Carleson Measure and Tent Spaces on the Siegel Upper Half Space

and Applied Analysis 3 The dilations on H are defined by δz δ z′, t δz′, δ2t , δ > 0, and the rotation on H n is defined by σz σ z′, t σz′, t with a unitary map σ of C. The conjugation of z is z z′, t z′,−t . The norm function is given by |z| ∣∣z′∣∣4 |t| )1/4 , z ( z′, t ) ∈ H, 1.5 which is homogeneous of degree 1 and satisfies |z−1| |z| and |zw| ≤ C |z| |w| for some absolute constant C. The distance function d z,w of point z and w in H is defined by d z,w |w−1z|. For z z′, t ∈ H or z x, y, t ∈ H for x x1, x2, , . . . , xn , y y1, y2, , . . . , yn , we define |z|∞ max { |x1|, . . . , |xn|, ∣y1 ∣∣, . . . , ∣yn ∣∣, √ |t| } , 1.6 where z′ z1, . . . , zn , and zk xk iyk, 1 ≤ k ≤ n. To define the cube in H, let x x1 , x2 , . . . , xn , where a denotes the largest integer less than or equal to a, and 〈x〉 x − x . We define the function F x, y by F ( x, y ) ∞ ∑ j 1 2 4j (〈 2x 〉([ 2y ] mod 2Z ) − 〈 2y 〉([ 2x ] mod 2Z )) , 1.7 and set I0 {( x, y, t ) ∈ H : 0 ≤ xk ≤ 1, 0 ≤ yk ≤ 1, 1 ≤ k ≤ n, 0 ≤ t − F ( x, y ) ≤ 1}, 1.8 Γ { m,n, l ∈ H : m,n ∈ Z; l ∈ Z}. Then H n ∑ γ∈Γ γ−1I ′ 0, ( disjoint union ) , 1.9 where I ′ 0 { x, y, t ∈ H : 0 ≤ xk < 1, 0 ≤ yk < 1, k 1, 2, . . . , n, 0 ≤ t − F x, y < 1} see 14 . Now a “cube” in fact, called “tile” more precisely with center w u, v, s and edge sidelength l is defined by I lw−1I0. Let |I| with the Lebesgue measure be the volume of I with length l I . It is easy to see that |I| l I 2n . Obviously, the diameter of I denoted by diam I is equal to cnl I , where cn is a constant depending only on n. The “dyadic cubes” on H can be defined by I 2−jγ−1I0, γ ∈ Γ, j ∈ Z. We also call the “cube” and “dyadic cube” as cube and dyadic cube, respectively. A ball in H with center w and radius r is denoted as B B w, r {z : |w−1z| < r}. The tent based on the set E ⊂ H is defined by T E {( ξ, ρ ) ∈ U : B(ξ, ρ) ⊂ E}. 1.10 The Schwartz class of rapidly decreasing smooth function on H will be denoted by S H . The dual of S H is S′ H , the space of tempered distributions on H. 4 Abstract and Applied Analysis 2. Hausdorff Capacity on Heisenberg Group In order to discuss the fractional Carleson measures on the Siegel upper half space, we introduce the p-dimensional Hausdorff capacity and Choquet integral on the Heisenberg group in the following. Definition 2.1. For p ∈ 0, 1 and E ⊂ H, the p-dimensional Hausdorff capacity of E is defined by