Theorem 3.1.

For 1 < p ≤ 2 , if f ∈ L p , p ′ ⁢ ( G × G ^ ) , then W ⁢ ( f ) ∈ B p ′ ⁢ ( L 2 ⁢ ( G ) ) and there exists C > 0 such that

Proof.

Define an operator T on L 1 + L 2 , 1 ⁢ ( G × G ^ ) by T ⁢ ( f ) = S W ⁢ ( f ) ⁢ ( n ) . Since the singular value sequence is a decreasing sequence, using [4, Corollary 1.35], 0 ≤ S A + B ⁢ ( 2 ⁢ n ) ≤ S A + B ⁢ ( 2 ⁢ n - 1 ) ≤ S A ⁢ ( n ) + S B ⁢ ( n ) for n ∈ ℕ and compact operators A and B. Therefore, for all n ∈ ℕ , we have

where ⌈ ⋅ ⌉ is the least integer function. Now, it can be shown that

for α 1 , α 2 > 0 .

Since the Weyl transform maps L 1 ⁢ ( ℝ 2 ⁢ n ) to ℬ ∞ ⁢ ( L 2 ⁢ ( ℝ n ) ) and L 2 ⁢ ( ℝ 2 ⁢ n ) to ℬ 2 ⁢ ( L 2 ⁢ ( ℝ n ) ) , similar to classical Marcinkiewicz interpolation theorem ([1, Theorem 4.13]), we can show that T maps L p , p ′ ⁢ ( G × G ^ ) to l p ′ ⁢ ( ℕ ) continuously and

Since ∥ W ⁢ ( f ) ∥ ℬ p ′ ⁢ ( L 2 ⁢ ( ℝ n ) ) = ∥ { S W ⁢ ( f ) ⁢ ( n ) } ∥ l p ′ , we get the desired result. ∎

Theorem 3.2.

Consider a positive function ϕ ∈ l 1 , ∞ ⁢ ( N ) . For 1 < p ≤ 2 and f ∈ L p ⁢ ( G × G ^ ) , we have

Proof.

Consider the measure ν on ℕ given by

For 1 < p ≤ ∞ , we let l p ⁢ ( ℕ , ν ) denote the space of all complex-valued sequences x = ( x n ) n ∈ ℕ such that

We now claim that if f ∈ L p ⁢ ( G × G ^ ) , then { S W ⁢ ( f ) ⁢ ( n ) ϕ ⁢ ( n ) } n ∈ ℕ ∈ l p ⁢ ( ℕ , ν ) . We will denote this correspondence by T and show that T is a bounded map. Our strategy here is to use the techniques involved in the Marcinkiewicz interpolation theorem. To do this, we first define sequence { P ⁢ ( f ) ⁢ ( n ) } n ∈ ℕ as

where ⌈ ⋅ ⌉ is the least integer function.

Similar to the previous theorem, for all n ∈ ℕ , we have

Thus

Now, we claim P is both weak type ( 2 , 2 ) and ( 1 , 1 ) with respect to measure ν.

The distribution function, in this case, is given by

To show that P is of weak type ( 1 , 1 ) , we prove that

Since for all n ∈ ℕ ,

we have

Hence

Now, let w = ∥ f ∥ 1 y . Then

Thus, for y > 0 , we have

Also, by using the Plancherel theorem for Weyl transform, it can be seen that T maps L 2 ⁢ ( G × G ^ ) continuously to l 2 ⁢ ( ℕ , ν ) since

This shows that P is weak type ( 2 , 2 ) . Now, for f ∈ L p ⁢ ( G × G ^ ) , we define

for suitable δ > 0 , so that f = f 0 α + f 1 α , and by (2), we have

Now the proof follows similar to classical Marcinkiewicz interpolation theorem [2, Theorem 1.3.2] and we have

or

Hence the proof. ∎