Q − a, Qa = Q Q−− a−1 is the coordinate of u corresponding to the center T of the Lie algebra G of G, z (u) is Article history: Received 19 March 2013 Accepted 29 April 2013 Communicated by Enzo Mitidieri corresponding to the orthogonal complement of T Let N (u) = (|z (u)|4 + t (u)2 ) be the homogeneous norm of u ∈ G, W (u) = |z (u)|−a be a weight The main purpose of this paper is to establish sharp constants for weighted Moser–Trudinger inequalities on domains of finite measure in G (Theorem 2.1) and on unbounded domains (Theorem 2.2) We also establish the weighted inequalities of Adachi–Tanaka type on the entire G (Theorem 2.3) Our results extend the sharp Moser–Trudinger inequalities on domains of finite measure in Cohn and Lu (2001, 2002) [13,14] and on unbounded domains in Lam et al (2012) [19] to the weighted case and improve the sharp weighted Moser–Trudinger inequality proved in Tyson (2006) [16] on domains of finite measure on G The usual symmetrization method (i.e., rearrangement argument) is not available on such groups and therefore our argument is a rearrangement-free argument recently developed in Lam and Lu (2012) [17,18] Our weighted Adachi–Tanaka type inequalities extend the nonweighted results in Lam et al (2012) [20] © 2013 Elsevier Ltd All rights reserved Keywords: Moser–Trudinger inequalities Weights Sharp constants Groups of Heisenberg type Introduction of groups of Heisenberg type Let Hn = R2n × R be the n-dimensional Heisenberg group with the noncommutative product law        (x, y, t ) · x′ , y′ , t ′ = x + x′ , y + y′ , t + t ′ + y, x′ − x, y′ , where x, y, x′ , y′ ∈ Rn , t , t ′ ∈ R, and ⟨·, ·⟩ denotes the standard inner product in Rn The Lie algebra of Hn is generated by the left-invariant vector fields T = ∂ , ∂t Xi = ∂ ∂ + 2yi , ∂ xi ∂t Yi = ∂ ∂ − 2xi , ∂ yi ∂t i = , , n Denote z = (x, y) ∈ R2n , ∗ ξ = (z , t ) ∈ Hn ,  1/4 ρ (ξ ) = |z |4 + t , Corresponding author Tel.: +86 13426196736 E-mail addresses: nguyenlam@wayne.edu (N Lam), rainthl@163.com (H Tang) 0362-546X/$ – see front matter © 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.na.2013.04.017 96 N Lam, H Tang / Nonlinear Analysis 89 (2013) 95–109 where ρ (ξ ) denotes the Heisenberg distance between ξ and the origin We use |∇Hn f | to express the norm of the subelliptic gradient of the function f : Hn → R:  1/2 n    2 |∇Hn f | = (Xi f ) + (Yi f ) i=1 ,Q Let Ω be an open set in Hn We use W0 ∥f ∥W 1,Q (Ω ) (Ω ) to denote the completion of C0∞ (Ω ) under the norm 1/Q  (|∇Hn f |Q + |f |Q )dξ = Ω It is known that the Heisenberg group is a special case of groups of Heisenberg type Let us recall some basic properties of the Heisenberg type (H-type) groups Let G be a finite-dimensional Lie algebra of step two, that is, [G, [G, G]] = Let ⟨·, ·⟩ be a given inner product on G × G and suppose T is the center of G, i.e the set of all elements T in G for which [T , V ] = for all V ∈ G Let Z be the orthogonal complement to the center T under the given inner product Following Kaplan [1], we can define the nilpotent Lie algebras of Heisenberg type as follows For each v ∈ Z, let adv : Z → T be the map given by adv (v ′ ) = [v, v ′ ] If Kv is the kernel of adv , we have the orthogonal decomposition Z = Kv + Zv G is said to be a Lie algebra of Heisenberg type (or H-type) if adv : Zv → T is a surjective isometry for every unit vector v ∈ Z Suppose now that G is a Lie algebra of H-type, with the decomposition G = Z + T Then for each T ∈ T the bilinear form taking Z × Z → R, defined by v, v ′ → ⟨T , [v, v ′ ]⟩ is skew-symmetric and nondegenerate for T ̸= Furthermore, there exists a linear isomorphism JT : Z → Z such that for v, v ′ ∈ Z and T ∈ T : (i) ⟨JT (v), v ′ ⟩ = ⟨T , [v, v ′ ]⟩, (ii) ⟨JT (v), v ′ ⟩ + ⟨v, JT (v ′ )⟩ = It also satisfies JT2 = −|T |2 I and ⟨JT (v), JT ′ (v)⟩ = ⟨T , T ′ ⟩|v|2 and thus |JT (v)| = |T ||v| Let G be the simply connected Lie group corresponding to G The exponential mapping from G to G is an analytic diffeomorphism We can define analytic mappings Z : G → Z, T : G → T by the condition that for all u ∈ G, u = exp(Z (u) + T (u)) Let {Xi }1≤i≤m be an orthonormal basis of Z and {Tk }1≤k≤q an orthonormal basis of the center T Let ∇G f (u) = (X1 f (u), , Xm f (u)) be the subelliptic gradient on G We also express Z (u) = z1 X1 + · · · + zm Xm , T (u) = t1 T1 + · · · + tq Tq , and z (u) = (z1 (u), , zm (u)), t (u) = (t1 (u), , tq (u)) Clearly, |Z (u)| = |z (u)| and |T (u)| = |t (u)| Given u = exp(Z (u) + T (u)), let N (u) = (|z (u)|4 + 16|t (u)|2 )1/4 be the homogeneous norm on G and ru = z (u) t (u) exp(rZ (u) + r T (u)) denote the dilation on G Set z ∗ (u) = N (u) , t ∗ (u) = N (u)2 and u∗ = (z ∗ (u), t ∗ (u)) Let us denote S (N ) = {u : N (u) = 1} the unit sphere with respect to the homogeneous norm N Then u∗ ∈ S (N ), when u ̸= Let Ω ⊂ G be an open set For p ≥ and a weight W on G, we say that f : Ω → R is in the weighted horizontal Sobolev space HW 1,p (Ω , W ) if f ∈ Lp (Ω , W ), ∇G f exists in the sense of distributions and |∇G f | ∈ Lp (Ω , W )  Denote ∥f ∥p,W ,Ω = G 1/p |∇G f |p Wdu We use 1,p HW0  Ω 1/p 1/p  1/p  and ∥∇G f ∥p,W = , ∥f ∥p,W = G |f |p Wdu |f |p Wdu , ∥∇G f ∥p,W ,Ω = U |∇G f |p Wdu (Ω , W ) to denote the completion of C0∞ (Ω ) under the norm 1/p   |∇G f |p Wdu |f |p Wdu + ∥f ∥HW 1,p (Ω ,W ) = Ω Ω We also use HW 1,p (G, W ) to denote the completion of C0∞ (G) under the norm ∥f ∥HW 1,p (G,W ) =  p |f | Wdu + G  p 1/p |∇G f | Wdu G A function Y : G → R is called first-layer symmetric if Y (u) depends on |z (u)| and t (u) A domain Ω is called first-layer symmetric if its characteristic function is first-layer symmetric 97 N Lam, H Tang / Nonlinear Analysis 89 (2013) 95–109 Moser–Trudinger inequalities and statement of the main theorems The Moser–Trudinger inequalities can be considered as the limiting case of Sobolev inequalities They were proposed independently by Yudovič [2], Pohožaev [3] and Trudinger [4] on Euclidean spaces Basically, the Sobolev inequalities claim ,p np that W0 (Ω ) ⊂ Lq (Ω ) for ≤ q ≤ p∗ = n−p where ≤ p < n and Ω ⊂ Rn , n ≥ is a bounded domain, while the ,n Moser–Trudinger inequalities deal with the borderline case p = n : W0 (Ω ) ⊂ Lϕn (Ω ) where Lϕn (Ω ) is the Orlicz space   associated with the Young function ϕn (t ) = exp β |t |n/(n−1) − for some β > It means that in this limiting case, the exponential integrability can be admitted: there exist constants αn > and Cn > depending only on n such that  sup 1,n f ∈W0  (Ω ), Ω |∇ f |n dx≤1 Ω  n exp αn |f | n−1  dx ≤ Cn |Ω | However, the best possible value of αn was not discovered until the 1971 paper of J Moser [5] Indeed, using a different approach, namely, the symmetrization, J Moser can set up the following result Theorem A (Moser–Trudinger Inequality) Let Ω be a domain with finite measure in Euclidean n-space Rn , n ≥ Then there −1 exists a sharp constant αn = nωnn− (where ωn−1 is the area of the surface of the unit n-ball) such that  |Ω |  n exp α |f | n−1 Ω  dx ≤ c0 < ∞  ,n for any α ≤ αn , any f ∈ W0 (Ω ) with Ω |∇ f |n dx ≤ This constant αn is sharp in the sense that if α > αn , then the above inequality can no longer hold with some c0 independent of f This result has been extended and complemented in many directions For instance, the singular Moser–Trudinger inequality which is an interpolation of the Hardy inequality and the Moser–Trudinger inequality was studied by Adimurthi and Sandeep in [6] When Ω has infinite volume, the subcritical Moser–Trudinger type inequalities for unbounded domains were first proposed by D.M Cao [7] when n = and J.M Ó [8] for the general case n ≥ These results were sharpened later by Adachi and Tanaka [9] in order to determine the best constant B Ruf [10] for the case n = 2, Y Li and B Ruf [11] for the general case n ≥ established a critical Moser–Trudinger type inequality for unbounded domains And more recently, it was extended in [12] to the singular Moser–Trudinger type inequality on the whole Euclidean space The sharp Moser–Trudinger inequality on the Heisenberg group was first established by Cohn and Lu [13] Indeed, the following was shown  Theorem B Let Hn be an n-dimensional Heisenberg group, Q = 2n + 2, Q ′ = Q /(Q − 1), and αQ = Q 2π n Γ Γ  Q −1  Γ  Q −1 Γ (n)−1 sup 1,Q f ∈W0 Q ′ −1 (Ω ), ∥∇H f ∥LQ ≤1 |Ω | Then there exists a constant C0 depending only on Q such that for all Ω ⊂ Hn , |Ω | < ∞,   exp αQ |f (ξ )|Q Ω ′  1 (2.1) dξ ≤ C0 < ∞ 1,Q If αQ is replaced by any larger number, the integral in (2.1) is still finite for any f ∈ W0 (Ω ), but the supremum is infinite This result was extended to the H-type groups by Cohn and Lu They proved in [14] the following Theorem C Let G be an H-type group with Z ⊂ G of dimension m ≥ and T ⊂ G of dimension q Set A(G) := Q   2q−1    Q −1 q+m π Γ Q +4 m , Q Γ (m/2)Γ (Q /2) where m + q is the topological dimension of G, Q = m + 2q is the homogeneous dimension of G Then there exist C0 < ∞ so that for all Ω ⊂ G, |Ω | < ∞, sup  |Ω |   exp AQ |f (u)| Ω Q Q −1  du : f ∈ ,Q W0  (Ω ), ∥∇G f ∥LQ ≤ ≤ C0 Furthermore, if AQ is replaced by any number greater than AQ , then the statement is false Balogh, Manfredi and Tyson [15] further extended the Moser–Trudinger inequalities to Carnot groups Moreover, the following weighted Moser–Trudinger inequality on a bounded domain on the H-type group was proved by J Tyson [16] 98 N Lam, H Tang / Nonlinear Analysis 89 (2013) 95–109 Theorem D Let G be an H-type group with Z ⊂ G of dimension m ≥ and T ⊂ G of dimension q Let W (u) = |z (u)|−a , ≤ a ≤ m − Set 1/(Q −a−1) A(G, W ) := (Q − a)cQ −2a , where m + q is the topological dimension of G, Q = m + 2q is the homogeneous dimension of G, and cp =  2π m/2 Γ ((m + p)/4) |∇G N |p dσ = 4q Γ (m/2)Γ ((Q + p)/4) S (N ) Then there exist C0 < ∞ so that    |F | sup  exp A(G, W ) ∥∇G F ∥Q −a,W ,Ω Ω ,F Ω W Ω Qa′  W ≤ C0 , where the supremum is taken over all nonconstant first-layer symmetric functions F in the weighted Sobolev space 1,Q −a (Ω , W ) on a first-layer symmetric domain Ω ⊂ G with finite Haar measure The supremum becomes infinite if A(G, W ) HW0 is replaced by any strictly large value Very recently, Lam and Lu [17,18] introduced a new idea, namely, a rearrangement-free argument, to study the Moser–Trudinger inequalities on Heisenberg groups and the Adams type inequalities for fractional derivatives This new method is also used to prove the subcritical Moser–Trudinger type and Adams type inequalities [19,20] In this paper, first we will set up the following sharp singular Moser–Trudinger inequality for the first-layer symmetric domains with finite Haar measure on G Theorem 2.1 Let G be an H-type group with Z ⊂ G of dimension m ≥ and T ⊂ G of dimension q Let W (u) = |z (u)|−a , ≤ a ≤ m − Where m + q is the topological dimension of G, Q = m + 2q is the homogeneous dimension of G Let Ω be a first-layer symmetric domain with finite measure, then we have sup W (Ω ) F if and only if 1− Q α −a  β Q ′ −1 (Q −a)(cQ a−2a ) ′ exp(β|F |Qa ) |N (u)|α Ω + α W < ∞, ≤ Here, cp = Q −a 2π m/2 Γ ((m+p)/4) 4q Γ (m/2)Γ ((Q +p)/4) , W (Ω ) = 1,Q −a first-layer symmetric functions F in the weighted Sobolev space HW0  W and the supremum is taken over all Ω (Ω , W ) with ∥∇G F ∥Q −a,W ,Ω ≤ Next, we will study the Moser–Trudinger type inequality for unbounded domains on H-type groups for the first-layer symmetric functions in the weighted Sobolev space HW 1,Q −a (G, W ) on the entire Heisenberg type group G: Theorem  any positive real number and [·] denote the integer part of a real number Then for any pair α ≥  2.2 Let τ be β [Q −a−1] + Q α−a ≤ 1, there holds (Q − a) − Q −a−1 , β > satisfying Q ′ −1 (Q −a)(cQ a−2a ) sup F  G N (u)α  Qa′ exp(β|F | ) − [Q −a−2] k=0  β k kQa′ |F | W ≤ ∞, k! where the supremum is taken over all first-layer symmetric functions F in the weighted Sobolev space HW 1,Q −a (G, W ) with ∥F ∥1,τ ,W ≤ If β Q ′ −1 (Q −a)(cQ a−2a ) + α Q −a  > 1, the supremum is infinite Here ∥F ∥1,τ ,W = G |∇G F |Q −a W + τ  G |F |Q −a W  Q −1 a Finally, we will establish the following inequalities of Adachi–Tanaka type on G under a much less restrictive assumption on the norm Namely, instead of restricting the norm by assuming ∥F ∥1,τ ,W = we only assume that ∥∇G F ∥Q −a,W ≤  a−1] Theorem 2.3 For any given (Q − a) − [QQ − −a−1 constant Cα,β such that  G N (u)α  Qa′ exp(β|F | ) − [Q −a−2] k=0   G |∇G F |Q −a W + τ  G |F |Q −a W Q ′ −1  Q −1 a ≤ 1, ≤ α < Q − a, then for any < β < (Q − a − α)(cQ a−2a ), there exists a  β k kQa′ Q −a−α |F | W ≤ Cα,β ∥F ∥Q −a,W , k! 99 N Lam, H Tang / Nonlinear Analysis 89 (2013) 95–109 for all first-layer symmetric functions F ∈ HW 1,Q −a (G, W ) with ∥∇G F ∥Q −a,W = And the restriction < β < (Q − a − Q ′ −1 Q ′ −1 1,Q −a α)(cQ a−2a ) is optimal That is: if β ≥ (Q − a − α)(cQ a−2a ), there exists a sequence {Fk (u)}∞ (G, W ) such that k=1 ⊂ HW ∥∇G Fk ∥Q −a,W = and    [Q  −a−2] k 1 α kQa′ Qa′ |F | exp(β|F | ) − W → ∞, a−α N (u)α k! ∥F ∥QQ − k=0 −a,W G as k → ∞ The paper is organized as follows We will provide some useful preliminaries in Section In Section 4, we will prove Theorem 2.1 Theorems 2.2 and 2.3 will be proved in Sections and respectively Preliminaries To prove these theorems, we first give some useful results The following version of the integration in polar coordinates can be found in the book of Folland and Stein [21] Proposition 3.1 There exists a Radon measure dσ on S (N ) such that for any f ∈ L1 (G) we have   f (u)du = G ∞  f (ru∗ )r Q −1 dσ (u∗ )dr S (N ) Let us recall the weighted convolution on an H-type group G Definition 3.1 Let W ∈ L1loc (G) be a weight The weighted convolution of two functions K , L : G → R with respect to W is defined to be K∗W L(u) :=  K (h)L(uh−1 )W (h)dh G at those u ∈ G when the integral is defined, where h−1 is the inverse of h and uh−1 denote group multiplication of u by h−1 Suppose F is a non-negative function defined on an H-type group G Define the non-increasing weighted rearrangement ∗ of F by FW (t ) = inf{s > : λF ,W (s) ≤ t }, where λF ,W (s) = W ({u ∈ G : F (u) > s}) In addition, define ∗∗ FW (t ) = t  −1 t ∗ FW (s)ds Using the generalized result of O’Neil [22] to measure spaces by Blozinski [23], we have the weighted O’Neil’s Lemma Lemma 3.1 Let W be a weight on an H-type group G For each pair of functions K , L : G → R with L first-layer symmetric, ∗∗ ∗∗ (K∗W L)∗W (t ) ≤ (K∗W L)∗∗ W (t ) ≤ tKW (t )LW (t ) +  ∞ ∗ (s)L∗W (s)ds KW t In Cohn and Lu’s paper [14], they get the one-parameter representation formulas on H-type groups, which will play an important role in our proofs Lemma 3.2 Suppose that β > −m + and f ∈ C0∞ (G) Then f (v) = −(cβ )−1 and −1 |f (v)| ≤ (cβ )   G G |z (u)|β−2 ⟨∇G f (v u−1 ), ∇G (N (u)4 )⟩du |N (u)|Q +β |z (u)|β−2 |∇G f (v u−1 )|du |N (u)|Q +β We notice that the weight takes the form W = |∇G N |−a N −a and |∇G N | = and K = |∇G N |Q −a−1 N Q −a−1 , we have |f (v)| ≤ (cQ −2a )−1 K∗W |∇G (f )| Finally, we state an important lemma, which can be found in [17] |z (u)| N on an H-type group Setting β = Q − 2a 100 N Lam, H Tang / Nonlinear Analysis 89 (2013) 95–109 Lemma 3.3 Let < α ≤ 1, < p < ∞ and a(s, t ) be a non-negative measurable function on (−∞, ∞) × [0, ∞] such that (a.e) a(s, t ) ≤ 1, sup t >0  when < s < t ,  + −∞ ∞ 1/p′ p′ a(s, t ) ds t = b < ∞ Then there is a constant c0 = c0 (p, b) such that if for φ ≥ 0,  then ∞ φ(s)p ds ≤ 1, −∞ ∞ e−Fα (t ) dt ≤ c0 , where Fα (t ) = α t − α  p ′ ∞ a(s, t )φ(s)ds −∞ Proof of Theorem 2.1 Q −a Let F ∈ HW0  (Ω , W ) be first-layer symmetric on a first-layer symmetric domain Ω ⊂ G, |Ω | < ∞, such that |∇G F |Q −a W ≤ Ω By an easy verification, L := |∇G F | is first layer symmetric By a direct calculation using the polar coordinate integration formula in Proposition 3.1, we compute the weighted distribution for K = λK ,W (s) = = =   {|∇G N |Q −a−1 >sN Q −a−1 } S (N )  |∇G N |Q −a−1 N Q −a−1 : |∇G N |−a N −a s−1/(Q −a−1) |∇G N | |∇G N |−a dσ r Q −a−1 dr Q −a  ′ s−Qa |∇G N |Q −2a dσ = S (N ) cQ −2a −Q ′ s a Q −a Here we have used the order zero homogeneity of |∇G N |, which follows from the order one homogeneity of N Then the ∗ weighted rearrangement and its average are given by KW (t ) = we have ∗ ∗∗ ∗∗ (K∗W L)W (t ) ≤ tKW (t )LW (t ) +  = Now assume that Since Q −a cQ −2a β −1/Qa′  Q ′ −1 (Q −a)(cQ a−2a ) λN −α (u),W (s) =  =  = then (N (u)−α )∗W (t ) =  + ∞  Q −a t cQ −2a −1/Qa′ ∗ KW (s)L∗W (s)ds t ′ (Q − a)t −1/Qa α Q −a  t L∗W (s)ds +  ∞ LQ −a Wdu = Ω ′ t ′ |∇G N |−a N −a du S (N ) c−a Q −a  Q −a t c−a  1 s α |∇G N |−a r Q −a−1 drdσ (s)− Q −a α − Q α−a  ′ ∗ ≤ Set U (u) = (K∗W L)(u), then (exp |U |Qa )∗W (t ) = exp{(UW (t ))Qa } N (u)−α >s   s−1/Qa L∗W (s)ds , Now we change variables by setting φ(s) = W (Ω ) Q −a L∗W (W (Ω )e−s )e  ∗∗ and KW (t ) = (Q − a)KW∗ (t ) By Lemma 3.1, W (Ω ) L∗W (t )Q −a dt =  ∞ φ(s)Q −a ds s − Q− a , then 101 N Lam, H Tang / Nonlinear Analysis 89 (2013) 95–109   α Setting A = (Q − a)(cQ−− 2a ) − Q −a , by the Hardy–Littlewood inequality, we have that 1− Q α −a W (Ω )  N (u)α Ω  ≤ 1− Q α −a W (Ω ) ′ exp(A|U |Qa ) Wdu W (Ω ) ′ ∗ exp(A(UW (t ))Qa ) − Q α−a    Q −a c−a t − Q α−a dt    α exp A(UW (W (Ω )e )) − − s ds = c−a Q −a    − Q α−a  ∞    W (Ω )e−s − 1′ s′ Q −a α (Q − a)(W (Ω )) Qa e Qa L∗W (τ )dτ ≤ 1− exp c−a Q − a 0  Qa′    W (Ω ) − 1′ α + τ Qa L∗W (τ )dτ sds − 1− − s Q −a W (Ω )e  − Q α−a  ∞    ∞   s α Q −a −z 1− Q − ′ a 1− = exp (Q − a)e Qa φ(z )e dz c−a Q −a s Qa′     s α sds − 1− φ(z )dz + Q −a − Q α−a  ∞    Q −a exp −F1− α (s) ds, = Q −a  Q −a c−a where F1− ∞ Qa′ −s ∗ (s) is the same as that in Lemma 3.3 with p = Q − a, and  if < s < t , 1, ′ a(s, t ) = (Q − a)e(t −s)/Qa , if t < s < ∞,  0, if − ∞ < s ≤  ∞ ∞ Q −a Since φ(s) ds ≤ 1, then by Lemma 3.3, it is clear that exp −F 1− α Q −a 1− Q α −a W (Ω )  ′ exp(A|U |Qa ) N (u)α Ω 1,Q −a 1− Q α −a W (Ω )  ′ exp(β|F |Qa ) Ω N (u)α Wdu ≤ 1− Q α −a W (Ω ) = 1− Q α −a W (Ω ) ≤ C0 < ∞, since β Q ′ −1 (Q −a)(cQ a−2a ) + α Q −a ds ≤ c0 Therefore, Wdu ≤ C0 Now, for all first-layer symmetric functions F ∈ HW0 domains Ω , using |F | ≤ (cQ −2a )−1 K∗W L, we have α Q −a   (s) (Ω , W ), with ∥∇G F ∥Q −a,W ,Ω ≤ and all first-layer symmetric   −Q ′ ′ exp(β cQ −a2a |U |Qa ) N (u)α Ω Wdu ′ exp(A|U |Qa ) Ω  ≤ and A = (Q − a)(cQ−−1 2a ) − α Q −a N (u)α  Wdu So the desired inequality is obtained Next we will show that the inequality in Theorem 2.1 does not hold if Ω = {u ∈ G : N (u) < 2R} For < l < R, let fl be the Moser function  1 R Qa′    , if ≤ N (u) ≤ l, log   l   log N (Ru) fl (u) = (cQ −2a )1/(Q −a)  , if l ≤ N (u) ≤ R,    R 1/(Q −a)  log   l  0, if R < N (u) β Q ′ −1 (Q −a)(cQ a−2a ) + α Q −a > Choose an R > and 102 N Lam, H Tang / Nonlinear Analysis 89 (2013) 95–109 It is easy to verify that Then and |∇G fl | =   log R l |∇G N |  N χB(R)\B(l) |∇fl |Q −a W = 1, Ω W (Ω ) 1− Q α −a  ′ exp(β(|fl |)Qa ) |N (u)|α Ω W ≥ = β Hence when Q ′ −1 (Q −a)(cQ a−2a ) fl  sup W (Ω ) 1− Q α −a α + Q −a Q −a log Qa′ Q −α−a− − Q− a Q −a c−a (2R) Q′ − Q −a a exp β(cQ −2a ) c−a (2R)Q −a Rβ(cQ −2a ) Q −a l R l  W B(l) |N (u)|α β ′ (cQ −2a )Qa −1 > 1, ′ exp β(|fl |)Qa |N (u)|α Ω W = +∞ That completes the proof of Theorem 2.1 Proof of Theorem 2.2 In order to establish the sharp weighted Moser–Trudinger inequality on the whole H-type group, we use the method as a−1] ) ≤ α < Q − a and τ > 0, Lam and Lu did in [18] It suffices to prove that for any α, τ satisfying (Q − a)(1 − [QQ − −a−1 there exists a constant C = C (α, τ , Q , a ) such that for all first-layer symmetric functions F ∈ C0∞ (G) \ {0}, F ≥ 0, and   Q −a Q −a |∇G F | W + τ G |F | W ≤ 1, there holds G  G N (u)α  ′ exp(βQ |F |Qa ) − [Q  −a−2] βQk k! k=0 Q ′ −1 kQ ′ |F | a  W ≤ C (α, τ , Q , a), where βQ = (Q − a − α)(cQ a−2a ) Set − (Q −a)(Q1 −a−1) A(F ) = τ Q −a ∥F ∥Q −a,W Ω (F ) = {u ∈ G : F (u) > A(F )} Then, it is clear that A(F ) < 1, and Ω (F ) is a first-layer symmetric domain Moreover,  |F |Q −a W ≥ G We have  |F |Q −a W ≥ Ω (F )  Ω (F ) a |A(F )|Q −a W = 2− Q −a−1 τ ∥F ∥QQ − −a,W |W (Ω )| |W (Ω )| ≤ Q −a−1 τ −a Since W (u) = |z (u)| , and Ω (F ) is first-layer symmetric, we know that Ω (u) is bounded and first layer symmetric Now we write  G N (u)α  ′ Qa exp(βQ |F | ) − k=0 where I1 =  Ω (F ) N (u)α  and I2 =  G\Ω (F ) [Q  −a−2] N (u)α Qa′ exp(βQ |F | ) − βQk k! kQa′ |F | [Q −a−2] k=0  Qa′ exp(βQ |F | ) −  βQk k! [Q  −a−2] k=0 W = I1 + I2 , kQa′ |F | βQk k! |F |  kQa′ W  W 103 N Lam, H Tang / Nonlinear Analysis 89 (2013) 95–109 First, we estimate I2 Since G \ Ω (F ) ⊂ {F (u) < 1}, we see that I2 ≤   N (u)α F (u)1 k=[Q −a−1]  + G If Q − a is an integer, |F |Q −a W + F (u)