Besov norms of the continuous wavelet transform in variable Lebesgue space
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Besov norms of the continuous wavelet transform in variable Lebesgue space Drema Lhamu1 · Sunil Kumar Singh2 Received: 10 April 2020 / Revised: 10 April 2020 / Accepted: 25 July 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, the continuous wavelet transform is studied in variable L p(·) spaces. Parseval relation and Reconstruction formula for the continuous wavelet transform in variable L p(·) spaces are derived. The characterization of variable Besov spaces associated with the continuous wavelet transform is also presented. Keywords Variable Lebesgue space · Continuous wavelet transform · Besov space Mathematics Subject Classification 65T60 · 46F12 · 30H25
1 Introduction In this section, we recall the definition and basic properties of Lebesgue variable spaces which will be used throughout this paper. A detailed discussion of properties of Lebesgue variable spaces may be found in [1,3,4]. Definition 1 (Exponent function) Suppose is a Lebesgue measurable subset of Rn . Let P() be the set of all Lebesgue measurable functions p(·) : → [1, ∞]. The elements of P() are called exponent functions. Given p(·) ∈ P(), we define the following numbers: p− = ess inf p(x) and p+ = ess sup p(x). x∈
B
x∈
Sunil Kumar Singh [email protected] Drema Lhamu [email protected]
1
Department of Mathematics, Jawaharlal Nehru College, Pasighat, Arunachal Pradesh 791102, India
2
Department of Mathematics, Mahatma Gandhi Central University, Motihari, Bihar 845401, India
D. Lhamu , S. K. Singh
For p(·) ∈ P(), we define three subset of as: ∞ ( p) = {x ∈ : p(x) = ∞} 1 ( p) = {x ∈ : p(x) = 1} ∗ ( p) = {x ∈ : 1 < p(x) < ∞} . The conjugate (dual) exponent function p (·) is defined by the formula 1 1 + = 1, for all x ∈ . p(x) p (x) Definition 2 Given ⊂ Rn , p(·) ∈ P(), the space L p(·) () denotes the space of all measurable functions f on such that the modular ρ p(·) ( f ) :=
\∞
| f (x)| p(x) d x < ∞
(1)
and ess sup | f (x)| < ∞.
(2)
x∈∞
The space L p(·) () is a Banach space with the norm f L p(·) = inf λ > 0 : ρ p(·) ( f /λ) ≤ 1 . In variable space we can also consider spaces where exponent function p(·) is unbounded. For example, we could take p(x) = 1 + |x|; x ∈ Rn , such spaces behave differently than the classical Lebesgue space. In this case we have L ∞ (Rn ) ⊂ L p(·) (Rn ). Given g ∈ L ∞ , for fix λ > g∞ then
Rn
|g(x)| λ
p(·)
dx ≤
Rn
||g(x)||∞ λ
1+|x| dx < ∞
and so g ∈ L p(·) (Rn ).
Theorem 1 (Hölder’s inequality) [1, p. 278] Let f ∈ L p(·) (), g ∈ L p (·) () and 1 ≤ p(x), p (x) ≤ ∞. Then
with k = sup
1 p(·)
+ sup
| f (x)g(x)|d x ≤ k f L p(·) g L p (·) ,
1 p (·) .
(3)
Besov norms of the continuous wavelet transform in variable… 1 Theorem 2 [1, p. 279] Let p(x) + supx∈\∞ r (x) < ∞. Then
1 q(x)
=
1 r (x) ;
p(x) ≥ 1, q(x) ≥ 1, r (x) ≥ 1 and let
f g L r (·) ≤ C f L p(·) g L q(·) , for all functions f ∈ L p(·) (), g ∈ L q(·) (), where C = sup\∞ sup\∞
r (x) q(x) .
(4) r (x) p(x)
+
Theo
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